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So my understanding is 12 notes was chosen in a chromatic scale cause in tuning, the twelfth root of two gives values quite close to simple ratios. For example for a fifth, the twelfth root of 2 raised to the 7th power gives a value really close to 3/2. However there are more accurate numbers right? But we don't use them cause that would've resulted in too many keys on the piano and the successive frequencies not being much discernible. Can you give a few examples of such numbers greater than 12 which also would give values close to the simple consonant ratios?

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    There are quite a few other well-known equally-tempered tunings that do so. 19 EDO has an almost perfect minor third, 31 EDO has an almost perfect major third and harmonic seventh, 53 EDO has a perfect fifth that is much closer to JI than in 12 EDO, et cetera.
    – El Ectric
    May 30 at 18:34
  • If you would like to see some of the mathematics behind this ("advanced" but not all that difficult), please allow me to recommend my article Arithmetic and music in twelve easy steps.
    – David
    May 31 at 0:31

4 Answers 4

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Before there were 12 notes one had 8 different notes (modulo octaves), formed by three different hexachords. Since ancient times musicology had the concept of tones and semitones, and these hexachords form notes with steps in tones and semitones. 12 is more or less what you get when you try to fill up the gaps so that all steps are semitones.

So this number 12 actually comes from the concept of a semitone. This is quite unrelated to the whole 12th root of 2 thing, which only came up because intervals defined by rational ratios just didn’t fit together as we wanted to use them.

Now, let’s get to your main question: What other equal divisions of the octave can approximate certain pure intervals similarly well or even better than the 12th root of 2?

Well, first of all that depends on your metric. Let’s say we're given a ratio p/q and a base step f. Then we want log_f(p/q)\ to be close to an integer. In other words if {.} := . - [.] is the distance to the next integer, we want

|{log_f(p/q)}|

to be small. This error is then measured in parts of step f. To make different f comparable we can express this in whole octaves by dividing this difference by log_f(2).

Now, if we fix a set of ratios we want to approximate well (such as 3/2, 4/3, 5/4, 6/5 (just fifth, fourth, maj third, min third) we’d thus get an error vector. This vector could be combined to a single metric in different ways, such as:

  • (weighted) sum of the components (1-Norm)
  • p-th root of the sum of the (weighted) p-th powers (p-Norm)
  • maximum of the components (sup-Norm)

Then we can simply try some options and select the ones that work well. (No need to apply complicated lattice approximations here).

Here for example we are looking at division of the octave into 1 to 1000 parts, with only those cases chosen that perform at least as well as the previous ones:

For 1-Norm (mean):

     n          3/2          4/3          5/4          6/5           E1
1    1 4.150375e-01 4.150375e-01 3.219281e-01 2.630344e-01 3.537594e-01
2    2 8.496250e-02 8.496250e-02 1.780719e-01 2.369656e-01 1.462406e-01
3    3 8.170417e-02 8.170417e-02 1.140524e-02 7.029893e-02 6.127812e-02
4    5 1.503750e-02 1.503750e-02 7.807191e-02 6.303441e-02 4.279533e-02
5    7 1.353393e-02 1.353393e-02 3.621381e-02 2.267988e-02 2.149039e-02
6   12 1.629167e-03 1.629167e-03 1.140524e-02 1.303441e-02 6.924495e-03
7   19 6.015132e-03 6.015132e-03 6.138621e-03 1.234889e-04 4.573094e-03
8   31 4.317339e-03 4.317339e-03 6.525503e-04 4.969890e-03 3.564280e-03
9   34 3.272793e-03 3.272793e-03 1.601317e-03 1.671477e-03 2.454595e-03
10  53 5.684034e-05 5.684034e-05 1.173378e-03 1.116538e-03 6.008990e-04
11 106 5.684034e-05 5.684034e-05 1.173378e-03 1.116538e-03 6.008990e-04
12 118 2.167380e-04 2.167380e-04 1.058034e-04 3.225414e-04 2.154552e-04
13 171 1.671791e-04 1.671791e-04 2.906680e-04 1.234889e-04 1.871288e-04
14 289 1.874142e-04 1.874142e-04 1.287869e-04 5.862729e-05 1.405607e-04
15 323 1.768182e-04 1.768182e-04 5.332926e-05 1.234889e-04 1.326136e-04
16 441 7.151288e-05 7.151288e-05 6.736997e-05 4.142919e-06 5.363466e-05
17 559 1.066565e-05 1.066565e-05 7.548293e-05 6.481728e-05 4.040788e-05
18 612 4.819540e-06 4.819540e-06 3.267005e-05 3.748959e-05 1.994968e-05

For the 2-Norm:

     n          3/2          4/3          5/4          6/5           E2
1    1 4.150375e-01 4.150375e-01 3.219281e-01 2.630344e-01 3.596307e-01
2    2 8.496250e-02 8.496250e-02 1.780719e-01 2.369656e-01 1.599215e-01
3    3 8.170417e-02 8.170417e-02 1.140524e-02 7.029893e-02 6.786597e-02
4    5 1.503750e-02 1.503750e-02 7.807191e-02 6.303441e-02 5.128550e-02
5    7 1.353393e-02 1.353393e-02 3.621381e-02 2.267988e-02 2.341021e-02
6   12 1.629167e-03 1.629167e-03 1.140524e-02 1.303441e-02 8.736183e-03
7   19 6.015132e-03 6.015132e-03 6.138621e-03 1.234889e-04 5.245511e-03
8   31 4.317339e-03 4.317339e-03 6.525503e-04 4.969890e-03 3.949825e-03
9   34 3.272793e-03 3.272793e-03 1.601317e-03 1.671477e-03 2.587489e-03
10  53 5.684034e-05 5.684034e-05 1.173378e-03 1.116538e-03 8.108535e-04
11 106 5.684034e-05 5.684034e-05 1.173378e-03 1.116538e-03 8.108535e-04
12 118 2.167380e-04 2.167380e-04 1.058034e-04 3.225414e-04 2.286799e-04
13 171 1.671791e-04 1.671791e-04 2.906680e-04 1.234889e-04 1.972531e-04
14 289 1.874142e-04 1.874142e-04 1.287869e-04 5.862729e-05 1.502260e-04
15 323 1.768182e-04 1.768182e-04 5.332926e-05 1.234889e-04 1.419708e-04
16 441 7.151288e-05 7.151288e-05 6.736997e-05 4.142919e-06 6.079486e-05
17 559 1.066565e-05 1.066565e-05 7.548293e-05 6.481728e-05 5.031517e-05
18 612 4.819540e-06 4.819540e-06 3.267005e-05 3.748959e-05 2.509610e-05
19 730 3.099387e-05 3.099387e-05 1.028667e-05 2.070720e-05 2.477824e-05

For the maximum norm:

     n          3/2          4/3          5/4          6/5         Emax
1    1 4.150375e-01 4.150375e-01 3.219281e-01 2.630344e-01 4.150375e-01
2    2 8.496250e-02 8.496250e-02 1.780719e-01 2.369656e-01 2.369656e-01
3    3 8.170417e-02 8.170417e-02 1.140524e-02 7.029893e-02 8.170417e-02
4    5 1.503750e-02 1.503750e-02 7.807191e-02 6.303441e-02 7.807191e-02
5    7 1.353393e-02 1.353393e-02 3.621381e-02 2.267988e-02 3.621381e-02
6   12 1.629167e-03 1.629167e-03 1.140524e-02 1.303441e-02 1.303441e-02
7   19 6.015132e-03 6.015132e-03 6.138621e-03 1.234889e-04 6.138621e-03
8   31 4.317339e-03 4.317339e-03 6.525503e-04 4.969890e-03 4.969890e-03
9   34 3.272793e-03 3.272793e-03 1.601317e-03 1.671477e-03 3.272793e-03
10  53 5.684034e-05 5.684034e-05 1.173378e-03 1.116538e-03 1.173378e-03
11 106 5.684034e-05 5.684034e-05 1.173378e-03 1.116538e-03 1.173378e-03
12 118 2.167380e-04 2.167380e-04 1.058034e-04 3.225414e-04 3.225414e-04
13 171 1.671791e-04 1.671791e-04 2.906680e-04 1.234889e-04 2.906680e-04
14 289 1.874142e-04 1.874142e-04 1.287869e-04 5.862729e-05 1.874142e-04
15 323 1.768182e-04 1.768182e-04 5.332926e-05 1.234889e-04 1.768182e-04
16 441 7.151288e-05 7.151288e-05 6.736997e-05 4.142919e-06 7.151288e-05
17 612 4.819540e-06 4.819540e-06 3.267005e-05 3.748959e-05 3.748959e-05
18 730 3.099387e-05 3.099387e-05 1.028667e-05 2.070720e-05 3.099387e-05

So we do not really see a difference in what metric we are using, apart from the 2-Norm and the maximum gaining n = 730, which is due to 612 being quite uneven (the 2-Norm puts more weight on larger values; the maximum norm only depends on the largest value. The 1-Norm on the other hand is most sensitive to small values).

We see that using n = 12 drastically improves the approximation as compared to the previous values, while later values only offer slight improvements. The next option would be n = 19, which does exist: https://en.wikipedia.org/wiki/19_equal_temperament

Then we get a big hole, and another slight improvement at n = 31, which does also exist: https://en.wikipedia.org/wiki/31_equal_temperament

You might also want to check out https://en.wikipedia.org/wiki/Equal_temperament

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  • 1
    53 is nice too.
    – ttw
    May 29 at 21:50
  • 2
    19 and 34 also have some nice ratios (in addition to 31 and 53)
    – user2395649
    May 30 at 5:34
  • Thanks that helped :). OK so basically I was creating a science project video about consonance dissonance and tuning systems. So is it wrong to like first explain pythagorean tuning then mention the flaws and then move onto equal temperament and show the maximum simple ratios are covered for 12, 19 etc. But 19 and 31 often lead to too many notes so 12 is used as the standard? Like to give a logical explanation is this a wrong approach?
    – Betelgeuse2051
    May 30 at 5:55
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    @Betelgeuse2051 I’d start with pythagorean tuning, gregorian chant (organa), the benefit of just fifths and and the disadvantage of unjust major thirds. Then go on to just tuning and its practical shortcomings. Then you can explain meantone tuning (which incidently naturally prefers the 19 tone ET, as 12 tone ET gives a rather bad approximation of the just major 3rd) and tempered tunings, leading to modern equal temperament. Then you can explore alternative divisions of the octave and their advantages and disadvantages.
    – Lazy
    May 30 at 6:33
  • 1
    Those tables would be a lot more readable if the errors where given in cents, rounded to whole.
    – leftaroundabout
    May 30 at 9:24
3

The 12-pitch chromatic scale preceded equal temperament, so the 12th root of 2 was chosen because there were already 12 pitches, not the other way around. Loosely speaking, as you increase the number of pitches, the ratios will grow increasingly closer to their ideal values.

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  • 8
    "as you increase the number of pitches, the ratios will grow increasingly closer to their ideal values": this is not at all true, as a few minutes with a spreadsheet will show.
    – phoog
    May 29 at 22:57
  • 3
    @phoog it seems mathematically obvious that as the divisions of the octave grow larger, the availability of pitches near specific ratios will increase. In the limit, there will be perfect correspondences.
    – Aaron
    May 29 at 23:09
  • 3
    @phoog It is a thing of lattice approximation. If the distance between two notes is 1/k octaves then any note can be approximated with at least a precision of 1/(2k). So as long as you keep increasing resolution the precision will get arbitrarily fine, which is what Aaron pointed at. What he is not claiming is that this is monotonic, so that having more notes willl always mean more precision.
    – Lazy
    May 30 at 6:27
  • 4
    @Lazy Or, rather, having more notes will eventually mean more precision; however, the precision will be different amongst different intervals.
    – Aaron
    May 30 at 6:30
  • 8
    "it seems mathematically obvious that as the divisions of the octave grow larger": yet 14-tone ET isn't a better approximation than 12-tone ET. @Lazy the "loosely speaking" isn't sufficient to clarify that, as your answer shows, there are only two finer equal divisions of the octave that offer closer approximations while being usable (and one of those only questionably so); the third has too many to be practical. In other words, increasing n 31 times, from 12 to 53, gives you 3 better systems and 28 that are not better. The general statement is meaningless in the realm of the practical.
    – phoog
    May 30 at 10:39
3

You didn't say specifically which simple ratios you want to approximate, so I'll focus on the three that are part of the 4:5:6 just major triad:

  • perfect fifth (P5) = 3/2
  • major third (M3) = 5/4
  • minor third (m3) = 6/5

Let n be the number of equal divisions of the octave (EDO). Let x = round(log2(3/2) * x) be the number of EDO steps assigned to a P5. Then, based on the Circle of Fifths, the M3 is 4x-2n EDO steps, while m3 is 2n-3x EDO steps. For example, with the familiar n=12 tuning, we get P5=7, M3=4, and m3=3 EDO steps.

I shall assume that n ≥ 7 (so that all the notes of the major scale can be distinct) and n ≤ 100 (for a reasonable but arbitrary upper bound).

To optimize the P5 interval, we wish to find n so that the fraction x/n is as close to the theoretical log2(3/2) ≈ 0.584963 as possible. It turns out that the best values of n are:

  1. 53 (error: 0.068 cent)
  2. 94 (0.173)
  3. 65 (0.417)
  4. 41 or 82 (0.484)
  5. 77 (0.656)
  6. 89 (0.831)
  7. 70 (0.902)
  8. 99 (1.075)
  9. 29, 58, or 87 (1.493)
  10. 12, 24, 36, 48, 60, 72, 84, or 96 (1.955)

To optimize M3, we want to find (4x-2n)/n as close to log2(5/4) ≈ 0.321928 as possible. The best values of n are:

  1. 31, 62, or 93 (0.783)
  2. 81 (1.129)
  3. 50 or 100 (2.314)
  4. 74 (2.875)
  5. 69 (3.705)
  6. 43 or 86 (4.384)
  7. 88 (4.496)
  8. 98 (5.523)
  9. 55 (6.414)
  10. 19, 38, 57, or 76 (7.366)

And to optimize the m3 interval, we want to find (2n-3x)/n as close to log2(6/5) ≈ 0.263034 as possible. The best values of n are:

  1. 19, 38, 57, or 76 (0.148)
  2. 88 (2.005)
  3. 69 (2.598)
  4. 64 (3.109)
  5. 50 or 100 (3.641)
  6. 45 (4.359)
  7. 81 (4.530)
  8. 31, 62, or 93 (5.964)
  9. 26 or 52 (7.436)
  10. 74 (7.533)

You can't simultaneously optimize all three intervals, so we'll have to compromise. How can we combine these results to get an "overall" optimum? One way is to simply add the errors for the three intervals, giving:

  1. 81 (11.317)
  2. 50 or 100 (11.910)
  3. 31, 62, or 93 (11.928)
  4. 69 (12.606)
  5. 88 (13.001)

If we use the sums of squared errors (in cents), then we get the slightly-different ranking:

  1. 81 (53.817)
  2. 50 or 100 (54.074)
  3. 69 (60.201)
  4. 31, 62, or 93 (63.022)
  5. 88 (66.485)

So, in conclusion:

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1

The primary problem in music tuning (temperament) is finding ratios of intervals that allow for combining 2 whole steps into a major third. The "ideal" whole step is (according to Just Tuning) 9/8 so a major third becomes (9/8)*(9/8) or 81/64 but the Just Tuning ratio for a major is 5/4 (80/64). Similar problems arise for other intervals (whether used melodically or harmonically).

Mathematically, one ends up trying to find numbers (A/B)^N=2. Taking A=18, B=17, and N=12 was suggested by Vincenzo Galilei for fretted or keyed instruments. No other B smaller than 18 is as good (with the same N). There are no exact solutions (proved about 2002 but guessed earlier).

Here are some papers on the subject.

https://www.whitman.edu/Documents/Academics/Mathematics/bartha.pdf https://www.parabola.unsw.edu.au/files/articles/2020-2029/volume-57-2021/issue-2/vol57_no2_1.pdf https://www.researchgate.net/profile/Jack-Douthett/publication/12488526_Construction_and_interpretation_of_equal-tempered_scales_using_frequency_ratios_maximally_even_sets_and_P-cycles/links/54203b350cf241a65a1b7740/Construction-and-interpretation-of-equal-tempered-scales-using-frequency-ratios-maximally-even-sets-and-P-cycles.pdf

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