Kite's thoughts on pergens: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-29 02:09:37 UTC</tt>.<br>

: The original revision id was <tt>625507585</tt>.<br>

The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. At least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.

For any 5-limit temperament, the boundaries of the sweet spot can be reasonably defined such that going beyond them makes the error of 3/2, 5/4 or 6/5 more than half a comma. Assuming untempered octaves, let ∆w be the error of 3/2 from just, and ∆y be the error of 5/4 from just, with positive values meaning sharp. The error for 6/5 is ∆w - ∆y. For a comma (a,b,c) of cents K, we have b·∆w + c·∆y + K = 0¢. One pair of limits, or range, is simply that ∆w must be between -K/2 and +K/2. Substitute ±K/2 for ∆y and solve for ∆w to get another range for ∆w. The error of 6/5 = ∆w - ∆y = ±K/2, thus ∆y = ∆w ± K/2. Substitute this for ∆y to get a third range. The intersection of the three ranges is the sweet spot. This gives us:

We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.

7-limit example: semaphore, 2.3.7 with 49/48 = 35.7¢, a = -4, b = -1, and c = 2. The sweet spot ranges are -1/2 to 1/2, 2 to 0, and -2 to 0. The sweet spot's range is very narrow, only one tuning: zero-comma, with a just fifth, and ∆w = 0¢. The 3-limit comma is (8,-5), and the tipping point is 5-edo's 5th = 720¢. At the tipping point, ∆w = 90.2¢ / 5 = 18.0¢. Semaphore won't tip over.

It may seem odd to have the sweet range so narrow, and it's possible that the sweet spot may not exist for certain commas. Relaxing the half-comma restriction to two-thirds-comma

The tipping point can be inferred directly from the way the comma is notated. Semaphore's 49/48 comma is a minor 2nd, and any m2 comma implies 5-edo. Since 720¢ is the sharpest possible 5th that heptatonic notation can support, no m2 comma will tip over. Porcupine's 250/243 comma is an A1, which always implies 7-edo, and a 685.7¢ tipping point. Any d2 comma implies 12-edo, with a 700¢ tipping point.

The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a &lt;u&gt;very&lt;/u&gt; unlikely 5th, and tipping is impossible. At least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.&lt;br /&gt;

For any 5-limit temperament, the boundaries of the sweet spot can be reasonably defined such that going beyond them makes the error of 3/2, 5/4 or 6/5 more than half a comma. Assuming untempered octaves, let ∆w be the error of 3/2 from just, and ∆y be the error of 5/4 from just, with positive values meaning sharp. The error for 6/5 is ∆w - ∆y. For a comma (a,b,c) of cents K, we have b·∆w + c·∆y + K = 0¢. One pair of limits, or range, is simply that ∆w must be between -K/2 and +K/2. Substitute ±K/2 for ∆y and solve for ∆w to get another range for ∆w. The error of 6/5 = ∆w - ∆y = ±K/2, thus ∆y = ∆w ± K/2. Substitute this for ∆y to get a third range. The intersection of the three ranges is the sweet spot. This gives us:&lt;br /&gt;

We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.&lt;br /&gt;

7-limit example: semaphore, 2.3.7 with 49/48 = 35.7¢, a = -4, b = -1, and c = 2. The sweet spot ranges are -1/2 to 1/2, 2 to 0, and -2 to 0. The sweet spot's range is very narrow, only one tuning: zero-comma, with a just fifth, and ∆w = 0¢. The 3-limit comma is (8,-5), and the tipping point is 5-edo's 5th = 720¢. At the tipping point, ∆w = 90.2¢ / 5 = 18.0¢. Semaphore won't tip over.&lt;br /&gt;

It may seem odd to have the sweet range so narrow, and it's possible that the sweet spot may not exist for certain commas. Relaxing the half-comma restriction to two-thirds-comma&lt;br /&gt;

The tipping point can be inferred directly from the way the comma is notated. Semaphore's 49/48 comma is a minor 2nd, and any m2 comma implies 5-edo. Since 720¢ is the sharpest possible 5th that heptatonic notation can support, no m2 comma will tip over. Porcupine's 250/243 comma is an A1, which always implies 7-edo, and a 685.7¢ tipping point. Any d2 comma implies 12-edo, with a 700¢ tipping point.&lt;br /&gt;