User:Overthink/Equal-step tunings, bimodular approximants, and the natave
In western music theory, we use 12-EDO, or 12 equal divisions of the octave. The advantage of using an equal step tuning is that one can modulate freely and play the same scales and chords from every key. In this article, we will analyze the math behind why some equal-step tunings work.
Dividing the natave
In an equal-step tuning, the ratio between consecutive notes is the same. Since an interval's size in cents is related exponentially to its frequency ratio, it is natural to consider equal divisions of Euler's number, e. As an interval being divided, it is often called the natave. This may not seem to make sense as e is an irrational number, but I promise that it will make sense. In n-EDe, each step has a frequency ratio of e^(1/n), and every interval in the system has a frequency ratio that is an integer power of this. The m-step interval has a frequency ratio of e^(m/n). This is an irrational number, but there is a formula to give a rational approximation of this interval, which gets more accurate the smaller m/n is. It is as follows: (2n+m)/(2n-m). If we let m/n=1/1, we get e^(1/1)≈(2×1+1)/(2×1-1)=3/1=1901.955¢, which is over 170 cents sharp of e=1731.234¢, not very accurate. However, if we let m/n=1/2, we get e^(1/2)≈(2×2+1)/(2×2-1)=5/3=884.359¢, only 18.7 cents sharp of e^(1/2)=865.617¢. Another example is e^(1/3)=577.078¢≈7/5=582.512¢, only sharp by 5.434¢!
However, our goal is to approximate rational intervals with an equal-step tuning, not the other way around. Luckily, we can use the inverse of the previously stated formula to estimate the width in nataves of a rational interval a/b. This formula is: a/b≈e^(2(a-b)/(a+b)). For example, if we plug 5/3 into this formula, we get 5/3≈e^(2(5-3)/(5+3))=e^(1/2), which matches our previous formula. Here is the approximate size in nataves derived from this formula for some just intervals:
| Interval | Cents | Exponential approximation | Cents |
|---|---|---|---|
| 1/1 | 0 | e^0 | 0 |
| 8/7 | 231.2 | e^(2/15) | 230.8 |
| 7/6 | 266.9 | e^(2/13) | 266.3 |
| 13/11 | 289.2 | e^(1/6) | 288.5 |
| 6/5 | 315.6 | e^(2/11) | 314.8 |
| 11/9 | 347.4 | e^(1/5) | 346.2 |
| 5/4 | 386.3 | e^(2/9) | 384.7 |
| 9/7 | 435.1 | e^(1/4) | 432.8 |
| 4/3 | 498.0 | e^(2/7) | 494.6 |
| 7/5 | 582.5 | e^(1/3) | 577.1 |
| 3/2 | 702.0 | e^(2/5) | 692.5 |
| 11/7 | 782.5 | e^(4/9) | 769.4 |
| 5/3 | 884.4 | e^(1/2) | 865.6 |
| 9/5 | 1017.6 | e^(4/7) | 989.3 |
| 2/1 | 1200 | e^(2/3) | 1154.2 |
One can see that narrower intervals are approximated very accurately, while wide intervals are not. The bimodular approximation of every just interval is flat, suggesting a stretch of the natave, often by nearly 70 cents from 1731.2¢ to 1800¢=2^(3/2). Overall, simpler ratios have simpler exponential approxiamtions, and vice versa. It can be noted that all-odd just ratios tend to have simpler exponential approximants, due to a cancellation of a factor of 2 in the formula. Similarly, powers of e with an even numerator have simpler bimodular approximants. We can obtain a equal temperament by taking an equal division of the natave, and accepting some of the bimodular approximations as actually representing that just interval. Note that since it is zero steps and has a size of 0 steps, the unison (1/1) is always accepted. We look at 12-EDe for an example. Accepted intervals are in bold.
| Steps | Cents | JI approximation (bimodular) | Error (cents) |
|---|---|---|---|
| 0 | 0 | 1/1 | 0 |
| 1 | 144.27 | 25/23 | -0.08 |
| 2 | 288.54 | 13/11 | -0.67 |
| 3 | 432.81 | 9/7 | -2.27 |
| 4 | 577.08 | 7/5 | -5.43 |
| 5 | 721.35 | 29/19 | -10.72 |
| 6 | 865.62 | 5/3 | -18.74 |
| 7 | 1009.89 | 31/17 | -30.19 |
| 8 | 1154.16 | 2/1 | -45.84 |
| 9 | 1298.43 | 11/5 | -66.58 |
| 10 | 1442.70 | 17/7 | -93.32 |
| 11 | 1586.96 | 35/13 | -127.65 |
| 12 | 1731.23 | 3/1 | -170.72 |
From our accepted intervals, we can calculate the number of steps of 3/1=5/3*7/5*9/7 as 6+4+3=13 steps. Note that we do not use the very inaccurate 12-step interval to represent 3/1 in this system. In the 3.5.7 subgroup, this temperament has a mapping of ⟨13 19 23], which is the equal-tempered Bohlen-Pierce scale! One must note that, in a pure-nataves tuning, the ~3/1 is 1875.5 cents, which is far too flat. We must therefore stretch the natave to about 1756 cents to get a pure tritave and an in-tune version of Bohlen-Pierce (see 13edt). We take a look at another example, 18-EDe:
| Steps | Cents | JI approximation (bimodular) | Error (cents) |
|---|---|---|---|
| 0 | 0 | 1/1 | 0 |
| 1 | 96.18 | 37/35 | -0.02 |
| 2 | 192.36 | 19/17 | -0.20 |
| 3 | 288.54 | 13/11 | -0.67 |
| 4 | 384.72 | 5/4 | -1.60 |
| 5 | 480.90 | 41/31 | -3.13 |
| 6 | 577.08 | 7/5 | -5.43 |
| 7 | 673.26 | 43/29 | -8.68 |
| 8 | 769.44 | 11/7 | -13.05 |
| 9 | 865.62 | 5/3 | -18.74 |
| 10 | 961.80 | 23/13 | -25.95 |
| 11 | 1057.98 | 47/25 | -34.90 |
| 12 | 1154.16 | 2/1 | -45.84 |
| 13 | 1250.34 | (omitted) | (omitted) |
| 14 | 1346.52 | ||
| 15 | 1442.70 | ||
| 16 | 1538.87 | ||
| 17 | 1635.05 | ||
| 18 | 1731.23 |
Using this, we get the val ⟨12 19 28 34 42 45] (12f) of 12et! In this case, for pure octaves, we need to stretch the natave by a much greater amount to 1800¢, though slightly less stretch is also viable. Note that this mapping is accurate in lower limits, but very inaccurate in higher ones. However, if we omit the mapping of 2/1, we get a 4/3.5/3.7/3.11/3.13/3 temperament with slightly smaller steps than 12et.