(c) 1998 Zeke Hoskin, SOCAN

In Heaven there is no Pythagorean Comma - Harper Tasche

Tuning Your Harp ... Playing in Modes ... Zeke Hoskin main page

WARNING: This article goes into great detail about tunings where the frequencies of notes are exactly or nearly in ratios of small whole numbers. If you stick to naked melodies, or at most two different notes at a time, almost any tuning can sound good. Using the tunings described here may make your music sound better to you. If you prefer something else, that's just fine.


Actually, it started well before that, but Pythagoras was the first one to write down an analysis that didn't get lost in the historical shuffle. Using a single-string instrument with a movable bridge (called a monochord, if you care), he decided that two notes sounded best together when the lengths of string were related by a small-number fraction like 1/1, 2/3, 3/2, 4/3, and maybe 5/4 and 6/5.

It turns out that when you fix the tension of a string, the frequency (that is, the number of times per second the string vibrates back and forth) depends on the length. If you double the length, the frequency is cut in half. (This only works perfectly for a perfectly flexible string, but the correction for stiffness was probably too small for Pythagoras to measure.) So what Pythagoras had found was that pairs of notes sound best together when their frequencies are related by exact small-number fractions. (That is partly because their overtones match, but I'm not going to go into overtones in this page.)

"Pitch" is the musical name for frequency. Where a physicist says the frequency doubles, a musician says the pitch goes up by an octave. Both statements mean exactly the same thing. It is helpful to have a table of names for pitch changes.

When a physicist says
the frequency goes up by a factor of
A musician says the pitch
goes up by
2 an octave
3/2 a fifth
4/3 a fourth
5/4 a major third
6/5 a minor third
5/3 a major sixth
8/5 a minor sixth
Suppose you tune a collection of pitches keeping the octaves and fifths exact. Start with a note with a frequency of 512. Now go up by a fifth to get 512 x 3/2 = 768. Go up by another fifth and get 1152. This is more than twice the frequency that we started, so divide by 2. (If you prefer: this is more than an octave higher, so go down an octave. Same thing.) This new note is 576.

Keep going till we have seven notes, add twice the original note and arrange them in order as
512, 576, 648, 729, 768, 864, 972, 1024.
Play these in a sequence. The fourth note doesn't sound quite right. Suppose we add another octave of notes, doubling each frequency of the first octave, so our collection is now
512, 576, 648, 729, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048.
Now we can try playing seven-note sequences starting in different places. The sequence starting at 768 sounds pretty good:

Frequency is 768 864 972 1024 1152 1296 1458 1536
which is 768 x 1 9/8 81/64 4/3 3/2 27/32 243/128 2
Most pairs of notes 4 steps apart are in the ratio of 3/2, so the fifths sound good. This collection of pitches is called the Pythagorean Scale, and much medieval music sounds best in this scale.


This works fine as long as the keynote of the melody is 768 (or 384, or 192). But remember that the tune sounded wrong when we started at 512. The fourth note was 729, which is not 4/3 above 512. So let's throw 682 2/3 into the scale, along with 341 1/3, 170 2/3, and so on (4/3 of 512, 256, 128, etc.) Now we can choose either 512 or 768 as keynotes. At this point it's easy to see why musicians speak of notes by letters instead of numbers. Here are the notes we have so far, with the corresponding letters: Note: C is NOT exactly 256. I chose these numbers to make the arithmetic work out to whole numbers. Actually, most people tune so A is 440.

256288324341.33 364.5384432486
512576648682.67 729768864972
1024 1152 1296 1365.33 1458 1536 1728 1944
We can play a tune in C (skipping the F# notes) or in G (skipping the F notes). But that leaves six other notes we might want to use. To play in F we need a Bb, to play in D we need a C#, to play in A we need a G#, E needs a D#, B needs an A#, and F# needs an E#. (I know there's no E# on your keyboard. Bear with me.) Adding these six notes:

C#D# E#G# A#Bb
2167/2048 19683/16384 177147/131072 6561/4096 59049/32768 16/9
Now we can play in eight scales. But what about the six notes we just added? To play in C# we need a B#, and so on. It seems that we can go on forever, adding notes so we can use existing notes as keynotes, then adding notes so we can use the new notes.

One answer is to stop at twelve notes, and arbitrarily say that we will use Bb when we want an A# and F when we want an E#. But we can see right away that Bb is not exactly equal to A#. This means that the scales of F, C, G, D, A, and E will be sound right, but the scales of Bb, B, C#, D#, F#, and G# will have wrong notes. It is impossible to maintain perfect fifths and octaves and still be able to transpose a tune to use any note you can play as the keynote.


In the Pythagorean scale, the interval from C to E is 81/64. This is a harsh-sounding interval, especially if a G note is also sounding, so medieval theorists called it a discord and ignored it. But a three-note chord with frequencies 1, 4/5, 3/2 sounds very pleasant, and post-Medieval music uses the sound heavily. There is an easy way to get pure thirds and fifths in the same scale, tuning the notes

1 9/8 5/4 4/3 3/2 5/3 15/8 2
Using this Just scale, we get a good 1 - 5/4 - 3/2 major chord starting on C, G, and F. We also get good 1 - 6/5 - 3/2 minor chords starting on A and E. But the chord starting on D sounds truly horrible, and the distance from C to D is larger than the distance for D to E, which makes melodies sound a bit odd to most people.


You may have enjoyed the example of Pythagorean tuning, using fractions like 177147/131072. Tough. For the rest of this article, I am going to express intervals in cents. Divide an octave into 1200 equal ratios (each = the 1200th root of 2). Here are some equivalents:

Octave 2/1 1200
Fifth 3/2 701.95
Fourth 4/3 498.05
Major Third 5/4 386.31
Pythagorean Third 81/64 407.82
Syntonic Comma (81/64)/(5/4) 21.51
Minor Third 6/5 315.64
Pythagorean Tone 9/8 203.91
12 Fifths (3/2)^12 8423.46
7 Octaves 128/1 = 2^7 8400
Pythagorean Comma difference 23.46

Theorists have given names to two places where the Pythagorean scale runs into trouble: the amount by which a Pythagorean third exceeds a good-sounding 5/4 is called the Syntonic Comma, and the amount by which 12 Pythagorean fifths miss coming back to the starting note (seven octaves up) is the Pythagorean Comma.


Let's try to make a scale where the tones are all the same size (hence the name "meantone") and the major thirds sound better. Note that we get to the third note of the scale via four jumps of a fifth (ending up two octaves higher, but we're ignoring octaves.) If we make the fifths a little smaller than 3/2 (701.95 cents), the third will be shortened by four times as much. Say we make the fifth smaller by 1/4 of the Syntonic Comma , so a quarter-comma fifth is 696.58 cents. Then the third will be smaller by an entire comma, which is perfect. Here is the beginning of a Quarter-Comma Meantone Scale

0 193.16 386.31 ? 696.58 889.74 ? 1200

Make F 386.31 below A, and B 386.31 above G, and we have a scale with three good major thirds and five of the note-to-note steps equal. One downside is that the fifths are very flat, 5.38 cents below their Pythagorean value. And just like the Pythagorean scale, when you add more notes to play in different keys, you never reach an end.
NoteCC# DD# ... B#
Pythagorean 0 113.68 203.91 317.59 ... 23.48 or 1223.48
Q-C Meantone 0 76.05 193.18 269.21 ... -41.06 or 1158.94

In each case, the distance from C to C# is unequal to the distance from C# to D.

One way to fix up the meantone scale is to sacrifice perfect major thirds for better fifths. Instead of taking a quarter of a comma off each fifth, take a fifth, or a sixth, or an eighth. As long as you take more than an eighth of a comma from the fifths, the error in the third is less than half a comma. Unless you add extra notes, some of the keys will sound strange. This is not necessarily a bad thing: if the various keys have different sounds, a composer can use this to control the sound of a melody or harmony.


Once you start tinkering with meantone scales, one obvious step is to decrease each fifth by 1/12 of a Pythagorean Comma. Now the fifth is exactly 700 cents, the major third is exactly 400 cents, and B# sounds exactly the same as C. We get a scale of 12 notes, each exactly 100 cents above the last. This Twelve-tone Equal-Tempered Scale is the simplest way to make a scale with decent fifths that can be transposed freely into all keys. The only real disadvantage is that the major third is 13.7 cents sharp and the minor third is 15.6 cents flat. But people soon get used to this sound, and twelve-equal is the overwhelming choice of Western music.

This is not the only equal-tempered scale. If you go up by 12 fifths, you get to B#, which in untempered Pythagorean scale is 23.48 cents above C. Suppose you keep adding notes until you get close to C again. After 7 more fifths (F##, C##, G##, D##, A##, E##, B##) we have gone up nineteen fifthss to reach B##, 137.14 cents above C. If we flatten every fifth by 1/19 of this amount, we get a fifth of 694.74 cents and a major third of 378.95 cents. This fifth is even worse than in the quarter-comma meantone scale, and the major third is 7.36 cents short of its sweetest value, much better than in twelve-equal but still not good. If you go by the notion that good fifths are strident and good thirds are soothing, nineteen-equal isn't a bad tuning for lullabies.

Keep going around and around. Five fifths after B## is A###, which is 46.96 cents above C. Divide that by 24 and we get the twelve-equal scale again. Seven fifths after A### is A####, 160.6 cents above C. We got here by taking thirty-one fifths in total, so divide by 31 to get 5.18. This gives a fifth of 696.77 and a major third of 387.1, only .79 cent away from a perfect 386.31. The value 5.18 is .22 of the syntonic comma, or about two ninths, so we could call this tuning two-ninths-comma meantone or thirty-one-tone equal tempered. Thirty-one has very good major thirds and a fifth that is a bit flat but better than quarter-comma. Thirty-one notes are a lot to cram into one octave on a keyboard, and challenging to notate.

The 12-equal scale has 5 long intervals of 2 steps and 2 short intervals of 1 step. 
The 19-equal  "  "     5   "     "      "  3 steps  "  2  "       "      "  2 steps
The 31-equal  "  "     5   "     "      "  5 steps  "  2  "       "      "  3 steps

We could follow the fifths around twelve more steps to get 43-equal, with intervals of 7 and 4 steps. This gets a fifth that's one cent better than thirty-one-equal - still 4.3 cents flat -- and a third that's four cents worse.

There is an interesting tuning that is not directly on this path. Fifty-three steps would lead to a fifth of 701.89 cents - close enough to perfect - and a major third of 407.55 cents - almost as bad as Pythagorean. Harry Partch noticed that the size of a step in 53-equal is 22.64 cents (making this essentially a "scale of commas"). Make the major third 17 steps instead of 18, and it works out to 384.91 cents, only 1.4 cents from perfect. The scale goes up by 9, 8, 5, 9, 9, 8, 5 steps, so a tune in the Partch scale would have intervals of nine and eight steps, just as the Just scale has intervals of 9/8 and 10/9. So this tuning is wonderful for harmonization, but imperfect for melodies. It is not a mean-tone scale.

The ratios between short and long steps in a scale don't have to be rational. Kornerrup decided that they should be in the Golden Ratio of (1 + square root of 5)/2, about 1.618, giving a long step of 192.4 cents, a short step of 118.9 cents, and a fifth of 696.2 cents. Charles Lucy uses a long step of 1 octave/2 pi, or 191.0 cents, giving a short step of 122.5 cents and a fifth of 695.5 cents. Both Lucytuning and Kornerrup Phi are close to quarter-comma meantone, but come from different theoretical bases.

Diatonic instruments like harps can be tuned to whatever scale sounds best for the piece in question, without worrying about transposition. Under these conditions, the best compromise is a meantone scale that gets the thirds and the fifths where you prefer them, somewhere between quarter-comma and eighth-comma. For instruments that have to transpose, we have to compromise between sweetness of sound and playability. Twelve-equal is the usual answer, but nineteen is feasible and thirty-one is possible on a keyboard.

Be aware that much ethnic music does NOT play well in twelve-equal. Use Pythagorean for medieval music, meantone for Baroque music, and listen really hard to what the singer does in Turkish and old Celtic music. I just don't have the knowledge to analyze the scales they use.

And notice that all of this analysis assumes that an octave is exactly 1200 cents. If you stretch the octave a bit, you can have longer fifths and/or shorter major thirds. Stretching the octave two cents lets you stretch the too-short fifths of quarter-comma meantone by one cent, keeping the good-sounding thirds. In thirty-one-tone equal, the good thirds go up by 20/31 of a cent and the fifths by 36/31 of a cent. Pianos are actually tuned with long octaves, which would help the sound of a meantone instrument but makes the long thirds of a 12-equal tuning even worse. (The long octaves are to deal with inharmonicities caused by string stiffness, and don't have anything to do with the tuning math we are considering.) Back to main page