**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.*

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 |

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 |

C | D | E | F | F# | G | A | B |
---|---|---|---|---|---|---|---|

256 | 288 | 324 | 341.33 | 364.5 | 384 | 432 | 486 |

512 | 576 | 648 | 682.67 | 729 | 768 | 864 | 972 |

1024 | 1152 | 1296 | 1365.33 | 1458 | 1536 | 1728 | 1944 |

C# | D# | E# | G# | A# | Bb |
---|---|---|---|---|---|

2167/2048 | 19683/16384 | 177147/131072 | 6561/4096 | 59049/32768 | 16/9 |

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.

C | D | E | F | G | A | B | C |
---|---|---|---|---|---|---|---|

1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2 |

Name | Ratio | Cents |
---|---|---|

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

C | D | E | F | G | A | B | C |
---|---|---|---|---|---|---|---|

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.

Note | C | C# | D | D# | ... | 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.

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