Music Theory for Conworlders: The Pythagorean Scale

[mylisten.com is having problems, so the stuff I was going to upload today still sits on my harddrive]

We have seen a few ways of constructing scales this far, and we'll now look at a rather specific example. It goes back to antiquity, and was quite predominant in European ecclesiastical music in medieval times, and also in late medieval and early renaissance art music. (I am not too sure on the exact times with regards to art music - we don't really have any art music separate from ecclesiastical ditto until late medieval times.) Other scales were in use in antiquity, and probably in medieval times as well, but the pythagorean scale has a certain reputation.

First of all, the perfect untempered fifth is, on most instruments with natural overtone series, the easiest interval to tune, except for the octave. Thus repeatedly tuning strings by jumping by fifths (C-G-D-A-E-...) will introduce little error, compared to what tuning by some less easily tuned interval will do. Tuning repeatedly by fifths therefore seems a rather natural approach. You can be pretty sure you've heard perfect untempered fifths at some rock concert, since several rock guitarists actually tune the E-A-D strings to a perfect untempered fifth + octave.

It is closely related, conceptually, to the modern western tuning as well, and is an important step in the historical development that lead up to the modern temperament. For a more mathematical look at the idea of repeatedly tuning by fifths, consider a 'period', a distance that basically is the 'space' within which we build our scale. We go with the octave for now, so either 2/1 or 1200 cents. Since we use fifths, we have the ratio 3/2, or 701.955000865 cents. We then build an octave-reduced stack of these. This can be done in two ways, giving the same result:

(3/2)⁰, 3/2,  (3/2)² / 2, (3/2)³ / 2, (3/2)⁴ / 4 ...

or

0,  701.96..., 701.96... * 2 mod 1200, 701.96... * 3 mod 1200,  701.96... * 4 mod 1200, ...

We repeat this for a while, and once we hit ^6 or *6, we reach a point where our scale looks like this:

8/8,  9/8, 81/64, 729/512, 3/2, 27/16, 243/128, 2/1

However, traditionally we divide each by 3/2, thus in effect rotating the cycle so what formerly was 3/2 now is 1/1. We have

8/8,  9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1

This is not particularly far off from our previous scale - the one built from stacking 4:5:6 in a similar fashion

8/8, 9/8, 80/64, 4/3, 3/2, 25/15, 240/128

our previous just intonation scale, rotated by dividing by 3/2, analogously to how our current scale's been rotated

We can see we hit pretty close -  81/64 and 80/64 differ by the syntonic comma 81/80 (about 21 cents or so), 9/8, 4/3, 3/2 are exactly the same, 240/128 and 243/128 differ by 243/240 = 81/80 too. 25/15 and 27/16 are a bit more cumbersome to work with, but differ by a comparable amount. It sounds fairly similar to the just intonation scale previously introduced, as well as to the twelve-tone equal temperament. Personally, I find the chords a bit more restless than just intonation, but the restlessness can give an appealing shimmer to the sound of it - depends a bit on the instrument used to play the song.

The familiar song, now in pythagorean intonation

So, our 4:5:6 ( = 64:80:96) chords are now instead represented by 64:81:96. A slight mistuning, but not all that terrible - a tenth of a wholetone.

This scale is the 'canonical' pythagorean scale. It has a structure of LLsLLLs (or in the form we first obtain, LLLsLLs). Most rotations of it - LsLLLsL, LLsLLsL - were in use as tonalities in their own right in medieval times. (With the exception of sLLsLLL, which very few compositions exist in, even today.) To 'rotate' a scale given in the L,s-notation, just take a symbol from either end and move it to the other end - repeat until the rotation you want is obtained.

There is a problem, however - a chain of 3/2s never, ever, will reach an octave of its starting point. We can easily prove it along these lines (only a sketch of the proof is given):

The last digit of successive powers of 2 form a repeating series of 2, 4, 8, 6, 2 ... and only the last digit affects the last digit of the next member in the series.

The last digit of successive powers of 3 form a repeating series of 3, 9, 7, 1, 3 ... and thus never form even numbers.

Thus 3^n/2^m never will equal 2^p, for integers n,m,p > 0.

Thus, by continuing up the series we get an infinite number of tones. However, we can decide to just terminate it at some point - such as building the scale above and being content at that point. Some cultures have used the same method and been content two steps earlier - 1/1 9/8 81/64 3/2 27/16 2/1 forms a very nice pentatonic scale, as does the rotation corresponding to the minor pentatonic 1/1 32/27 4/3 3/2 16/9.

The musicians of several cultures that used this scale went to 12 steps, because they found that the twelfth iteration is fairly close to an octave of the starting point. (About an eight of a wholetone off.) (3/2)¹² ≃ 2⁷. Thus, the eleven first fifths were perfectly in tune, but the twelfth got reduced by the amount of the error. Using the twelfth fifth was therefore avoided, and keys that included it were not used all that much. Of course, that particular fifth could be avoided or used sparingly in compositions in keys that contained it.

The scale that is obtained thus is fairly evenly spaced - should be something like, given in cents ~ 113, 91, 113, 91, 113, 91, 91, 113, 91, 113, 91, 91 if I count correctly - LsLsLssLsLss. L is the large Pythagorean semitone of 2187/2048 , s the small Pythagorean semitone of 256/243. This means different keys will sound slightly differently when playing the same composition in them - let's introduce a meta-notation: L, s. These are the steps of a scale that is a subset of the above cromatic scale. Now, some L will consist of Ls, some will consist of ss. Some s will consist of L, some of s.
Thus, when we make a scale that is LLsLLLs, the first rotation gives [Ls, Ls, L, ss, Ls, Ls, s]; if we start from the second note and build the same scale, we get [Ls, Ls, s, Ls, Ls, ss, L]. From the third tone, we get [Ls, Ls, s, Ls, Ls, Ls, s], etc. This will lend the scale some variation, of course - each key will now have a slightly unique sound with regards to some interval. However, providing examples of this requires some work. I will probably do it some weekend.

Both of the facts given above kind of might give a musician reason to think that hey, why not take the average of seven 91-cent steps and five 113 cent steps or alternatively 'hey, why not average out the error of the last fifth on all twelve of them - not a single one would be all that badly damaged, and the bad one would be greatly improved'. (Other solutions could also have occurred to him if he used some form of extended pythagorean, where the gamut goes past 12 fifths.)

 [Historically, we actually have some reason to think Pythagoras may have used a period of 4/3 instead of the octave, but for some mathematical reasons, the scale you get by adding two fifths into the frame of a fourth gives [9/8, 81/64, 4/3]. This was then placed on top of 3/2 as well, giving the exact same scale as the stacking method (after rotating it back one step).]

Now, what could a culture have done differently? It could possibly use a different interval, say 5/4, to generate its tuning - a thing even late renaissance and early baroque European music did! It could go further than twelve iterations to find a tuning with smaller granularity. It could use this method over part of its gamut, and some other method elsewhere. It could potentially use a different period.

We will look at using multiple generators next, after which we'll start looking at tempering things.