Music, Tuning

Pythagorean Comma

This post contains audio examples of a discrepancy in musical tuning known as the Pythagorean Comma. I’d like to do a full discussion of this topic at some point, but my aim here is just to share some illustrations I made; in this post I’ll assume the reader already understands the underlying theory.

To explore the Pythagorean Comma, I wrote a toy piece (view score) that traverses a series of twelve ascending perfect fifths with some octave skips along the way.  The trick was to keep the example within a constrained pitch range, and to make sure that each interval is sustained long enough that its quality can be judged by ear.  (Since something unexpected happens at the end, I wanted to give the listener a chance to verify that there’s no subterfuge employed along the way!)  The interval palette is limited to ascending fifths and descending octaves; there are no descending fourths.  The pattern works like this: (1) play a note, (2) hold the note and play its fifth, (3) play the two notes again together, (4) play the fifth along with its instance an octave below. Now repeat, choosing the upper or lower instance of the fifth from the previous step as the new starting point — make this choice so that the next fifth will fall in the range C4-C5. For example, we start at C4, then play G4, then C4 and G4 together, then G3 and G4 together — that’s one iteration of the cycle — then we take G3 as our next starting note, play D4 above it, and so on. After doing this twelve times we should land right on C5 if we’re using equal temperament.

First listen to the clip in equal temperament. Pay attention to the end and notice how everything sounds “in tune” as the sequence leads us back to the same note we started with. The clip is rendered using a “Bright Pad” soundfont, and you might ask why I’ve chosen a cheesy 80’s synth for these examples.  No, it’s not my secret love of 80’s music.  Actually, I experimented with lots of high-quality instrument samples but all of the instruments I tried that were capable of long sustain also had artifacts that make it harder to discern whether the tuning is pure.  I tried some beautifully sampled organ stops, for example, but found that some of the notes had a slight wobble or vibrato that could cause intervals to sound like they’re beating.  I’m still looking for other sounds for these examples, but Bright Pad works for now as it has good sustain, no wobble, and lots of overtone content that can help with judging the tuning.  As a side note, if you listen very closely to this clip in equal temperament, you might be able to notice a slight contrast in the texture of the fifths and the octaves: the fifths are just a touch more busy.

Ascending Fifths — Equal Temperament:

Now listen to the same material in Pythagorean tuning and you’ll hear the jarring effect of the comma near the end of the sequence. I’ve tuned this second clip so that each “new” note is a pure perfect fifth above the previous note, meaning that their frequencies always form a 3/2 ratio.  (If you listen closely you might notice there’s less of a contrast in texture between the fifths and octaves here than in the previous clip, but it’s a very subtle difference.)  As we traverse the series of pure fifths, the notes drift increasingly sharp of their equal-tempered counterparts, because a pure fifth is roughly 2 cents wider than a tempered fifth.  However, when the sequence reaches its penultimate step at F, we break the tuning pattern for the sake of comparison: instead of playing a C tuned as a pure fifth above our F, we play an octave-equivalent of our original C — the same note we started with — and now this C sounds flat! That’s the crux of the demo. In order to complete the sequence with pure interval, we’d need a second version of C, one tuned a Pythagorean Comma (roughly 23.46 cents or 1/4 of a semitone) sharp of the C we started with.  The C needs to be sharp because the F we’ve arrived at through the series is itself sharp.  Since we reuse our original C instead, the final F-C interval is too narrow and sounds mistuned in comparison to the pure fifths we’ve been hearing all along.  Here’s the clip:

Ascending Fifths — Pythagorean: 

The thing to remember is that both clips start and end on the very same C: in equal-temperament that C still sounds “right” at the end of the sequence, but in Pythagorean tuning it sounds “wrong” because sequence of pure fifths has led us to expect a sharper C.

Another way to observe the gradual pitch drift created by the pure fifths is to play both versions simultaneously, and listen to how the discrepancy between versions grows with each ascending fifth: you will hear this as increasing wobble or beating in the sound.  To save you the trouble of trying to start the two previous clips at the same time, I’ve prepared one clip that superimposes them.  I made one slight modification at the end of the pure fifths version, so that it concludes with the sharp C that the sequence demands, instead of returning to original C; that original C will be heard in the equal-tempered version playing simultaneously.

Ascending Fifths — Pythagorean and Equal Temperament Superimposed:

Here’s another example (view score) that works much like above, except that it follows a series of descending fifths. This time, our penultimate note is a G that’s significantly flat of its 12ET counterpart. Right before the sequence ends, we break the tuning pattern as above. Instead of descending from our flat G to the flat C that would make pure fifth with it, we descend to our original C, and now this C sounds sharp!  Again we’re left with a fifth that’s too narrow and sounds wobbly.

Descending Fifths — Equal Temperament:

Descending Fifths — Pythagorean:

In summary, if you pick a starting note and depart on a series of ascending pure fifths, your starting note will begin to sound flat; if you use descending pure fifths instead, your starting note will begin to sound sharp.  Remarkable!

Appendix

Here are the tunings I used in the examples.  They are set up slightly differently from what’s commonly called “Pythagorean Tuning,” where you take a series of six fifths above the home note and five fifths below.  For the ascending fifths example, I arrived at the tuning by traversing eleven fifths above the home note, and for the descending fifths example I traversed eleven fifths below the home note.

Ascending Fifths:

Note Ratio Cents Deviation from 12ET
C 1/1 0 0
C#/Db 2187/2048 113.69 13.69
D 9/8 203.91 3.91
D#/Eb 19683/16384 317.6 17.6
E 81/64 407.82 7.82
F 177147/131072 521.51 21.51
F#/Gb 729/512 611.73 11.73
G 3/2 701.96 1.96
G#/Ab 6561/4096 815.64 15.64
A 27/16 905.87 5.87
A#/Bb 59049/32768 1019.55 19.55
B 243/128 1109.78 9.78

Descending Fifths:

Note Ratio Cents Deviation from 12ET
C 1/1 0 0
C#/Db 1024/972 90.22 -9.78
D 1048576/944784 180.45 -19.55
D#/Eb 64/54 294.13 -5.87
E 65536/52488 384.36 -15.64
F 4/3 498.04 -1.96
F#/Gb 4096/2916 588.27 -11.73
G 4194304/2834352 678.49 -21.51
G#/Ab 256/162 792.18 -7.82
A 262144/157464 882.40 -17.6
A#/Bb 16/9 996.09 -3.91
B 16384/8748 1086.31 -13.69

If you reorder these tables so the notes ascend (or descend) in fifths, you’ll notice that the deviations from 12ET then increase with each successive fifth away from the home note: 0, 1.96, 3.91, 5.87, 7.82, 9.78, 11.73, 13.69, 15.64, 17.6, 19.55, 21.51.

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