C Majority
February 24, 2006 6:51 PM Subscribe
Why are C and F the notes without corresponding flat keys on the piano?
Or, conversely, why are B sharp and E sharp enharmonic with C natural and F natural?
I have tried to find the answer on line and through my musicologist friends, but to no avail. Everyone's perfectly willing to explain whole steps and half steps and the rudiments of music theory, but what I want to know is why the exceptions to the rule are C and F? Why not D and G? Was there some moment in music history when the Catholic Church declared C flat the devil's accidental? Was it an arbitrary decision among keyboard makers? Is it based on some ancient Greek tombstone?
Or, conversely, why are B sharp and E sharp enharmonic with C natural and F natural?
I have tried to find the answer on line and through my musicologist friends, but to no avail. Everyone's perfectly willing to explain whole steps and half steps and the rudiments of music theory, but what I want to know is why the exceptions to the rule are C and F? Why not D and G? Was there some moment in music history when the Catholic Church declared C flat the devil's accidental? Was it an arbitrary decision among keyboard makers? Is it based on some ancient Greek tombstone?
The seven-note scale surely predates designating those notes with letters.
posted by S.C. at 7:03 PM on February 24, 2006
posted by S.C. at 7:03 PM on February 24, 2006
Cause a c-flat is a b, and an f flate is an e?
If you'll excuse me, my ear couldn't manufacture a c-flat or and f-flat. If it could, the the whole idea of octaves-- which are completely intuitive to untrained ears-- would be be destroyed.
Trust me, a c-flat and an f-flat make sense to your ear, if not to your intellect. Try listening to a b and a c one octave above. You'll hear the difference.
posted by gesamtkunstwerk at 7:13 PM on February 24, 2006
If you'll excuse me, my ear couldn't manufacture a c-flat or and f-flat. If it could, the the whole idea of octaves-- which are completely intuitive to untrained ears-- would be be destroyed.
Trust me, a c-flat and an f-flat make sense to your ear, if not to your intellect. Try listening to a b and a c one octave above. You'll hear the difference.
posted by gesamtkunstwerk at 7:13 PM on February 24, 2006
After trying to read this, I think the best answer is "because".
posted by SkelPaff at 7:27 PM on February 24, 2006
posted by SkelPaff at 7:27 PM on February 24, 2006
Maybe this will help...ever heard of a chromatic scale? this is a scale made up of all the notes-for example the twelve half-step notes between middle C and high C (obviously you could start with any note. )
So basically when you translate that into an eight note scale with varying amounts of sharps or flats, the arrangement is simply one of practicality. An e sharp is the same as an f simply because that is how it is arranged on a keyboard. It's the same note, just a different name is all. You call it one or the other simply based on what scale you are using-bearing in mind an e sharp is a total rarity. Who the heck writes music with 6 or 7 sharps?
posted by konolia at 8:04 PM on February 24, 2006
So basically when you translate that into an eight note scale with varying amounts of sharps or flats, the arrangement is simply one of practicality. An e sharp is the same as an f simply because that is how it is arranged on a keyboard. It's the same note, just a different name is all. You call it one or the other simply based on what scale you are using-bearing in mind an e sharp is a total rarity. Who the heck writes music with 6 or 7 sharps?
posted by konolia at 8:04 PM on February 24, 2006
Semi-informed answer: the pattern of whites and blacks on a standard keyboard gives you a C Major and A Minor scale on the whites, so it's a good start. The fact that the semitones are B->C and E->F is arbitrary/historical.
posted by signal at 8:18 PM on February 24, 2006
posted by signal at 8:18 PM on February 24, 2006
i'm thinking 'arbitrary decision among keyboard makers'
in addition to conventions relating to nomenclature.
since there are, by convention (in western music) 7 letter names with the requirement that they be divided (in some manner) into 12 chromatic tones, we have to "leave out" so-to-speak some sharp/flat combinations.
for example, if a scale were constructed with the constraint that we use each single letter name ~in addition~ to its sharp to indicate a set of distinct tones, we'd need to have 14 of these distinct tones (c c# d d# e e# f f# g g# a a# b b#). but since there really are only 12 chromatic tones, we must leave some letter-name/accidental combinations out of the construct, or more accurately, we end up making e# equivalent to f and b# equivalent to c. it's really only naming convention coupled with the visual design of the piano keyboard. note how the keyboard is divided so we see plainly the e-f and b-c adjacency (white key next to white key)...you ~could~ design a keyboard without such landmarks but it would'nt be nice.. it'd be a lot harder to maintain orientation.
posted by The_Auditor at 9:10 PM on February 24, 2006
in addition to conventions relating to nomenclature.
since there are, by convention (in western music) 7 letter names with the requirement that they be divided (in some manner) into 12 chromatic tones, we have to "leave out" so-to-speak some sharp/flat combinations.
for example, if a scale were constructed with the constraint that we use each single letter name ~in addition~ to its sharp to indicate a set of distinct tones, we'd need to have 14 of these distinct tones (c c# d d# e e# f f# g g# a a# b b#). but since there really are only 12 chromatic tones, we must leave some letter-name/accidental combinations out of the construct, or more accurately, we end up making e# equivalent to f and b# equivalent to c. it's really only naming convention coupled with the visual design of the piano keyboard. note how the keyboard is divided so we see plainly the e-f and b-c adjacency (white key next to white key)...you ~could~ design a keyboard without such landmarks but it would'nt be nice.. it'd be a lot harder to maintain orientation.
posted by The_Auditor at 9:10 PM on February 24, 2006
...pretty much what konolia said (i shoulda previewed!)
posted by The_Auditor at 9:14 PM on February 24, 2006
posted by The_Auditor at 9:14 PM on February 24, 2006
A few answers to your question, billtron. First: I suspect the most thorough and accurate answer is that it was an arbitrary decision. After all, why do we use the FIRST seven letters of the alphabet? Couldn't we just as easily have used the LAST (or, for that matter, ANY) seven letters? I've taught music to non-English speakers who have already had SOME basic music instruction in their native language, and they had no idea what a, b, c, etc., meant - because they were more accustomed to do-re-mi-fa-so-la-ti-do. Then your question becomes, "why is there no flat do or flat fa?" I think it's just arbitrary. (In fact, now that I think about it, wouldn't it have made more sense for the A scale to have no sharps and flats - that is to say, a half step between c-d and also between g-a?)
Secondly, there ARE, in fact, b sharps, and e sharps, as well as f flats and c flats. It's all about context. The seventh scale degree of the c# major scale is a b# - NOT a c natural, a b#. And the third scale degree of the same scale is a e#. Now, given that the SOUNDS came BEFORE the piano keyboard, again it seems arbitrary to me to decide where to leave out the black keys.
Finally, it's also true that, on most instruments, there is no color differentation between sharps/flats and naturals. In fact, when a lot of this was worked out, people were just comparing different lengths of string (ie Pythagoras et al.).
Methinks it's arbitrary.
posted by fingers_of_fire at 9:43 PM on February 24, 2006
Secondly, there ARE, in fact, b sharps, and e sharps, as well as f flats and c flats. It's all about context. The seventh scale degree of the c# major scale is a b# - NOT a c natural, a b#. And the third scale degree of the same scale is a e#. Now, given that the SOUNDS came BEFORE the piano keyboard, again it seems arbitrary to me to decide where to leave out the black keys.
Finally, it's also true that, on most instruments, there is no color differentation between sharps/flats and naturals. In fact, when a lot of this was worked out, people were just comparing different lengths of string (ie Pythagoras et al.).
Methinks it's arbitrary.
posted by fingers_of_fire at 9:43 PM on February 24, 2006
I'd been working on a long answer to this quesiton, but its better to just point you here.
posted by gsteff at 9:50 PM on February 24, 2006
posted by gsteff at 9:50 PM on February 24, 2006
jeez. it is not arbitrary. The white keys result from simple harmonics of C.
First take C. 3 times its frequency is G' and five times its frequency is E'. This gives the triad C-E-G.
Now go up a 5th to G (which is more strongly related to C than E). This gives the triad G-B-D.
Then go down a 5th to F (F is to C as C is to G). This gives the triad F-A-C.
These 3 triads define all the white notes.
posted by MonkeySaltedNuts at 9:59 PM on February 24, 2006
First take C. 3 times its frequency is G' and five times its frequency is E'. This gives the triad C-E-G.
Now go up a 5th to G (which is more strongly related to C than E). This gives the triad G-B-D.
Then go down a 5th to F (F is to C as C is to G). This gives the triad F-A-C.
These 3 triads define all the white notes.
posted by MonkeySaltedNuts at 9:59 PM on February 24, 2006
Of course, the letters are arbitrary, the actual notes are not.
I am amazed that people seem to be answering without realizing the mathematical difference between E and an F is the same as, for example, a D and a D#, a half step
The seven note system is pretty convoluted, but sounds pretty good. I took a music theory class once. Promptly forgot all of it, sadly.
posted by delmoi at 10:52 PM on February 24, 2006
I am amazed that people seem to be answering without realizing the mathematical difference between E and an F is the same as, for example, a D and a D#, a half step
The seven note system is pretty convoluted, but sounds pretty good. I took a music theory class once. Promptly forgot all of it, sadly.
posted by delmoi at 10:52 PM on February 24, 2006
I am amazed that people seem to be answering without realizing the mathematical difference between E and an F is the same as, for example, a D and a D#, a half step
Er, some people. Most people are getting it right, of course.
posted by delmoi at 10:53 PM on February 24, 2006
Er, some people. Most people are getting it right, of course.
posted by delmoi at 10:53 PM on February 24, 2006
My explaination would be:
Someone, a long time ago, decided to themselves "hey, there are eight notes in a major scale, root to octave, so let's give them eight names". Completely ignoring the fact that there are actually 12 notes between the root and the octave. To my mind, this has confused things greatly, and made music, and keys, much more difficult to understand than they need to be.
Keyboards were built to reflect the "8 notes to a scale...with all those extra ones inbetween" idea, and are confusing compared to the simplicity of other stringed instruments. What's more, they decided to make the simplest scale - the all white notes one - start on C rather than something sensible like A. Note that there is no such problem on a guitar - just twelve frets making up an octave. When you play a guitar, or a violin, you get a feeling for the fact that "scales" are made up of certain intervals out of the twelve notes - tones, or semi-tones - rather than thinking of things in the rather silly terms of "black" and "white" keys. As others have said, B# is C. There are certain scales that require the note C to be flattened, in which case you play B. Others require B to be sharp, in which case you play C. This is easy to understand on a guitar, but piano keyboards add an extra level of confusion by making it look like there are two sets of notes - black and white.
posted by Jimbob at 12:08 AM on February 25, 2006
Someone, a long time ago, decided to themselves "hey, there are eight notes in a major scale, root to octave, so let's give them eight names". Completely ignoring the fact that there are actually 12 notes between the root and the octave. To my mind, this has confused things greatly, and made music, and keys, much more difficult to understand than they need to be.
Keyboards were built to reflect the "8 notes to a scale...with all those extra ones inbetween" idea, and are confusing compared to the simplicity of other stringed instruments. What's more, they decided to make the simplest scale - the all white notes one - start on C rather than something sensible like A. Note that there is no such problem on a guitar - just twelve frets making up an octave. When you play a guitar, or a violin, you get a feeling for the fact that "scales" are made up of certain intervals out of the twelve notes - tones, or semi-tones - rather than thinking of things in the rather silly terms of "black" and "white" keys. As others have said, B# is C. There are certain scales that require the note C to be flattened, in which case you play B. Others require B to be sharp, in which case you play C. This is easy to understand on a guitar, but piano keyboards add an extra level of confusion by making it look like there are two sets of notes - black and white.
posted by Jimbob at 12:08 AM on February 25, 2006
I am amazed that people seem to be answering without realizing the mathematical difference between E and an F is the same as, for example, a D and a D#, a half step
Not quite, delmoi. Those intervals are really only "equivalent" because today, by convention, we tune them that way.
I'm oversimplifying horribly, but if you tune strictly according to the harmonic series, you'll find that many notes we now consider "the same" don't quite match up. If you run around the circle of fifths, you won't quite make it back to where you started. The remaining difference, incidentally, is the hilariously named Pythagorean comma.
If the music you're playing mostly features pitches based on overtones of your tonic, arranged in independent melodic lines or stacked in fourths or fifths, then tuning according to the harmonic series works fine.
But "just" (harmonic) tuning isn't good if your music involves vertical harmonies. And if your music "modulates", or jumps around from one key to the next, you'll constantly have to stop and retune.
Western music since the Renaissance has emphasized triads and lots of tonal relationships and modulation. To get around the inconsistencies in "just" tuning, musicians in the Western traditions have used various evening-out schemes called "temperaments".
You've heard of the Well-Tempered Clavier? Bach's "well-tempered" scale allowed for much nimbler modulation than had previously been possible. There's some debate about how it worked; the writer in the Math Forum link SkelPaff posted above thinks that it "retained pure or nearly pure fifths and thirds in several keys, while sacrificing some of the purity in other keys."
It also helps, of course, if you stay within closely related families of keys. As MonkeySaltedNotes describes above, these are still the white keys of the piano.
The common scheme today is "equal temperament". This divides the octave into twelve identical semitones so that when you modulate your intervals all stay exactly the same size. You can modulate all you want and transpose seamlessly. There's a cost, of course. What's lost is a) some thrillingly perfect intervals, and b) different scales with subtly different characteristics.
If you're interested, that Math Forum link is relatively clear. I'd recommend it above the seven or eight Wikipedia articles, all of which seem to assume a lot of prior knowledge. Reginald Bain's site isn't bad either.
posted by tangerine at 12:10 AM on February 25, 2006
Not quite, delmoi. Those intervals are really only "equivalent" because today, by convention, we tune them that way.
I'm oversimplifying horribly, but if you tune strictly according to the harmonic series, you'll find that many notes we now consider "the same" don't quite match up. If you run around the circle of fifths, you won't quite make it back to where you started. The remaining difference, incidentally, is the hilariously named Pythagorean comma.
If the music you're playing mostly features pitches based on overtones of your tonic, arranged in independent melodic lines or stacked in fourths or fifths, then tuning according to the harmonic series works fine.
But "just" (harmonic) tuning isn't good if your music involves vertical harmonies. And if your music "modulates", or jumps around from one key to the next, you'll constantly have to stop and retune.
Western music since the Renaissance has emphasized triads and lots of tonal relationships and modulation. To get around the inconsistencies in "just" tuning, musicians in the Western traditions have used various evening-out schemes called "temperaments".
You've heard of the Well-Tempered Clavier? Bach's "well-tempered" scale allowed for much nimbler modulation than had previously been possible. There's some debate about how it worked; the writer in the Math Forum link SkelPaff posted above thinks that it "retained pure or nearly pure fifths and thirds in several keys, while sacrificing some of the purity in other keys."
It also helps, of course, if you stay within closely related families of keys. As MonkeySaltedNotes describes above, these are still the white keys of the piano.
The common scheme today is "equal temperament". This divides the octave into twelve identical semitones so that when you modulate your intervals all stay exactly the same size. You can modulate all you want and transpose seamlessly. There's a cost, of course. What's lost is a) some thrillingly perfect intervals, and b) different scales with subtly different characteristics.
If you're interested, that Math Forum link is relatively clear. I'd recommend it above the seven or eight Wikipedia articles, all of which seem to assume a lot of prior knowledge. Reginald Bain's site isn't bad either.
posted by tangerine at 12:10 AM on February 25, 2006
Condensed answer: the equal tempered diatonic scale, of course, is not one long run of notes. Some are skipped, and that's what makes it the scale it is.2 To play that scale in a given key, you play that pattern of skipped notes starting from that note. We could have just as easily named the notes A-L and never strictly had to use accidentals, but we chose to label the notes from the perspective of one key, C, with the skipped notes referred to as sharps or flats.1
1See this question as to why C was picked.
2 The scale itself is made from the notes of six successive fifths - F-C-G-D-A-E-B. (Though it also consists of three triads fifths apart, that's not the underpinning of the scale). The fifth (a 1/3 vibration) is the first harmonic after the 1/2 octave harmonic, and hence arguably the most basic to build a scale out of.
posted by abcde at 12:13 AM on February 25, 2006
1See this question as to why C was picked.
2 The scale itself is made from the notes of six successive fifths - F-C-G-D-A-E-B. (Though it also consists of three triads fifths apart, that's not the underpinning of the scale). The fifth (a 1/3 vibration) is the first harmonic after the 1/2 octave harmonic, and hence arguably the most basic to build a scale out of.
posted by abcde at 12:13 AM on February 25, 2006
(And yes, that's a fairly amusing formatting error.)
posted by abcde at 12:15 AM on February 25, 2006
posted by abcde at 12:15 AM on February 25, 2006
The really mysterious question isn't what notes are on the keyboard, but what frequencies are assigned to those notes.
The equal tempered scale makes the ratio between each half-step a constant (this is why modern Western music can be played in any key, without it sounding odd.) This is also part of the reason why the modern keyboard design works.
In a mean tone scale, several fifths have unexpected (and unpleasant) musical effects. As such, they would sometimes add seperate keys for a "single" note. (One key for D# and another for Eb), in order to avoid the wolf fifth.
For what it's worth, the major diatonic is just (in half-steps)
0 2 4 5 7 9 11 12, so having a keyboard that lays them out like that seems fairly natural.
Drunken rambling over.
posted by I Love Tacos at 12:31 AM on February 25, 2006
The equal tempered scale makes the ratio between each half-step a constant (this is why modern Western music can be played in any key, without it sounding odd.) This is also part of the reason why the modern keyboard design works.
In a mean tone scale, several fifths have unexpected (and unpleasant) musical effects. As such, they would sometimes add seperate keys for a "single" note. (One key for D# and another for Eb), in order to avoid the wolf fifth.
For what it's worth, the major diatonic is just (in half-steps)
0 2 4 5 7 9 11 12, so having a keyboard that lays them out like that seems fairly natural.
Drunken rambling over.
posted by I Love Tacos at 12:31 AM on February 25, 2006
On post, oh crap, that's what I get for reading my e-mail mid-post. I got (largely) BEAT!
posted by I Love Tacos at 12:32 AM on February 25, 2006
posted by I Love Tacos at 12:32 AM on February 25, 2006
[Note: written on the fly late at night, hope it's not too confusing.]
It is indeed not arbitrary. While many things in western music and music notation are somewhat convoluted because of the weight of their history, they all exist for good reasons.
In this case, the pattern of whole/half steps that make up western music's primary mode (diatonic--specifically, major) cause half steps to fall between the third/fourth and seventh/eighth (octave) scale degrees, with a scale having seven notes before an octave is reached.
That interval pattern has been in use for hundreds of years--see this Wiki article on the Diatonic scale, which defines it as "a seven-note musical scale comprising five whole-tone and two half-tone steps, in which the half tones are maximally separated," and was derived from the overtone series as mentioned above.
Each of the notes in such a pattern can serve as a starting point. The resulting scales are known as modes (or 'church modes'), and are named: ionian, dorian, phrygian, lydian, mixolydian, aeolian, and locrian. So, if you take a C major scale and play it starting on D, you get a dorian scale. (For more, see this terrific article on the church modes, with samples of each.)
In the Renaissance, this pattern became standard and widely practiced, and around the late renaissance/early baroque, ionian was settled upon as the principal--or "major"--mode, with aeolian used as the basis for the minor scale. From then to the the early 20th century, it's been the basis of western musical composition.
HOWEVER, if one moves by half-steps from note to note, one finds that there are in fact 12 notes in an octave, the seven of a diatonic scale pattern plus five others not used. (since the diatonic scale uses whole steps more than half, this accounts for the extra notes "in between".)
Those extra notes are used to allow a composer to "modulate," or move from one key to another. (If I want to go from C major to F major, I need a B-flat, which isn't in the C major scale. Plus, my choices of harmony in any key would be pretty frickin boring without the other, chromatic, notes included.)
You ask why is it that the first note (the "tonic") of the primary, all-white-key pattern is named "C" rather than something more obviously sensible like, perhaps, "A"? Because of Guido of Arrezo, who is "regarded as the inventor of modern musical notation (staff notation) that replaced neumatic notation".
Also from the article: "He developed new technologies for teaching, including the staff notation and the 'do-re-mi' (diatonic) scale, in which the name of the single notes were taken from the initial syllables of the lines of the first stanza of a then-popular hymn, Ut queant laxis."
The text of that stanza is:
Ut queant laxis
resonare fibris
Mira gestorum
famuli tuorum,
Solve polluti
labii reatum, Sancte Ioannes.
Do those syllables look familiar? The are indeed our modern solfege syllables, used to name pitches throughout the world. "Ut" was replaced with "Do" at some point, likely because 'ut' is a closed sound and horrible to sing, and 'do' is open (and many think is from 'dominus'). The seventh scale degree was not used in the hymn, nor was it commonly used in Guido's time at all, so thus is unnamed. We call it 'si' today. Or 'ti' if you're using moveable 'do' solfege. So that gives us the familiar 'do re mi fa sol la si/ti do'. In fact, in 'moveable do' solfege, the tonic is always called 'do', no matter what key you're in. In 'fixed do' solfege naming, 'do' = 'C', because...
...the original hymn was most commonly sung in the key of, you guessed it, the pitch (frequency) that became known as 'do' or 'C'. So that's why that's the name of the first note of the all-white-key version of the diatonic scale is 'do'. Now, as to how 'do' came to be called 'C' instead of 'A', I don't know. I'll have to dig around to find out.
See also: here, here or here for more specific topics along these lines.
posted by LooseFilter at 12:38 AM on February 25, 2006 [1 favorite]
It is indeed not arbitrary. While many things in western music and music notation are somewhat convoluted because of the weight of their history, they all exist for good reasons.
In this case, the pattern of whole/half steps that make up western music's primary mode (diatonic--specifically, major) cause half steps to fall between the third/fourth and seventh/eighth (octave) scale degrees, with a scale having seven notes before an octave is reached.
That interval pattern has been in use for hundreds of years--see this Wiki article on the Diatonic scale, which defines it as "a seven-note musical scale comprising five whole-tone and two half-tone steps, in which the half tones are maximally separated," and was derived from the overtone series as mentioned above.
Each of the notes in such a pattern can serve as a starting point. The resulting scales are known as modes (or 'church modes'), and are named: ionian, dorian, phrygian, lydian, mixolydian, aeolian, and locrian. So, if you take a C major scale and play it starting on D, you get a dorian scale. (For more, see this terrific article on the church modes, with samples of each.)
In the Renaissance, this pattern became standard and widely practiced, and around the late renaissance/early baroque, ionian was settled upon as the principal--or "major"--mode, with aeolian used as the basis for the minor scale. From then to the the early 20th century, it's been the basis of western musical composition.
HOWEVER, if one moves by half-steps from note to note, one finds that there are in fact 12 notes in an octave, the seven of a diatonic scale pattern plus five others not used. (since the diatonic scale uses whole steps more than half, this accounts for the extra notes "in between".)
Those extra notes are used to allow a composer to "modulate," or move from one key to another. (If I want to go from C major to F major, I need a B-flat, which isn't in the C major scale. Plus, my choices of harmony in any key would be pretty frickin boring without the other, chromatic, notes included.)
You ask why is it that the first note (the "tonic") of the primary, all-white-key pattern is named "C" rather than something more obviously sensible like, perhaps, "A"? Because of Guido of Arrezo, who is "regarded as the inventor of modern musical notation (staff notation) that replaced neumatic notation".
Also from the article: "He developed new technologies for teaching, including the staff notation and the 'do-re-mi' (diatonic) scale, in which the name of the single notes were taken from the initial syllables of the lines of the first stanza of a then-popular hymn, Ut queant laxis."
The text of that stanza is:
Ut queant laxis
resonare fibris
Mira gestorum
famuli tuorum,
Solve polluti
labii reatum, Sancte Ioannes.
Do those syllables look familiar? The are indeed our modern solfege syllables, used to name pitches throughout the world. "Ut" was replaced with "Do" at some point, likely because 'ut' is a closed sound and horrible to sing, and 'do' is open (and many think is from 'dominus'). The seventh scale degree was not used in the hymn, nor was it commonly used in Guido's time at all, so thus is unnamed. We call it 'si' today. Or 'ti' if you're using moveable 'do' solfege. So that gives us the familiar 'do re mi fa sol la si/ti do'. In fact, in 'moveable do' solfege, the tonic is always called 'do', no matter what key you're in. In 'fixed do' solfege naming, 'do' = 'C', because...
...the original hymn was most commonly sung in the key of, you guessed it, the pitch (frequency) that became known as 'do' or 'C'. So that's why that's the name of the first note of the all-white-key version of the diatonic scale is 'do'. Now, as to how 'do' came to be called 'C' instead of 'A', I don't know. I'll have to dig around to find out.
See also: here, here or here for more specific topics along these lines.
posted by LooseFilter at 12:38 AM on February 25, 2006 [1 favorite]
...as I think about it, my guess would be that somehow, when letters were first used as note names correlated with the 'do re mi' naming of Guido's system, the natural minor (aeolian) was preferred, and thus the first note of that pattern was named 'A', which in the Guidonian system would have been the pitch commonly called 'la': our modern 'A'.
That seems most likely: a preference for the minor mode at some point. But I could just be making shit up at this point.
posted by LooseFilter at 12:44 AM on February 25, 2006
That seems most likely: a preference for the minor mode at some point. But I could just be making shit up at this point.
posted by LooseFilter at 12:44 AM on February 25, 2006
To be clear: the reason we have "accidentals" or chromatic notes is that when pitches were first named, the music in practice was more simple w/r/t pitch--so musicians named only the pitches of the most common pattern, the diatonic scale. Because that's an uneven pattern in terms of distance between each pitch in it (whole and half steps), the possibility was left open to "fill in" the half steps not included in the diatonic pattern, by simply moving only in that smallest unit of the half step.
That's called a "chromatic scale" and those notes simply weren't used until later (when composers started wanting to modulate). Because they weren't inititally used, they weren't named--why bother? When musical language evolved to begin using the chromatic, or in between, notes, a naming problem was encountered--and, as Guido's naming system had already been in use for a looong time by that point, a compromise had to be made, and thus we have "accidentals".
Sort of like building an airport with 10 gates, each 700 feet apart, somehow indelibly numbered 1-10, and then realized you really needed 20 gates, each 350 apart. You'd have to go with 1.5, 2.5, etc.
Also also, C-flat and E-sharp (etc.) are used plenty. Plenty plenty.
posted by LooseFilter at 12:58 AM on February 25, 2006
That's called a "chromatic scale" and those notes simply weren't used until later (when composers started wanting to modulate). Because they weren't inititally used, they weren't named--why bother? When musical language evolved to begin using the chromatic, or in between, notes, a naming problem was encountered--and, as Guido's naming system had already been in use for a looong time by that point, a compromise had to be made, and thus we have "accidentals".
Sort of like building an airport with 10 gates, each 700 feet apart, somehow indelibly numbered 1-10, and then realized you really needed 20 gates, each 350 apart. You'd have to go with 1.5, 2.5, etc.
Also also, C-flat and E-sharp (etc.) are used plenty. Plenty plenty.
posted by LooseFilter at 12:58 AM on February 25, 2006
In the beginning was the tonic pitch. And sexual dimorphism said, Let there be puberty; and there was puberty; and there were two voices, that could produce two frequencies in a 2:1 ratio. And the ear heard the 2:1 ratio, and it was good; and in fact, the two pitches seemed to have the same meaning. And a space there was betwixt them; and later on the Greeks got around to naming it the octave. And the unison and the octave were the first day.
But the octave was without form, and void, and music brooded upon the face of the audible spectrum. And music said, Let there be notes, and let them be for melodies to divide the unison below from the octave above; and it was so. And the frequencies of the notes were in ratios of small whole numbers to the frequency of the tonic; and the ear heard these notes that they were good. And there were Greeks, as mentioned, and the Greeks chose the six simplest of these ratios, and placed them betwixt the tonic and its octave to be for melodies. And these mathematically perfect frequency relations were called just intonation, and the scales that could be made of them were called modes; and they sufficed unto the middle ages. And the just intonation and the modes were the second day.
And music said, Let there be harmony; and there was harmony; and forthwith a multitude of problems arose. Waiting for the seventh day and the serpent had apparently gone out of style. For behold, we needed some more notes. We wanted to play in more than one key, and we wanted to construct chords of three and more notes atop more than one base pitch. But lo, if ye stick to the strict mathematical ratios as ye derive a second generation of pitches, they do clash one with another. You are given notes vexatiously close together without matching, and you are given larger intervals whose frequency ratio reduceth not to a neat fraction, wherein consonance fails. And again, the multitude of notes is too great. Verily, verily, I say unto you, Cain and Abel were not half so much trouble.
But there were musical giants on the earth in those days, and they tried a number of different solutions to this problem, all of which involved fudging some of those problem pairs to a compromise, and avoiding some others. All of these systems were called temperaments; they slightly adulterated the tonic intervals, in order to tame the others. Hence the Well-Tempered part of Bach's famous Clavier. And the ear heard well temperament, and all its contemporary competitors; and they were OK. But they were not yet perfectly systematic; some intervals were better-tuned in some keys than in others. And thus it came to pass that each key had its own idiosyncratic interval tunings, and each key had its own mood; and Bach and the sons of Bach did write music, for each key according to its mood. And well temperament and quarter-comma meantone and all the rest of them were the third day.
(It really is like tempering steel: You are judiciously reducing the mathematical/crystalline perfection of some of the material in order to make the rest hang together better. Not that they knew so much micrometallurgy back in the day.)
And from the first to the third days there were instinct, platonism, and craft; but on the fourth day there arose science. And science laid hold upon musical tunings, and it brought forth equal temperament. For somebody had noticed that you could start from a tonic and take twelve successive just fifths (the 3:2 ratio), and you'd end up astonishingly close to your original tonic note. (Well, OK, you also have to divide by 128 / drop down seven octaves, but you get to do that for free.) My calculator informs me that the ratio of the two is 1:1.01364... This is too close to even be a discord, as you can verify (I did!) with the tone generator in Audacity. Try 440 and 446.003 Hz. A trained amateur can hear the difference when they're played in sequence, as well as hearing the (not unpleasant) 6 Hz vibrato when they're played together; but I can say, either as a singer or a listener, that a lot of other things would have to be pretty much perfect before it became unwelcome. And science took that 0.01364... of error and distributed it equally across all twelve jumps; and the new tempered fifths generated exactly eleven other notes, and they cycled back around to the tonic. And behold, as it happens, six of those notes (plus the tonic) are very close to those old just notes of the Greek/medieval system, and the other five slot in among them in a mostly-alternating pattern; thus the white keys and the black keys. The tuning is close enough to just that all the old music can still be played; but now you can play in any one of twelve keys, modulating among them freely, and they all have the same mood. All major thirds sound the same, regardless of which pitch they start from, and so on. And the equal temperament and the modulations and the consonant chords and all were the fourth day.
And in the fifth day, the author confessed that the above was all oversimplified. For behold, a multitude of the unequal temperaments were 12-note systems even before 12-equal temperament came along, even if their intervals were based on a subset of 19-equal or some such monstrosity. And the author did handwave his omission, claiming that this simply shows that solutions to the temperament problem converge. And the confession and the handwave were the fifth day.
And the author did confess again, for verily, only keyboard instruments, electronic instruments, fretted strings, and some singers were in fact equal-tempered. Yea, fretless strings tend to play just when they can; and the strived-for 'ring' of barbershop and a capella comes by singing just-intoned chords; for the ear liketh just intonation better than it liketh temperament. And behold, valved brass play just within any given position, but when the valves or the slide move this author knoweth not exactly what is happening with the tuning. Yea, and the acoustics and fluid dynamics of the woodwinds are more subtil than every brass of the band. And they are all approximated by the equal temperament, and that is generally enough. And the confession and the approximation were the sixth day.
And the author said, Go to, seek out the article Tuning, Tonality, and 22-Tone Temperament; and seek ye out also any basic text on musical acoustics; for from thence have I learned all that I know; and I shall rest from this essay. And the citation and the rest were the seventh day; and they were good.
On preview: What tangerine said, and what LooseFilter said, and then both things I Love Tacos said.
posted by eritain at 2:01 AM on February 25, 2006 [8 favorites]
But the octave was without form, and void, and music brooded upon the face of the audible spectrum. And music said, Let there be notes, and let them be for melodies to divide the unison below from the octave above; and it was so. And the frequencies of the notes were in ratios of small whole numbers to the frequency of the tonic; and the ear heard these notes that they were good. And there were Greeks, as mentioned, and the Greeks chose the six simplest of these ratios, and placed them betwixt the tonic and its octave to be for melodies. And these mathematically perfect frequency relations were called just intonation, and the scales that could be made of them were called modes; and they sufficed unto the middle ages. And the just intonation and the modes were the second day.
And music said, Let there be harmony; and there was harmony; and forthwith a multitude of problems arose. Waiting for the seventh day and the serpent had apparently gone out of style. For behold, we needed some more notes. We wanted to play in more than one key, and we wanted to construct chords of three and more notes atop more than one base pitch. But lo, if ye stick to the strict mathematical ratios as ye derive a second generation of pitches, they do clash one with another. You are given notes vexatiously close together without matching, and you are given larger intervals whose frequency ratio reduceth not to a neat fraction, wherein consonance fails. And again, the multitude of notes is too great. Verily, verily, I say unto you, Cain and Abel were not half so much trouble.
But there were musical giants on the earth in those days, and they tried a number of different solutions to this problem, all of which involved fudging some of those problem pairs to a compromise, and avoiding some others. All of these systems were called temperaments; they slightly adulterated the tonic intervals, in order to tame the others. Hence the Well-Tempered part of Bach's famous Clavier. And the ear heard well temperament, and all its contemporary competitors; and they were OK. But they were not yet perfectly systematic; some intervals were better-tuned in some keys than in others. And thus it came to pass that each key had its own idiosyncratic interval tunings, and each key had its own mood; and Bach and the sons of Bach did write music, for each key according to its mood. And well temperament and quarter-comma meantone and all the rest of them were the third day.
(It really is like tempering steel: You are judiciously reducing the mathematical/crystalline perfection of some of the material in order to make the rest hang together better. Not that they knew so much micrometallurgy back in the day.)
And from the first to the third days there were instinct, platonism, and craft; but on the fourth day there arose science. And science laid hold upon musical tunings, and it brought forth equal temperament. For somebody had noticed that you could start from a tonic and take twelve successive just fifths (the 3:2 ratio), and you'd end up astonishingly close to your original tonic note. (Well, OK, you also have to divide by 128 / drop down seven octaves, but you get to do that for free.) My calculator informs me that the ratio of the two is 1:1.01364... This is too close to even be a discord, as you can verify (I did!) with the tone generator in Audacity. Try 440 and 446.003 Hz. A trained amateur can hear the difference when they're played in sequence, as well as hearing the (not unpleasant) 6 Hz vibrato when they're played together; but I can say, either as a singer or a listener, that a lot of other things would have to be pretty much perfect before it became unwelcome. And science took that 0.01364... of error and distributed it equally across all twelve jumps; and the new tempered fifths generated exactly eleven other notes, and they cycled back around to the tonic. And behold, as it happens, six of those notes (plus the tonic) are very close to those old just notes of the Greek/medieval system, and the other five slot in among them in a mostly-alternating pattern; thus the white keys and the black keys. The tuning is close enough to just that all the old music can still be played; but now you can play in any one of twelve keys, modulating among them freely, and they all have the same mood. All major thirds sound the same, regardless of which pitch they start from, and so on. And the equal temperament and the modulations and the consonant chords and all were the fourth day.
And in the fifth day, the author confessed that the above was all oversimplified. For behold, a multitude of the unequal temperaments were 12-note systems even before 12-equal temperament came along, even if their intervals were based on a subset of 19-equal or some such monstrosity. And the author did handwave his omission, claiming that this simply shows that solutions to the temperament problem converge. And the confession and the handwave were the fifth day.
And the author did confess again, for verily, only keyboard instruments, electronic instruments, fretted strings, and some singers were in fact equal-tempered. Yea, fretless strings tend to play just when they can; and the strived-for 'ring' of barbershop and a capella comes by singing just-intoned chords; for the ear liketh just intonation better than it liketh temperament. And behold, valved brass play just within any given position, but when the valves or the slide move this author knoweth not exactly what is happening with the tuning. Yea, and the acoustics and fluid dynamics of the woodwinds are more subtil than every brass of the band. And they are all approximated by the equal temperament, and that is generally enough. And the confession and the approximation were the sixth day.
And the author said, Go to, seek out the article Tuning, Tonality, and 22-Tone Temperament; and seek ye out also any basic text on musical acoustics; for from thence have I learned all that I know; and I shall rest from this essay. And the citation and the rest were the seventh day; and they were good.
On preview: What tangerine said, and what LooseFilter said, and then both things I Love Tacos said.
posted by eritain at 2:01 AM on February 25, 2006 [8 favorites]
Awesome question, awesome answers.
posted by Civil_Disobedient at 4:04 AM on February 25, 2006
posted by Civil_Disobedient at 4:04 AM on February 25, 2006
Best answer: Wow, many interesting answers although I think they don't really address the question at hand. The placement of black keys on the piano has to do with the Greater and Lesser Perfect Systems: the scales developed in Greek music theory that formed the basis for Medieval music theory. Letter names were eventually applied by Medieval scholars to the greek names for the various scale degrees. The Greek names had nothing to do with absolute pitch, but rather relationships of intervals within tetrachords (groups of four notes.) The lowest, called Proslambanomenos, came to be called A (although different writers use different letter names until they became standardized (not sure when)). The GPS gamut went from A to the A two octaves above, although this is different depending on the author.
The G and LPS were built by stacking tetrachords, either adjacent, or in the case of LPS one overlapping. This overlapping tetrachord, built on the note an octave about the proslambanomenos (A) created the B-flat that characterized the LPS. This was useful in justifying different modes and eliminating the tritone between B and F.
Keyboards originally used only the GPS (as in Roman water organs). The B-flat of the LPS was added next, but as a thinner key, not a black one. Other accidentals were added later. See here
Okay, but why no black note for C or F? Well, they correspond to the original half-steps in the GPS (B-C, E-F, etc.)
(Wow, I've lurked for a couple years now and it takes this sort of nonsense to draw me out of the shadows.)
posted by imposster at 10:29 AM on February 25, 2006 [1 favorite]
The G and LPS were built by stacking tetrachords, either adjacent, or in the case of LPS one overlapping. This overlapping tetrachord, built on the note an octave about the proslambanomenos (A) created the B-flat that characterized the LPS. This was useful in justifying different modes and eliminating the tritone between B and F.
Keyboards originally used only the GPS (as in Roman water organs). The B-flat of the LPS was added next, but as a thinner key, not a black one. Other accidentals were added later. See here
Okay, but why no black note for C or F? Well, they correspond to the original half-steps in the GPS (B-C, E-F, etc.)
(Wow, I've lurked for a couple years now and it takes this sort of nonsense to draw me out of the shadows.)
posted by imposster at 10:29 AM on February 25, 2006 [1 favorite]
Oh, and here is a nice diagram of the G and LPS with Greek note names. Not sure where the Roman letters come from, but I do remember that in some sources the letter names assigned to the Greek names went up to letter P (thank God they decided to just repeat after G.)
Note that in German, B refers to B-flat while H refers to B-natural. I think it has something to do with all this, but I've already spent too much time writing. Maybe someone knows off the top of their head.
posted by imposster at 10:35 AM on February 25, 2006
Note that in German, B refers to B-flat while H refers to B-natural. I think it has something to do with all this, but I've already spent too much time writing. Maybe someone knows off the top of their head.
posted by imposster at 10:35 AM on February 25, 2006
P.S. Former musicologist, current ethnomusicologist.
(Okay, now I'm done. Seriously.)
posted by imposster at 10:39 AM on February 25, 2006
(Okay, now I'm done. Seriously.)
posted by imposster at 10:39 AM on February 25, 2006
eritain: Respect. What a hilarious, thoughtful post.
FWIW: valved brass play just within any given position, but when the valves or the slide move this author knoweth not exactly what is happening with the tuning.
Brass instruments are just open pipe that can play up and down the overtone series of a given fundamental. When one presses a valve combination, the fundamental is changed--lowered--and a new overtone series is then available. The "key" of a brass instrument (trumpet in B-flat, horn in F, etc.) comes from the pitch that is the fundamental of the open pipe with no valves pressed.
So, for a trumpet in B-flat, the overtone series from the fundamental up, with no valves pressed, would be:
Bb--Bb (oct.)--F--Bb--D--F--Ab--Bb--C--D--etc. by ever-decreasing intervals on up the overtone series. (Of course, the fundamental of Bb is WRITTEN as a 'do' or C, which is why a B-flat trumpet is a transposing instrument.)
On any valved brass instrument, pressing the middle (second) valve lowers the fundamental by one half-step; first valve by a whole step; first & second together 1 and a half; second & third by 2 whole steps; first & third by two and a half steps (or a perfect 4th). Etc. For a trombone, slide positions roughly equate to valve combinations, with the same function: 1st position (shortest) increases the length by the same amount that pressing the second valve on a valved instrument does.
So the valves enable a brass instrument to basically be a bunch of open pipes of differing length all tied together and able to be switched between incredibly quickly. It was an enormous technological innovation with fantastic consequences in musical composition--as I'm sure you know--and pretty frickin clever besides.
Woodwinds work roughly the same way, but the keys allow them to manipulate a single pipe's 'length' by opening and closing holes along it in various ingenious combinations.
posted by LooseFilter at 12:17 PM on February 25, 2006
FWIW: valved brass play just within any given position, but when the valves or the slide move this author knoweth not exactly what is happening with the tuning.
Brass instruments are just open pipe that can play up and down the overtone series of a given fundamental. When one presses a valve combination, the fundamental is changed--lowered--and a new overtone series is then available. The "key" of a brass instrument (trumpet in B-flat, horn in F, etc.) comes from the pitch that is the fundamental of the open pipe with no valves pressed.
So, for a trumpet in B-flat, the overtone series from the fundamental up, with no valves pressed, would be:
Bb--Bb (oct.)--F--Bb--D--F--Ab--Bb--C--D--etc. by ever-decreasing intervals on up the overtone series. (Of course, the fundamental of Bb is WRITTEN as a 'do' or C, which is why a B-flat trumpet is a transposing instrument.)
On any valved brass instrument, pressing the middle (second) valve lowers the fundamental by one half-step; first valve by a whole step; first & second together 1 and a half; second & third by 2 whole steps; first & third by two and a half steps (or a perfect 4th). Etc. For a trombone, slide positions roughly equate to valve combinations, with the same function: 1st position (shortest) increases the length by the same amount that pressing the second valve on a valved instrument does.
So the valves enable a brass instrument to basically be a bunch of open pipes of differing length all tied together and able to be switched between incredibly quickly. It was an enormous technological innovation with fantastic consequences in musical composition--as I'm sure you know--and pretty frickin clever besides.
Woodwinds work roughly the same way, but the keys allow them to manipulate a single pipe's 'length' by opening and closing holes along it in various ingenious combinations.
posted by LooseFilter at 12:17 PM on February 25, 2006
LooseFilter: "...the original hymn was most commonly sung in the key of, you guessed it, the pitch (frequency) that became known as 'do' or 'C'. So that's why that's the name of the first note of the all-white-key version of the diatonic scale is 'do'. Now, as to how 'do' came to be called 'C' instead of 'A', I don't know. I'll have to dig around to find out."
To follow up: Guido developed the hexachord system (6 pitches) around the Greek tetrachord system (4 pitches), adding a pitch above and below (starting on A, our friend the proslambanomenos). The pitch below A (even Guido used letter names at this point), the first note of the GPS, was indicated by Guido with a gamma sign. Thus it came to be called "gamma ut," later to become our "gamut" in the sense of a range of pitches.
Back to the hexachord: patterned as such (WS = whole step, HS = half step) WS WS HS WS WS. If students could remember this pattern, with the aid of a mnenomic device based on the chant cited above, they could sing any plainchant. The main thing was to remember where the half step was (mi-fa in Guido's syllables). Transposing the the hexachord (staring on different pitch levels) could account for all of the pitches necessary to perform plainchant.
The chant used by Guido was not necessarily performed on the frequency we think of a C - there was no comparable idea at the time. Rather, the pitches outlined by the syllables corresponded to the hexachord which could be combined, as the Greeks had with tetrachords, to remember the position of half-step in the gamut (formerly GPS/LPS). Notice, the hexchord built on G and C have the same pattern of WS and HS. C was chosen in notating the chant because it was in the middle of the gamut, roughly corresponding to mens' voice range. It wasn't that melodies did not use more than six notes, in fact most chants are sung over the range of an octave. Transposing (mutating) hexachords allowed for keeping track of the intervals. Only when the hexachord is built on C was it called ut/do.
It gets more confusing from there, not that it was particularly clear to begin with.
posted by imposster at 2:33 PM on February 25, 2006
To follow up: Guido developed the hexachord system (6 pitches) around the Greek tetrachord system (4 pitches), adding a pitch above and below (starting on A, our friend the proslambanomenos). The pitch below A (even Guido used letter names at this point), the first note of the GPS, was indicated by Guido with a gamma sign. Thus it came to be called "gamma ut," later to become our "gamut" in the sense of a range of pitches.
Back to the hexachord: patterned as such (WS = whole step, HS = half step) WS WS HS WS WS. If students could remember this pattern, with the aid of a mnenomic device based on the chant cited above, they could sing any plainchant. The main thing was to remember where the half step was (mi-fa in Guido's syllables). Transposing the the hexachord (staring on different pitch levels) could account for all of the pitches necessary to perform plainchant.
The chant used by Guido was not necessarily performed on the frequency we think of a C - there was no comparable idea at the time. Rather, the pitches outlined by the syllables corresponded to the hexachord which could be combined, as the Greeks had with tetrachords, to remember the position of half-step in the gamut (formerly GPS/LPS). Notice, the hexchord built on G and C have the same pattern of WS and HS. C was chosen in notating the chant because it was in the middle of the gamut, roughly corresponding to mens' voice range. It wasn't that melodies did not use more than six notes, in fact most chants are sung over the range of an octave. Transposing (mutating) hexachords allowed for keeping track of the intervals. Only when the hexachord is built on C was it called ut/do.
It gets more confusing from there, not that it was particularly clear to begin with.
posted by imposster at 2:33 PM on February 25, 2006
As much as I like early music history, there was a European paridigm shift after the work of Guido of Arezzo on musical notation and the first modern clavichord keybord layout [I don't know when]. This means that early music history is largely irrelevant because later musicians worked within a new parigidm.
As I explained above, the layout of the white keys is a minimal rational representation of harmonics. I should have also said that black keys are needed to provide 3rds to all of the white keys whose 3rds are not white.
Anyway, at the time of the shift, the new chromatic keyboard made so much sense that most of the European music since then has been notated on staves that reflect the black and white clavichord keyboard.
posted by MonkeySaltedNuts at 7:17 PM on February 25, 2006
As I explained above, the layout of the white keys is a minimal rational representation of harmonics. I should have also said that black keys are needed to provide 3rds to all of the white keys whose 3rds are not white.
Anyway, at the time of the shift, the new chromatic keyboard made so much sense that most of the European music since then has been notated on staves that reflect the black and white clavichord keyboard.
posted by MonkeySaltedNuts at 7:17 PM on February 25, 2006
eritain, that may be the single best post I've ever seen in *.metafilter.com.
I see why Matt linked this question from the front page!
posted by Malor at 7:24 PM on February 25, 2006
I see why Matt linked this question from the front page!
posted by Malor at 7:24 PM on February 25, 2006
Malor: I'm not sure I follow. Why would C be chosen as the basis for the minimal representation of harmonics? For instance, why don't we call 'C,' the note around which some much modern musical thinking revolves, 'A'?
Perhaps I need to clarify my earlier point: the modern-day letter names (as I understand it) come from medieval interpretations of Greek (via Rome) music theory. Often it was interpreted incorrectly and did not necessarily describe medieval music practice, but scholars were keen on the intellectual foundation of Greek theory.
There some good history here, but again, the author does not explain why we would have a half-step between the letter-names B and C, as well as E-F.
It all goes back to the proslambanomenos.
(Maybe I just enjoy saying "proslambanomenos.")
posted by imposster at 8:05 PM on February 25, 2006
Perhaps I need to clarify my earlier point: the modern-day letter names (as I understand it) come from medieval interpretations of Greek (via Rome) music theory. Often it was interpreted incorrectly and did not necessarily describe medieval music practice, but scholars were keen on the intellectual foundation of Greek theory.
There some good history here, but again, the author does not explain why we would have a half-step between the letter-names B and C, as well as E-F.
It all goes back to the proslambanomenos.
(Maybe I just enjoy saying "proslambanomenos.")
posted by imposster at 8:05 PM on February 25, 2006
and yeah, I forgot to a snark about white key 3 chord rockwhich has infested us at least since blues, and became popular from the 1950s on.
White key chords: I, IV, V. No matter how you transpose it these are still white.
posted by MonkeySaltedNuts at 8:07 PM on February 25, 2006
White key chords: I, IV, V. No matter how you transpose it these are still white.
posted by MonkeySaltedNuts at 8:07 PM on February 25, 2006
Err, not Malor, rather MSN.
Although, I'd be happy to hear Malor's thoughts on the proslambanomenos.
(Mmmmm.)
posted by imposster at 8:07 PM on February 25, 2006
Although, I'd be happy to hear Malor's thoughts on the proslambanomenos.
(Mmmmm.)
posted by imposster at 8:07 PM on February 25, 2006
Response by poster: Imposster, from one ethnomusicologist to another, I find your proslambanomenos explanation more convincing than MonkeySaltedNuts' rather thinly supported paradigm shift theory. But what do I know?
Perhaps they are both correct, and the standardization of Guido's system and the European paradigm shift could be one in the same.
And I would have highlighted eritain's clever response but it didn't answer the question.
posted by billtron at 11:44 PM on February 25, 2006
Perhaps they are both correct, and the standardization of Guido's system and the European paradigm shift could be one in the same.
And I would have highlighted eritain's clever response but it didn't answer the question.
posted by billtron at 11:44 PM on February 25, 2006
imposter: they suck. Worst part of the male anatomy. :)
posted by Malor at 2:12 AM on February 26, 2006
posted by Malor at 2:12 AM on February 26, 2006
howard goodall has a great musicology series called "howard goodall's big bangs" on the ovation network, only about 8 episodes but they air frequently. he has a whole show about this issue. interesting and if i remember correctly answers this question well.
posted by hardyboy at 8:07 AM on February 26, 2006
posted by hardyboy at 8:07 AM on February 26, 2006
Good answers here. I have little to add except to emphasize one point:
As others have said, B# is C.
That's only true if you're playing a piano. For instruments without fixed pitch, like the voice or a violin, B# is not necessarily the same as C, and neither are G# and Ab equivalent, etc.
posted by ludwig_van at 9:51 AM on February 26, 2006
As others have said, B# is C.
That's only true if you're playing a piano. For instruments without fixed pitch, like the voice or a violin, B# is not necessarily the same as C, and neither are G# and Ab equivalent, etc.
posted by ludwig_van at 9:51 AM on February 26, 2006
[i'm going to use "note" for a single frequency and "tone" for the sound made when you play a particular note on an instrument]
i'm really repeating the above, but i don't see how anyone is going to understand what's there if they don't already know the answer.
sound is waves. different notes have different length waves.
in fact, except for very simple synthesizers (and even then, only approximately) any "real" tone played by an instrument has a mix of different wavelengths (notes), but normally one is stronger than the others, which gives the note.
so when you "play a note" you're really generating a tone compsed of many different notes - the named note that you are "playing" is the strongest one.
from the way sound is made in most instruments it turns out that the different wavelengths of the notes in a tone are related. they're related in that the lengths are simple ratios of each other. so if the dominant note has a wavelength of 1 metre, say (i have no idea what note that would be, or if it is even audible - it doesn't matter for this illustration) then there will probably also be a note with a wavelength 2/3 of a metre, and another 1/2 a metre, etc.
ok, so a single tone, played on an instrument, is a sound composed of a dominant wavelength (note) and other wavelengths (notes) with simple numerical relationships (this is what the greeks famously found out).
now, it turns out that some tones seem to sound good together. i don't know why, but they seem to, even across cultures (although not always). and it turns out that the tones that sound good together have dominant notes with the same kinds of relationships as the notes that appear when you play a single tone.
so maybe they sound good together because you're hearing the same notes again, but with a different one being dominant. or maybe they sound good together because of something in the way the ear or brain works. i don't know.
but for some reason, notes that are related by appearing in the same tone on an instrument sound good when played as different tones, together.
now if you play a tone with a dominant note of C, the other notes in the tone - and so the other tones that sound good together - are the white keys.
almost.
almost because if you make a piano that is exactly like that, you can only play music in C. if you try starting with any other note, some of the related notes are missing.
now it turns out, quite by chance, that if you divide an octave (a doubling of frequency) from C into twelve steps, eight of those end up being pretty damn close to the eight white notes (including the octave note, which is C an octave higher - hence 8 notes rather than 7 if you're being pedantic). not quite, but close enough.
this is the important point. twelve steps in an octave is arbitrary. the reason it's used is because of this lucky accident - eight of the twelve notes are very close to the naturally-related notes you get when you play a tone on a real, physical instrument (i'm simplifying slightly - there's not exactly eight related notes, but it's the general idea) (and, again, those 8 notes have wavelengths that are related by simple numerical ratios).
why bother using this odd approximation? because then you can start on another tone (D, or A, or whatever) and find the equivalent "white notes" (this time using some of the black notes - they're only black notes for C, if you see what i mean).
again, they don't quite match up, but, again, they're good enough.
so that's the basic reason we have 12 notes rather than 8: so that whatever note we start on (whatever key we write music in), we can pick out the eight basic notes that sound good with it (or at least, good enough approximations).
and that's called "equal temperament". you can have it on a piano because each key is tuned separately. however, in an instrument where the same physical "thing" is used to generate all the notes, you get the "real" 8 notes, not the almost-the-same-but-not-quite approximation you get from picking the nearby ones when you divide things equally into twelve. which is what people are discussing above at one point (that's not quite true - you can bend things a bit for some kinds of instruments).
now obviously there's a huge amount of cultural history / arbitrary choices behind this, and it's not as clear as i make out (why exactly eight notes, for example?). but that's the basic idea - a piano is tuned so that moving from one key to the next always goes up the same amount, and after twelve you've gone up an octave. the white keys pick up the notes that (are close enough to those that) "sound good" when played with C (and which also occur in the tone you hear when you play C).
posted by andrew cooke at 11:49 AM on February 26, 2006
i'm really repeating the above, but i don't see how anyone is going to understand what's there if they don't already know the answer.
sound is waves. different notes have different length waves.
in fact, except for very simple synthesizers (and even then, only approximately) any "real" tone played by an instrument has a mix of different wavelengths (notes), but normally one is stronger than the others, which gives the note.
so when you "play a note" you're really generating a tone compsed of many different notes - the named note that you are "playing" is the strongest one.
from the way sound is made in most instruments it turns out that the different wavelengths of the notes in a tone are related. they're related in that the lengths are simple ratios of each other. so if the dominant note has a wavelength of 1 metre, say (i have no idea what note that would be, or if it is even audible - it doesn't matter for this illustration) then there will probably also be a note with a wavelength 2/3 of a metre, and another 1/2 a metre, etc.
ok, so a single tone, played on an instrument, is a sound composed of a dominant wavelength (note) and other wavelengths (notes) with simple numerical relationships (this is what the greeks famously found out).
now, it turns out that some tones seem to sound good together. i don't know why, but they seem to, even across cultures (although not always). and it turns out that the tones that sound good together have dominant notes with the same kinds of relationships as the notes that appear when you play a single tone.
so maybe they sound good together because you're hearing the same notes again, but with a different one being dominant. or maybe they sound good together because of something in the way the ear or brain works. i don't know.
but for some reason, notes that are related by appearing in the same tone on an instrument sound good when played as different tones, together.
now if you play a tone with a dominant note of C, the other notes in the tone - and so the other tones that sound good together - are the white keys.
almost.
almost because if you make a piano that is exactly like that, you can only play music in C. if you try starting with any other note, some of the related notes are missing.
now it turns out, quite by chance, that if you divide an octave (a doubling of frequency) from C into twelve steps, eight of those end up being pretty damn close to the eight white notes (including the octave note, which is C an octave higher - hence 8 notes rather than 7 if you're being pedantic). not quite, but close enough.
this is the important point. twelve steps in an octave is arbitrary. the reason it's used is because of this lucky accident - eight of the twelve notes are very close to the naturally-related notes you get when you play a tone on a real, physical instrument (i'm simplifying slightly - there's not exactly eight related notes, but it's the general idea) (and, again, those 8 notes have wavelengths that are related by simple numerical ratios).
why bother using this odd approximation? because then you can start on another tone (D, or A, or whatever) and find the equivalent "white notes" (this time using some of the black notes - they're only black notes for C, if you see what i mean).
again, they don't quite match up, but, again, they're good enough.
so that's the basic reason we have 12 notes rather than 8: so that whatever note we start on (whatever key we write music in), we can pick out the eight basic notes that sound good with it (or at least, good enough approximations).
and that's called "equal temperament". you can have it on a piano because each key is tuned separately. however, in an instrument where the same physical "thing" is used to generate all the notes, you get the "real" 8 notes, not the almost-the-same-but-not-quite approximation you get from picking the nearby ones when you divide things equally into twelve. which is what people are discussing above at one point (that's not quite true - you can bend things a bit for some kinds of instruments).
now obviously there's a huge amount of cultural history / arbitrary choices behind this, and it's not as clear as i make out (why exactly eight notes, for example?). but that's the basic idea - a piano is tuned so that moving from one key to the next always goes up the same amount, and after twelve you've gone up an octave. the white keys pick up the notes that (are close enough to those that) "sound good" when played with C (and which also occur in the tone you hear when you play C).
posted by andrew cooke at 11:49 AM on February 26, 2006
ludwig_van: if B# really isn't the same as C when you're singing, wouldn't you be off-pitch? B# and C are both 523.28Hz (or 1046.56Hz, or...)
posted by zsazsa at 11:56 AM on February 26, 2006
posted by zsazsa at 11:56 AM on February 26, 2006
ludwig_van: if B# really isn't the same as C when you're singing, wouldn't you be off-pitch? B# and C are both 523.28Hz (or 1046.56Hz, or...)
It's not that simple, though. All of this stuff can get rather complicated and messy. When you're talking about performance practices, you can't so easily reduce things to frequencies and math. Psychoacoustics is a whole branch of study based on how people perceive sound. And the fact that I generally hear repeated is that the human ear can't detect differences in pitch less than 5 Hz.
Musically, it depends on context. Different players interpret things differently, but for example if you're in A major and G# is the leading tone, it will probably be sung sharper than it would if it was the third in F minor, written as Ab natural. It comes down to the system of equal temperment explained by many posters above. Enharmonic pitches are actually not equivalent, they have just been made equivalent as a tradeoff so that one can play in all keys and have them all sound equally in tune (or equally out of tune).
Pitch is really less standardized than one might think. Many modern orchestras have been making their tuning A higher and higher; I believe some have gone up to 446 Hz.
posted by ludwig_van at 12:15 PM on February 26, 2006
It's not that simple, though. All of this stuff can get rather complicated and messy. When you're talking about performance practices, you can't so easily reduce things to frequencies and math. Psychoacoustics is a whole branch of study based on how people perceive sound. And the fact that I generally hear repeated is that the human ear can't detect differences in pitch less than 5 Hz.
Musically, it depends on context. Different players interpret things differently, but for example if you're in A major and G# is the leading tone, it will probably be sung sharper than it would if it was the third in F minor, written as Ab natural. It comes down to the system of equal temperment explained by many posters above. Enharmonic pitches are actually not equivalent, they have just been made equivalent as a tradeoff so that one can play in all keys and have them all sound equally in tune (or equally out of tune).
Pitch is really less standardized than one might think. Many modern orchestras have been making their tuning A higher and higher; I believe some have gone up to 446 Hz.
posted by ludwig_van at 12:15 PM on February 26, 2006
Er, I meant Ab, not Ab natural!
posted by ludwig_van at 12:15 PM on February 26, 2006
posted by ludwig_van at 12:15 PM on February 26, 2006
now, it turns out that some tones seem to sound good together. i don't know why, but they seem to, even across cultures (although not always). and it turns out that the tones that sound good together have dominant notes with the same kinds of relationships as the notes that appear when you play a single tone.
But andrew, aren't you explaining why they sound good together? Perhaps "good" isn't the right word here - they sound consonant together because, as you've just described, their frequencies are mathematically related (approximately, at least).
I think musical concepts can seem difficult to grasp because they're all so inter-related. It's hard to explain one idea without referencing 10 other ideas.
I'll add some technical terms to what andrew just explained - the "named note" that one plays, which is the pitch we perceive, is known as the fundamental. The other frequencies which accompany the fundamental in all physical instruments (and which, through their particular combination, give the instrument its timbre) are known as overtones. On stringed instruments, one can touch a spot which lies at a certain fraction of the string's length (these spots are called nodes) to prevent the fundamental note from being heard while allowing the overtones to sound. This is known as playing harmonics. There are natural harmonics, which are played on an open string, and artificial harmonics, which involve a stopped string. You can google for a lot of reading about how to produce harmonics on different instruments.
posted by ludwig_van at 12:25 PM on February 26, 2006
But andrew, aren't you explaining why they sound good together? Perhaps "good" isn't the right word here - they sound consonant together because, as you've just described, their frequencies are mathematically related (approximately, at least).
I think musical concepts can seem difficult to grasp because they're all so inter-related. It's hard to explain one idea without referencing 10 other ideas.
I'll add some technical terms to what andrew just explained - the "named note" that one plays, which is the pitch we perceive, is known as the fundamental. The other frequencies which accompany the fundamental in all physical instruments (and which, through their particular combination, give the instrument its timbre) are known as overtones. On stringed instruments, one can touch a spot which lies at a certain fraction of the string's length (these spots are called nodes) to prevent the fundamental note from being heard while allowing the overtones to sound. This is known as playing harmonics. There are natural harmonics, which are played on an open string, and artificial harmonics, which involve a stopped string. You can google for a lot of reading about how to produce harmonics on different instruments.
posted by ludwig_van at 12:25 PM on February 26, 2006
Oh, and one more related bit of info - this business of overtones is why orchestrations students are taught to avoid writing closely spaced notes in the bass register. In an ensemble like an orchestra, the overtones from the bass instruments will blend with and support the sound of all the instruments playing in higher registers. However, overtones of notes that are a second or a third apart will clash with each other, making everything sound muddy. That's why you see more fifths and octaves in the bass.
posted by ludwig_van at 12:29 PM on February 26, 2006
posted by ludwig_van at 12:29 PM on February 26, 2006
Two more unsolicited cents (but isn't that the whole point of the internet?)
Lv: In an ensemble like an orchestra, the overtones from the bass instruments will blend with and support the sound of all the instruments playing in higher registers. However, overtones of notes that are a second or a third apart will clash with each other, making everything sound muddy. That's why you see more fifths and octaves in the bass.
I think this has more to do with critical bandwidth and the physiology of the basilar membrane. Low frequencies stimulate a wider swath on the BS meaning that two low sounds close in frequency will interfere at a smaller musical interval.
That's all I can come up with without looking up the info (God forbid!). Anyone else into the inner ear?
posted by imposster at 1:38 PM on February 26, 2006
Lv: In an ensemble like an orchestra, the overtones from the bass instruments will blend with and support the sound of all the instruments playing in higher registers. However, overtones of notes that are a second or a third apart will clash with each other, making everything sound muddy. That's why you see more fifths and octaves in the bass.
I think this has more to do with critical bandwidth and the physiology of the basilar membrane. Low frequencies stimulate a wider swath on the BS meaning that two low sounds close in frequency will interfere at a smaller musical interval.
That's all I can come up with without looking up the info (God forbid!). Anyone else into the inner ear?
posted by imposster at 1:38 PM on February 26, 2006
ludwig_van, that's very interesting. It's probably the reason why singers that use autotune sound so lifeless.
posted by zsazsa at 2:05 PM on February 26, 2006
posted by zsazsa at 2:05 PM on February 26, 2006
What ludwig_van is talking about is the heart of pure intonation, the pre-well-tempered system. When music is in a key, it has a tonal center around which all other pitches are organized: in C major it's C, Bb major it's Bb, etc. An important acoustical phenomenon of sound is that any pitch is a fundamental vibration + a series of faster, sympathetic vibrations; these sympathetic vibrations occur at specific ratios to the fundamental, and are called 'overtones'. (It's what I mentioned in how brass instruments work.)
The overtone series above a fundamental 'C', for instance, would be the following pitches:
C (fund) - C (oct) - G - C - E - G - Bb - C - D - E - etc.
...at increasingly smaller intervals, even unto half-steps, but one is usually out of the realm of audibility by that point. Each point in that overtone series is called a 'partial', so the list above names the first 10 partials of the C overtone series.
You see, a tonic note (the tonal center, or key note) exerts a perceptual gravity over all other pitches in that scale. This is why any listener feels a pull from the 5th scale degree (the dominant) to the first (the tonic), and that cadence, from V to I, is the most fundamental in western music. The reason one sound can, when placed in a certain context, feel to be 'incomplete' or 'moving' or 'needing resolution' is that the context--a tonal center--causes our perception to orient all tones to that particular fundamental.
Interestingly, the notes in the overtone series--which occur spontaneously in nature, in this consistent, predictable pattern--don't fit into an evenly tempered system, which is where the compromises in intonation detailed by other posters come about. The minor differences must be evened out to allow modulation.
So, as a wind player, I learned that I needed to play the third of a major chord 14 cents lower than its position in an even-tempered system, for that chord to sound truly in tune. Other common adjustments: the fifth above a root must be raised 2 cents; a minor third above root must be raised 12 cents; a minor seventh above a root (such as in the very common dominant seventh harmony) must by lowered THIRTY-ONE cents to sound "in tune".
And "in-tune-ness" is about two things: matching similar frequencies (unisons or octaves); and bringing other notes in line with the overtone series of the root pitch of a given harmony, or of a given tonal center. Thus, while a G# in the key of A must be raised to fit the eleventh partial, a G# in the key of E must in fact be lowered to fit the fifth partial.
(in-tune-ness is also about tone, balance, and blend, but that would be completely off topic even by the standards of this post.)
posted by LooseFilter at 4:00 PM on February 26, 2006 [1 favorite]
The overtone series above a fundamental 'C', for instance, would be the following pitches:
C (fund) - C (oct) - G - C - E - G - Bb - C - D - E - etc.
...at increasingly smaller intervals, even unto half-steps, but one is usually out of the realm of audibility by that point. Each point in that overtone series is called a 'partial', so the list above names the first 10 partials of the C overtone series.
You see, a tonic note (the tonal center, or key note) exerts a perceptual gravity over all other pitches in that scale. This is why any listener feels a pull from the 5th scale degree (the dominant) to the first (the tonic), and that cadence, from V to I, is the most fundamental in western music. The reason one sound can, when placed in a certain context, feel to be 'incomplete' or 'moving' or 'needing resolution' is that the context--a tonal center--causes our perception to orient all tones to that particular fundamental.
Interestingly, the notes in the overtone series--which occur spontaneously in nature, in this consistent, predictable pattern--don't fit into an evenly tempered system, which is where the compromises in intonation detailed by other posters come about. The minor differences must be evened out to allow modulation.
So, as a wind player, I learned that I needed to play the third of a major chord 14 cents lower than its position in an even-tempered system, for that chord to sound truly in tune. Other common adjustments: the fifth above a root must be raised 2 cents; a minor third above root must be raised 12 cents; a minor seventh above a root (such as in the very common dominant seventh harmony) must by lowered THIRTY-ONE cents to sound "in tune".
And "in-tune-ness" is about two things: matching similar frequencies (unisons or octaves); and bringing other notes in line with the overtone series of the root pitch of a given harmony, or of a given tonal center. Thus, while a G# in the key of A must be raised to fit the eleventh partial, a G# in the key of E must in fact be lowered to fit the fifth partial.
(in-tune-ness is also about tone, balance, and blend, but that would be completely off topic even by the standards of this post.)
posted by LooseFilter at 4:00 PM on February 26, 2006 [1 favorite]
(There are 100 'cents' in the semitone of an equal temperament system, so 1200 cents to the octave. Yes, a trained ear can hear one tenth of a half-step difference--or less. In rehearsals, my students commonly perceive differences of as little as 4 or 5 cents.)
posted by LooseFilter at 4:08 PM on February 26, 2006
posted by LooseFilter at 4:08 PM on February 26, 2006
(And yes, a G# as the third of the V chord in A major would be lowered, while a G# played melodically in the same key would be raised a hair. At least, that's what I was taught, and that's what makes my ears happy.)
(If the G# was in the melody AND part of the V chord? You have to go with what you're presenting as most important--melody or harmonic motion. For me, depends on the composer. But regardless, a good performer knows the context.)
posted by LooseFilter at 4:15 PM on February 26, 2006
(If the G# was in the melody AND part of the V chord? You have to go with what you're presenting as most important--melody or harmonic motion. For me, depends on the composer. But regardless, a good performer knows the context.)
posted by LooseFilter at 4:15 PM on February 26, 2006
If I had to guess, I'd say there' s some link to the hexachord system. It can't be just a coincidence that the original hexachords were C and F hexachords.
posted by fvox13 at 4:49 PM on February 26, 2006
posted by fvox13 at 4:49 PM on February 26, 2006
I think this has more to do with critical bandwidth and the physiology of the basilar membrane. Low frequencies stimulate a wider swath on the BS meaning that two low sounds close in frequency will interfere at a smaller musical interval.
I don't know anything about the inner ear. But like I said, as I understand it it's due to the overtone series. The overtones of the bass instruments are crucial to the sound of the ensemble because of the way they blend with the instruments whose fundamentals are in the upper register. But the overtone series of notes that are a third apart are much less consonant than those a fifth or an octave apart. When you've got bass instruments giving you discordant overtones, it makes everything sound muddy. Of course this has also been used intentionally by some modern orchestrators to create such an effect.
posted by ludwig_van at 4:49 AM on February 27, 2006
I don't know anything about the inner ear. But like I said, as I understand it it's due to the overtone series. The overtones of the bass instruments are crucial to the sound of the ensemble because of the way they blend with the instruments whose fundamentals are in the upper register. But the overtone series of notes that are a third apart are much less consonant than those a fifth or an octave apart. When you've got bass instruments giving you discordant overtones, it makes everything sound muddy. Of course this has also been used intentionally by some modern orchestrators to create such an effect.
posted by ludwig_van at 4:49 AM on February 27, 2006
The pythagoreans would take it as necessarily true that intervals which are the lowest-factored ratios are the ones we naturally like to hear. But most of us these days are neither pythagoreans or platonists1; so although this probably nevertheless satisfies our intuition, rigorously we want an explanation that goes beyond "because it is".
Since this is a matter of psychological preference, we'll wonder if it's innate or learned. Let's assume for the sake of argument that it's innate.
If it's innate, then we're stuck with the very difficult question of why the biology of psychoacoustics in humans is as it is. That will probably have a mechanical answer in combination with an evolutionary answer, the former far more discoverable than the latter.
1. Actually, I've come to suspect that most people are intuitively platonists.
posted by Ethereal Bligh at 4:55 AM on February 27, 2006
Since this is a matter of psychological preference, we'll wonder if it's innate or learned. Let's assume for the sake of argument that it's innate.
If it's innate, then we're stuck with the very difficult question of why the biology of psychoacoustics in humans is as it is. That will probably have a mechanical answer in combination with an evolutionary answer, the former far more discoverable than the latter.
1. Actually, I've come to suspect that most people are intuitively platonists.
posted by Ethereal Bligh at 4:55 AM on February 27, 2006
ludwig_van: that's right--the orchestration principles come from the overtone series. Also, our ear really loves the ratios in that pattern, so the scoring of chords often follows that pattern, which also accounts for the 'space' among pitches in bass instruments.
EB: I agree that it's likely intuitive--western tonality is, at its core, a discovered system.
Personally, I think it's because humans love patterns, and the overtone series is an aural pattern we're immersed in daily, whether we realize it or not.
posted by LooseFilter at 10:29 AM on February 27, 2006
EB: I agree that it's likely intuitive--western tonality is, at its core, a discovered system.
Personally, I think it's because humans love patterns, and the overtone series is an aural pattern we're immersed in daily, whether we realize it or not.
posted by LooseFilter at 10:29 AM on February 27, 2006
Shouldn't have read this thread, now I'm going to be humming "Ut queant laxis" all day...
posted by gillyflower at 1:43 PM on February 27, 2006
posted by gillyflower at 1:43 PM on February 27, 2006
Here's a semitone worth of two cents.
LF: I think we are in agreement but coming from different directions. While consonance is subjective, research in the Western music tradition has demonstrated that the judgement of consonance and dissonance can be tied to physiology. Of course this can be conditioned by habituation and other cultural factors, but bear with me.
Donald Greenwood found that the perceived dissonance of two pure tones was related to frequency as measured in critical bands. That is, a frequency moves in a wave along the basilar membrane reaching a point of maximum excitation. The distance of the critical band on the BS is about 1mm: if two frequencies fall within 1mm sensory dissonance is perceived. Perceived dissonance reaches a maximum point at .4mm, or 40% of a CB.
To confuse things, while CBs are constant along the BS, the range of frequencies they encompass are not. A CB on the part of the BS that respond to low frequencies (the bass) includes a wider range of frequencies. I was slightly inaccurate in my previous post, low sounds do not activate more of the BS, but rather low frequencies are stimulated closer together, falling within a CB, creating a sensation of sensory dissonance. So a major 3rd may sound perfectly fine in a high range, but sound muddy and dissonant in a low range because at that point they fall within a critical band.
When complex tones interact, it is not only their fundamental frequencies that can generate sensory dissonance, but also the overtones. An interval or chord will sound more or less dissonant depending on if the overtones fall within a critical band of each other. In an example I found by David Huron, the first ten harmonics of two tones in the interval of a tritone produce 5 intervals that fall within a critical bandwidth - helping to produce the sense of dissonance associated with this interval. In a similar example for a major 5th, only two intervals are produced that fall within a CB.
At the same time, the intensity of the overtones can also impact sensory dissonance. The significant dissonances for the tritone occur on lower harmonics - which will be more intense creating more sensory dissonance. The dissonances in the overtones of the 5th occur much higher in the series - being lesser in intensity and producing less sensory dissonance.
This helps to explain why it sounds better to have pitches in the bass further apart, while it makes less difference for higher frequencies.
Of course, all of this must be contextualized culturally. (Always have to throw that in.)
posted by imposster at 10:51 AM on February 28, 2006
LF: I think we are in agreement but coming from different directions. While consonance is subjective, research in the Western music tradition has demonstrated that the judgement of consonance and dissonance can be tied to physiology. Of course this can be conditioned by habituation and other cultural factors, but bear with me.
Donald Greenwood found that the perceived dissonance of two pure tones was related to frequency as measured in critical bands. That is, a frequency moves in a wave along the basilar membrane reaching a point of maximum excitation. The distance of the critical band on the BS is about 1mm: if two frequencies fall within 1mm sensory dissonance is perceived. Perceived dissonance reaches a maximum point at .4mm, or 40% of a CB.
To confuse things, while CBs are constant along the BS, the range of frequencies they encompass are not. A CB on the part of the BS that respond to low frequencies (the bass) includes a wider range of frequencies. I was slightly inaccurate in my previous post, low sounds do not activate more of the BS, but rather low frequencies are stimulated closer together, falling within a CB, creating a sensation of sensory dissonance. So a major 3rd may sound perfectly fine in a high range, but sound muddy and dissonant in a low range because at that point they fall within a critical band.
When complex tones interact, it is not only their fundamental frequencies that can generate sensory dissonance, but also the overtones. An interval or chord will sound more or less dissonant depending on if the overtones fall within a critical band of each other. In an example I found by David Huron, the first ten harmonics of two tones in the interval of a tritone produce 5 intervals that fall within a critical bandwidth - helping to produce the sense of dissonance associated with this interval. In a similar example for a major 5th, only two intervals are produced that fall within a CB.
At the same time, the intensity of the overtones can also impact sensory dissonance. The significant dissonances for the tritone occur on lower harmonics - which will be more intense creating more sensory dissonance. The dissonances in the overtones of the 5th occur much higher in the series - being lesser in intensity and producing less sensory dissonance.
This helps to explain why it sounds better to have pitches in the bass further apart, while it makes less difference for higher frequencies.
Of course, all of this must be contextualized culturally. (Always have to throw that in.)
posted by imposster at 10:51 AM on February 28, 2006
On bass notes:
One thing I learned when I was tuning an old piano - the overtones of the bass strings are somewhat off from the theoretical model (I forget by how much and in what direction but it was something like the 2nd overtone might be 2.05 times the fundamental.)
So anyway, when you tune a bass string you don't tune the fundamental to agree with the other strings but you tune its ovetones. This means that a simple pitch meter which measures the fundamental is worthless for tuning bass strings.
posted by MonkeySaltedNuts at 12:13 PM on March 1, 2006
One thing I learned when I was tuning an old piano - the overtones of the bass strings are somewhat off from the theoretical model (I forget by how much and in what direction but it was something like the 2nd overtone might be 2.05 times the fundamental.)
So anyway, when you tune a bass string you don't tune the fundamental to agree with the other strings but you tune its ovetones. This means that a simple pitch meter which measures the fundamental is worthless for tuning bass strings.
posted by MonkeySaltedNuts at 12:13 PM on March 1, 2006
LooseFilter: That wasn't exactly what I meant when I said I don't know what's going on with the brass and the winds. Yes, valves and positions alter the fundamental of the air column in a brass instrument, in ways you can describe with half and whole steps. But such a description doesn't tell you anything about the intonation with which those notes are played, which is the question I was addressing.
A thought experiment will clarify: If I'm blowing a trumpet on its fundamental B-flat, and then I depress the second valve, I add about two inches to the air column and lower its pitch. The exact magnitude of the change in pitch is determined by the ratio of lengths; if new length over old is equal to the twelfth root of two, then the pitch drops exactly one equally-tempered semitone, whereas if it's some other nearby value we get a semitone in some other intonation. Now suppose I'm blowing a trumpet with the first valve already down -- a considerably longer base length -- and, again, I depress the second valve to lower the pitch. I have added the same length to the air column that I did before, which means that I have not multiplied its length by the same factor that I did before. The proportional change is smaller, and therefore the semitone drop is smaller. Similar arguments apply no matter what fixed length you add to which two base lengths, and show that additive combinations of a small number of valves (less than 12) cannot generate 12-equal temperament among the fundamentals.
Compound that situation with the fact that the overtone series stops matching 12-equal well at about the seventh partial, and the fact that a horn with a bell isn't really acoustically equivalent to a cylinder with a sharply-defined end, and, well -- I am awestruck that the inventor(s) of modern brass managed to make any several notes tune well with 12-equal, much less optimize for the notes that are most usually played. The situation is also improved a lot by a good player who knows where to lip up or down / when to employ the third-valve slide / how to fudge the official trombone positions / what to do with hand inside bell. But perfect 12-equal on all notes is still not bloomin' likely.
Ditto for the woodwinds: The abstraction says that you calculate pitch from the length of the air column down to the first open hole, and anything after that doesn't count; but in point of fact that ain't so. Even the straightforward soprano recorder shows two obvious exceptions -- the right pinky can affect the pitch by the size of hole it opens, which the abstraction doesn't allow for, and it's also impossible to play a decently-tuned F without stopping a hole below the first open hole. And the subtleties only multiply from there as instruments grow in size, tonal richness, and physical complexity. If the abstraction held true, you could get perfect 12-equal out of any woodwind by drilling holes (of uniform size!) in a pure exponential spacing -- but it doesn't, and you can't.
Ethereal Bligh: As to the mechanics of consonance: The cochlea does a neat little bit of spectrum analysis by ducting the sound along its length and then back again, the membrane that separates the incoming sound from the outgoing sound being thicker at one end of the cochlea than the other. So for any given frequency, there is exactly one point on the cochlea where the incoming wave, outgoing wave, and mechanical properties of the membrane will really drive it back and forth (or in and out, I forget). At that spot, its motion perturbs hair cells, which generate electrical input to a nerve fiber. So each fiber in the auditory nerve responds strongly to one sound frequency. But even a pure tone isn't going to stimulate just one fiber; it's going to alias at other locations. A 300-Hz signal looks like a steady signal to the 300-Hz detector, but it also looks like a spotty signal to the 600-Hz detector, the 900-Hz detector, and so on up; plus, it can drive the membrane (imperfectly) at the 100-Hz and 60-Hz locations, and so on down. (If I remembered the physiology better, I'd tell you whether it can also drive the 150-Hz and 75-Hz locations, but I don't.) Which is to say, you're continuously perceiving weak harmonics of whatever you hear, whether they are in fact there or not. If that's the case, the sensation had better not be irritating -- in fact, it's good if it's slightly pleasant, because that way you'll notice if, say, you lose your high frequency hearing, and you can compensate for the handicap. To my mind, the real question is, 'Why aren't frequency combinations pleasant when they're close together?' Like you, I guess there's an evolutionary answer, something having to do with threats to reproduction that make broad-spectrum noise (river rapids, grass fires, rockslides?) or that produce similar but non-identical pitched sounds (pack hunting animals?). Doesn't explain why ocean waves are soothing, either, since those are more likely than not to produce lots of adjacent frequencies ... hmmm.
posted by eritain at 3:37 PM on March 2, 2006
A thought experiment will clarify: If I'm blowing a trumpet on its fundamental B-flat, and then I depress the second valve, I add about two inches to the air column and lower its pitch. The exact magnitude of the change in pitch is determined by the ratio of lengths; if new length over old is equal to the twelfth root of two, then the pitch drops exactly one equally-tempered semitone, whereas if it's some other nearby value we get a semitone in some other intonation. Now suppose I'm blowing a trumpet with the first valve already down -- a considerably longer base length -- and, again, I depress the second valve to lower the pitch. I have added the same length to the air column that I did before, which means that I have not multiplied its length by the same factor that I did before. The proportional change is smaller, and therefore the semitone drop is smaller. Similar arguments apply no matter what fixed length you add to which two base lengths, and show that additive combinations of a small number of valves (less than 12) cannot generate 12-equal temperament among the fundamentals.
Compound that situation with the fact that the overtone series stops matching 12-equal well at about the seventh partial, and the fact that a horn with a bell isn't really acoustically equivalent to a cylinder with a sharply-defined end, and, well -- I am awestruck that the inventor(s) of modern brass managed to make any several notes tune well with 12-equal, much less optimize for the notes that are most usually played. The situation is also improved a lot by a good player who knows where to lip up or down / when to employ the third-valve slide / how to fudge the official trombone positions / what to do with hand inside bell. But perfect 12-equal on all notes is still not bloomin' likely.
Ditto for the woodwinds: The abstraction says that you calculate pitch from the length of the air column down to the first open hole, and anything after that doesn't count; but in point of fact that ain't so. Even the straightforward soprano recorder shows two obvious exceptions -- the right pinky can affect the pitch by the size of hole it opens, which the abstraction doesn't allow for, and it's also impossible to play a decently-tuned F without stopping a hole below the first open hole. And the subtleties only multiply from there as instruments grow in size, tonal richness, and physical complexity. If the abstraction held true, you could get perfect 12-equal out of any woodwind by drilling holes (of uniform size!) in a pure exponential spacing -- but it doesn't, and you can't.
Ethereal Bligh: As to the mechanics of consonance: The cochlea does a neat little bit of spectrum analysis by ducting the sound along its length and then back again, the membrane that separates the incoming sound from the outgoing sound being thicker at one end of the cochlea than the other. So for any given frequency, there is exactly one point on the cochlea where the incoming wave, outgoing wave, and mechanical properties of the membrane will really drive it back and forth (or in and out, I forget). At that spot, its motion perturbs hair cells, which generate electrical input to a nerve fiber. So each fiber in the auditory nerve responds strongly to one sound frequency. But even a pure tone isn't going to stimulate just one fiber; it's going to alias at other locations. A 300-Hz signal looks like a steady signal to the 300-Hz detector, but it also looks like a spotty signal to the 600-Hz detector, the 900-Hz detector, and so on up; plus, it can drive the membrane (imperfectly) at the 100-Hz and 60-Hz locations, and so on down. (If I remembered the physiology better, I'd tell you whether it can also drive the 150-Hz and 75-Hz locations, but I don't.) Which is to say, you're continuously perceiving weak harmonics of whatever you hear, whether they are in fact there or not. If that's the case, the sensation had better not be irritating -- in fact, it's good if it's slightly pleasant, because that way you'll notice if, say, you lose your high frequency hearing, and you can compensate for the handicap. To my mind, the real question is, 'Why aren't frequency combinations pleasant when they're close together?' Like you, I guess there's an evolutionary answer, something having to do with threats to reproduction that make broad-spectrum noise (river rapids, grass fires, rockslides?) or that produce similar but non-identical pitched sounds (pack hunting animals?). Doesn't explain why ocean waves are soothing, either, since those are more likely than not to produce lots of adjacent frequencies ... hmmm.
posted by eritain at 3:37 PM on March 2, 2006
P.S.: Yeah, I didn't really answer the question. I did get as far as why there have to be two spots in the octave cycle with adjacent natural notes, and why they have to be at the distance from one another that they are, but at that point the question reduces to 'Why did we call that note A and not this one?', which I still didn't touch, mostly because I know absolutely nothing about it. Random speculation: The letter names were assigned by someone who, like me, found the minor scale a whole lot tastier than the major, and wished to privilege it. (Compare the maxim, attributed to Tom Jobim, 'Sadness is more beautiful than happiness.') But I suspect the real answer has to do with someone's hexachords (also the source of the flat and natural symbols, according to Wikipedia).
And yeah, the equations that pull pure harmonic motion out of a plucked string include assumptions that the string's cross-section is uniform, that it can assume any angle at the point where it's anchored, and so on, which explains the business with the bass end of the piano.
posted by eritain at 4:08 PM on March 2, 2006
And yeah, the equations that pull pure harmonic motion out of a plucked string include assumptions that the string's cross-section is uniform, that it can assume any angle at the point where it's anchored, and so on, which explains the business with the bass end of the piano.
posted by eritain at 4:08 PM on March 2, 2006
Yes, valves and positions alter the fundamental of the air column in a brass instrument, in ways you can describe with half and whole steps. But such a description doesn't tell you anything about the intonation with which those notes are played...
Mine was perhaps an overly simplistic explanation, but I thought it a relevant point to the original question since tonality is as much discovered as invented. I find in my teaching that an understanding of the overtone series is a key concept to help ground young music majors' learning, and provides some connection among the disparate threads of their learning (practical and theoretical); I've also found that understanding such basics is valuable to non-musicians as well.
The basic ideas underpinning harmony, harmonic "motion", western music's primary pitch system, etc., are best first understood--in my experience--from the perspectives of construction of the diatonic scale (for horizontal uses of pitch) and of the overtone series, esp. demonstrating the most basic (easily illustrated to even the most unintiated) tonal tendency of V-I, and why it is so. Conceptual foundation building, I suppose. Apologies if my post assumed less than you know.
posted by LooseFilter at 10:24 AM on March 5, 2006
Mine was perhaps an overly simplistic explanation, but I thought it a relevant point to the original question since tonality is as much discovered as invented. I find in my teaching that an understanding of the overtone series is a key concept to help ground young music majors' learning, and provides some connection among the disparate threads of their learning (practical and theoretical); I've also found that understanding such basics is valuable to non-musicians as well.
The basic ideas underpinning harmony, harmonic "motion", western music's primary pitch system, etc., are best first understood--in my experience--from the perspectives of construction of the diatonic scale (for horizontal uses of pitch) and of the overtone series, esp. demonstrating the most basic (easily illustrated to even the most unintiated) tonal tendency of V-I, and why it is so. Conceptual foundation building, I suppose. Apologies if my post assumed less than you know.
posted by LooseFilter at 10:24 AM on March 5, 2006
To the original poster, an attempt at a simple, direct answer to your original question (which I'm not sure has been accomplished yet by me or anyone else):
The fundamental patterns upon which western music centers (centered, in some instances) form what's most basically called a 'diatonic scale'. These scales contain uneven intervals between notes. Which allows the possibility of notes between the bigger intervals, but not the smaller ones.
So, if you set down a diatonic scale pattern in white keys, five of the spaces (where there are 'whole' steps between two notes in the scale) allow for a note in between. Two of the spaces (where there are 'half' steps) do not--those two pairs of notes already have the smallest space in between them. (The smallest space for mainstream western tonality, that is.)
Those two pairs of notes with the half steps in between are B & C and E & F. Thus, if you raise an E by one half step, and call it E#, it's the same note as F. But if you raise an F by one half step, and call it an F#, you haven't made it to G yet. (Conversely, lowering a G one half step yields a spelling of Gb, even though that's--basically--the same pitch as F#.)
Spellings unfamiliar to those for whom music might be a hobby or non-professional passion, may not have encountered much music that would use them. When studying/performing music of the western canon, spellings like Cb, B#, E#, and Fb are encountered fairly often. (Even counter-intuitive things like Bbb--'B double flat'--do occur. It's an old system and, like old code, circuitous methods of doing things do pile up.)
posted by LooseFilter at 10:37 AM on March 5, 2006
The fundamental patterns upon which western music centers (centered, in some instances) form what's most basically called a 'diatonic scale'. These scales contain uneven intervals between notes. Which allows the possibility of notes between the bigger intervals, but not the smaller ones.
So, if you set down a diatonic scale pattern in white keys, five of the spaces (where there are 'whole' steps between two notes in the scale) allow for a note in between. Two of the spaces (where there are 'half' steps) do not--those two pairs of notes already have the smallest space in between them. (The smallest space for mainstream western tonality, that is.)
Those two pairs of notes with the half steps in between are B & C and E & F. Thus, if you raise an E by one half step, and call it E#, it's the same note as F. But if you raise an F by one half step, and call it an F#, you haven't made it to G yet. (Conversely, lowering a G one half step yields a spelling of Gb, even though that's--basically--the same pitch as F#.)
Spellings unfamiliar to those for whom music might be a hobby or non-professional passion, may not have encountered much music that would use them. When studying/performing music of the western canon, spellings like Cb, B#, E#, and Fb are encountered fairly often. (Even counter-intuitive things like Bbb--'B double flat'--do occur. It's an old system and, like old code, circuitous methods of doing things do pile up.)
posted by LooseFilter at 10:37 AM on March 5, 2006
Response by poster: LooseFilter:
"Those two pairs of notes with the half steps in between are B & C and E & F."
This is the step in your explanation that my original question was about. Why are the two pairs between these note name pairs? Why not between D & E and G & A, which would make just as much sense according to the tuning system of even-tempered keyboard instruments.
Can you explain why the half steps are in between these note name pairs, and not between other note name pairs?
Imposster has come closest to answering the question. In fact, only you and imposster have even attempted to answer the question. Most posters argued about tuning. My question is not about tuning. It is about nomenclature.
posted by billtron at 1:25 PM on March 6, 2006
"Those two pairs of notes with the half steps in between are B & C and E & F."
This is the step in your explanation that my original question was about. Why are the two pairs between these note name pairs? Why not between D & E and G & A, which would make just as much sense according to the tuning system of even-tempered keyboard instruments.
Can you explain why the half steps are in between these note name pairs, and not between other note name pairs?
Imposster has come closest to answering the question. In fact, only you and imposster have even attempted to answer the question. Most posters argued about tuning. My question is not about tuning. It is about nomenclature.
posted by billtron at 1:25 PM on March 6, 2006
That's the part I have yet to find confirmation of, surprisingly.
My best guess is that that, when letter names were settled on, the natural minor mode was considered more central.* In the diatonic scale, the 'natural minor' is the aeolian mode, and the all white key version of that scale would start on 'A'.
But I don't know if it's true--just my best guess.
*There is some evidence from ancient humans to support this--the oldest pitch instruments yet discovered (40-80,000 years old, IIRC) are based upon the natural minor mode, not the major mode.
posted by LooseFilter at 11:50 PM on March 6, 2006
My best guess is that that, when letter names were settled on, the natural minor mode was considered more central.* In the diatonic scale, the 'natural minor' is the aeolian mode, and the all white key version of that scale would start on 'A'.
But I don't know if it's true--just my best guess.
*There is some evidence from ancient humans to support this--the oldest pitch instruments yet discovered (40-80,000 years old, IIRC) are based upon the natural minor mode, not the major mode.
posted by LooseFilter at 11:50 PM on March 6, 2006
This thread is closed to new comments.
posted by ParisParamus at 6:57 PM on February 24, 2006