E-F B-C, why no semitone?

August 14, 2008 10:08 AM

I was asked the other day why there are no semitones between e-f and b-c. My answer was that it had to do with the frequencies of the notes but i'd like to be able to give a more in-depth answer as this doesn't really cover it.
posted by chelegonian to media & arts (16 comments total)

e to f and b to c ARE semitones.
posted by stubby phillips at 10:14 AM on August 14, 2008 [2 favorites]

In short, they're named this way because they belong to the major scale beginning on C, which is a note that has singular prominence in Western music (the root causes of this lie in the scales used in Medieval church music, called the modes, but that's another discussion).

The interval sequence of the major scale is whole tone, whole tone, half tone, whole tone, whole tone, whole tone, half tone. Given that it's more convenient to not have to refer to any note in the most common major scale with a sharp or a flat, and that it's convenient to call them each by a unique letter starting from C, E and B have no sharp distinct from F and C.
posted by invitapriore at 10:23 AM on August 14, 2008 [1 favorite]

let's put it this way: the notes we currently use (C#, Ab, etc) are based on the diatonic scale superimposed on a domain of 12 possible notes. diatonic scales are 8 note scales with identical sequences of whole tones and semi tones. there are 12 of them, one based on each of the 12 possible notes (enharmonic inequality makes this a little more complicated, but let's not piss off dnab).

the diatonic scales are very familiar to us: the olden church modes and the modern major and natural minor scales are both diatonic.

in order to get 8 notes to fit in a 12 note universe, two of the intervals would have to be half-steps (semitones). (slightly off topic: the whole tone scale only has 6 notes).

why do we care about the diatonic scale? because it's the first 8 tones of the overtone series (rearranged and divided by powers of two (and slightly tempered nowadays)). the overtone series is an actual physical phenomenon, so it all makes sense in a way. and it sounds natural to the ear.

does this help or confuse?
posted by stubby phillips at 10:23 AM on August 14, 2008

From this website :

The most audible overtones of any one note add up to its major chord, when played out loud rather than as overtones: Tonic, Fifth & Third.

The most audible overtones of a tonic or keynote all have simple ratios, like 2:1 (octave), or 3:2 (fifth note of scale), and the 4th note of the scale, whose first different overtone is the given tonic, has a ratio of 4:3 . In fact these three notes are present in virtually every musical scale known on earth.

If you write out the overtones of these three notes and string out the three most audible ones of each within the span of an octave, you will get the major scale:
Tonic C: Overtones are: C, G, E, and Bb (I've left out the additional octave overtones as redundant and too high and weak to be noticed within the framework of average human hearing.)
Fifth G: Overtones are G, D, B, and F
Fourth F: Overtones are F, C, A, and Eb.
posted by Bearman at 10:30 AM on August 14, 2008

I think my question was the wrong one. Conventions for naming the notes aside what is it about the frequencies of the notes and their relationship to one another that determines which notes belong to which scales? is there some physics explanation for it or is it to do with our preference as listeners?
posted by chelegonian at 10:45 AM on August 14, 2008

This page might help
posted by harmfulray at 11:03 AM on August 14, 2008

It's important to understand that the notes in the diatonic scale have been tempered so that a cycle of fifths brings us around to the root instead of to the harmonic natural. That is, there is a partial "physics explanation" for the scales we use, and our natural "preference as listeners" would actually conform to that explanation, but for notational and performative convenience we (at least in the West) have slightly re-tuned the scales in a way that divorces them from the underlying overtones and harmonics.
posted by nicwolff at 11:04 AM on August 14, 2008

what is it about the frequencies of the notes and their relationship to one another that determines which notes belong to which scales?

As explained above by stubby phillips and bearman, the basis is the overtone series that results from a vibrating string or column of air. For a more thorough explanation, read this article on relating tuning and timbre.
posted by ludwig_van at 11:06 AM on August 14, 2008

Paraphrased from the site I posted above:

The most pleasing intervals are the simplest ratios of vibrations. An octave(C to C ) is 2:1,
a fifth (C to G) is 3:2, and a fourth (C to F) is 4:3. These are the simplest ratios, and these three notes (again in the key of C - C, F, G) are found in most known scales on the planet.

The most audible overtones of any note add up to the major chord.
For C, you get C E G
For F, you get F A C
For G, you get G B D

If you put all of these notes together, you get the major scale:

C D E F G A B C
posted by Bearman at 11:07 AM on August 14, 2008

it's a combination of both - we "naturally?" (i think it is natural, but perhaps even this is cultural) hear certain tones as "going together" when their frequencies have simple relationships (for example, twice the frequency is an actave and we call those "the same note").

the must common ratios (things like 3/2, 4/3 etc) give you a bunch of notes related to an initial frequency. and if you start with C then those notes (C major) are the other named notes (the white keys on a piano).

almost.

it turns out that if you do that, you end up only being able to play in the key of C.

but if you do a bit of as hack and divide an octave into 12 equal divisiions then you get numbers that almost match the frequencies you get from simple ratios (you also get some extra frequencies that don't match - they're the black keys, the sharps and flats).

this approximation sounds "good enough" - the notes almost match. AND from the way the maths works out, you can also start anywhere else and get "close enough" to the notes in other keys too (so you can play muic in A major, D, makor etc, without retuning your instrument).

that hack is called equal temperament.

hope i've got that right - i'm no musician, just found this kind of thing interesting a while back. there's a free book kicking round on the 'net somewhere that goes into the maths behind music in quite some detail.
posted by not sure this is a good idea at 11:08 AM on August 14, 2008

I might be misunderstanding, but are you asking why Western music uses semitones rather than, say, quartertones?
posted by lekvar at 12:12 PM on August 14, 2008

thanks everyone this has been really helpful. apologies to s.p and bearman for not reading your answers more throughly and thanks to harmfulray for the link, it explained overtones. That wasn't what i had in mind lekvar but why is that? or is it covered by invitapriore at the top of the page?
posted by chelegonian at 12:41 PM on August 14, 2008

As I understand it, quartertones aren't used in Western music simply because that isn't how we've commonly measured tone. Basically, convention. Because of this convention, instruments that can produce quartertones are rare. Some Middle-Eastern music does use quartertones.

But I'm sure someone can do a better job explaining it than I can.
posted by lekvar at 12:52 PM on August 14, 2008

It's worth noting that the notes E# and Cb both exist, in the keys of F# Major and Gb Major respectively.
posted by benzo8 at 5:11 PM on August 14, 2008

As I understand it, quartertones aren't used in Western music simply because that isn't how we've commonly measured tone. Basically, convention.

The answer to this question is the same as the answer to the original question. The overtone series produced by a vibrating string approximately creates our major scale, which contains no quarter tones.

It's worth noting that the notes E# and Cb both exist, in the keys of F# Major and Gb Major respectively.

Those notes are found in those major scales, but that isn't the only time they'll pop up. Among others, C# major and A+ chords are spelled with an E#, and Db7, Fdim, and Ddim7 chords contain a Cb.
posted by ludwig_van at 6:27 PM on August 14, 2008

One way to explain this is to think in terms of preserving perfect 5ths--that's the interval between C & G or a frequency ratio of 3:2.

In making a scale, you are trying to reconcile notes that match up with this ratio (3:2) together with the fact that pretty much everyone agrees that notes that match up via the ratio of 2:1 (octaves) sound alike.

Well if you mess around with the math on this for a while--or take some strings or pipes or tone boards, or whatever and try to tune them up--you soon discover that you can't do it.

You just can't get a group of notes that all relate together perfectly via the 3:2 ratio AND the 2:1 ratio.

The chart below show powers of 3/2 and 2. You can see that the two never meet--there is no number (greater than 1) that is some integer power of 3/2 and also an integer power of 2:

 
 1 1.5 2.25 3.38 5.06 7.60 11.4 17.1 25.6 38.4 57.7 86.5 129.7 etc.
 1     2        4        8        16     32      64       128   etc.

What this means in musical terms is this: There is no possible way to create a collection of notes where all 5ths are perfectly in tune and all octaves are perfectly in tune.

However, you can get pretty close. 7.6 almost equals 8. 17.1 almost equals 16. 129.7 almost equals 128.

What this means in musical terms is this: You can find some collections of notes where the 5ths and the octaves will be pretty close to being in tune. These will have 5 notes (7.6 ~ 8), 7 notes (17.1 ~ 16), or 12 notes (129.7 ~ 128).

However, you're going to have some fudge factor to deal with.

Dealing with the fudge factor is where the answer to your question comes in--some notes in a scale are squished in a bit closer together, and some are spread wider apart. This is done to create the impression of a scale where everything is in tune, when--as we just discovered--that is actually impossible.

Along the way, through some thousands of years of messing around with this problem while making music, musicians also discovered that messing around with the arrangement of these wide and narrow spots within the scale can create different scales, each with a different "feel" or "sound".

5 note scales
The smallest amount of notes you can put together that works, sort of, with some fudge factor, to reconcile the 3:2 vs 2:1 problem is 5. This is known as a pentatonic scale.

(Of course you have the trivial solution--the one-note scale. We'll skip an extended discussion of that one.)

What you find with a pentatonic scale is you have some "nice" perfect 5ths, that are close to being in tune. But no matter what you do you are going to end up with some terribly out of tune 5ths. There is no perfect way to fix this problem but there are different ways of shifting things around to try to make them acceptable. That is why there are a bunch of different types of pentatonic scales--different ways of shifting and adjusting the notes.

The thing all of these different types of pentatonic scale have in common is this: Since the 3:2 ratios didn't work out perfectly, you have a certain amount of out-of-tune-ness to deal with. You can spread the out-of-tune-ness around with perfect equality. In that case EVERY harmonic interval in the scale will sound just horribly, wretchedly out of tune.

Or, you can make most of the intervals in the scale pretty much perfect, and shunt of ALL of the out-of-tuneness into just one or two intervals. (Then in practice you just avoid those horrible sounding intervals and use all the others, which are good sounding.)

(Now getting down to the answer to your question.) The result of this is a scale where the 5 notes are not evenly distributed throughout the octave. Instead you have some notes a little closer together and some a little further apart.

You can observe this with the black notes on a piano, which are a type of pentatonic scale. Notice that some of the black notes have more space between them than others. (in particular, some black notes have just one white note in between, while others have two black notes).

So you started up with the problem of how to resolve the 3:2 and 2:2 ratios (or, less mathematically, how to make a musical instrument that has a managably small number of different notes and still sounds reasonably in tune) and we ended up with a scale where the gaps between some notes are wider and some, narrower.

There are different ways to handle the narrower & wider gaps, and each of these will give you a scale with a different "feel" or "sound" (an idea that goes back at least as far as the ancient Greeks), depending on where the wide and narrow gaps fall.

The 7-note scale (our major & minor scales)
All that is for a 5-note scale. What about for a 7-note scale?

Like the 5-note scale, the 7-note scale isn't perfect. Again you can spread the out-of-tune-ness out equally and end up with a horrible mess, or you can make most intervals sound good at the expense of a few really horrible ones. But (again) you end up with a scale with uneven gaps.

In the C major scale, those uneven gaps, smaller than the rest, are E-F and B-C. And that of course is what brought you to ask this question.

In specific terms, making a scale this way allows almost all the 5ths in the scale to be pretty much perfectly in tune:

C-G, D-A, E-B, F-C, G-D, and A-E are all good.

However, B-F is "terrible". In fact it is so terrible that nowadays we actually give it a completely different name than those other 5ths (diminished 5th rather than perfect 5th).

Now the major scale is one way to solve this problem (a 7-note scale that preserves 3:2 and 2:1 frequency ratios as well as possible). There are lots of other ways, though and you have heard some of them--for instance, minor scales and modes.

Just as with the pentatonic scale, each of these different solutions gives a scale with a different "sound" or "feel" to it.

The 12-note scale
Now if you are not satisfied with the 5 note or 7 note scales and keep trying to find a solution to this problem, you find out that the next scale that works out better has 12 notes.

Still the solution doesn't work perfectly. However, the difference between theory and reality is now small enough that for the first time you can realistically make the choice to evenly distributing the out-of-tune-ness (a relatively modern invention known as equal temparement).

Or you can choose to do as we did before--keep some 5ths more perfectly in tune and other less perfectly, with the result of a scale where frequency difference between some adjacent notes is a little wider and some a little narrower. However, now the differences are so small that we really consider them just to be slight adjustments in tuning.

There are different ways of making these slight adjustments, referred to as different "temperaments".

Interestingly, one of the oldest temperaments is the so-called meantone temperament, where all the 5ths are tuned perfectly except for one, which is just terrible and is known as the "wolf interval".

Musicians all knew about the wolf interval and simply made a practice of avoiding it--mostly by avoiding certain keys and chords where it would figure prominently.

More than 12 notes
Well, you don't have to stop with 12--the quarter-tone scale, with 24 notes per octave, has already been mentioned. Some of these scales have been developed, used, and have their proponents.

But the 5-, 7-, and 12-note scales are the most commonly heard--not only in "western" music but (with various sorts of local flavor) in music around the world.

More apropos to your question, that is why you look down at a modern piano/organ/keyboard and see (repeated in each octave) a pattern of 5 black notes having some wide/narrow gaps, a pattern of 7 white notes having some wide/narrow gaps, and pattern of 12 black & white notes all of which seem to be evenly spaced.
posted by flug at 6:57 PM on August 14, 2008 [3 favorites]