Kite's thoughts on pergens: Difference between revisions
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As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to "tip over". That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. | As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to "tip over". That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. | ||
Does the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which contains a tipping point, but it remains an open question whether one exists. | Does the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which contains a tipping point, but it remains an open question whether one exists. | ||
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. | 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 the 5th from a just 3/2, 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: | |||
* ∆w must range between -1/2 comma and 1/2 comma | |||
* ∆w also must range between -(c+2)/2b comma and (c-2)/2b comma | |||
* ∆w also must range between -(c+2)/(2b+2c) comma and (c-2)/(2b+2c) comma | |||
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. | |||
The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 | Another example: for porcupine, the comma is 250/243 = 49¢, b = -5 and c = 3. Second range: 1/2 comma to -1/10 comma, again within the first range. Third range: 5/4 comma to -1/4 comma. Total range: -1/10 comma to 1/2 comma. | ||
The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). Let K' be the cents of this comma, which may be descending, hence K' may be negative. At exactly the tipping point, ∆w = -K'/(b+4c). | |||
For porcupine, the 3-limit comma is (-11, 7) = A1 = 114¢. The tipping point is ∆w = -114/7 = -16¢. Expressed as a fraction of the vanishing comma 250/243, ∆w = -(114¢)/(7·49¢) comma = -1/3 comma. This falls outside of the sweet spot, and porcupine won't tip over. | |||
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | ||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||
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As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to &quot;tip over&quot;. That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.<br /> | As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to &quot;tip over&quot;. That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.<br /> | ||
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Does the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which contains a tipping point, but it remains an open question whether one exists. <br /> | Does the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which contains a tipping point, but it remains an open question whether one exists.<br /> | ||
<br /> | <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 <u>very</u> 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.<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 <u>very</u> 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.<br /> | ||
<br /> | <br /> | ||
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 the 5th from a just 3/2, 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:<br /> | |||
<ul><li>∆w must range between -1/2 comma and 1/2 comma</li><li>∆w also must range between -(c+2)/2b comma and (c-2)/2b comma</li><li>∆w also must range between -(c+2)/(2b+2c) comma and (c-2)/(2b+2c) comma</li></ul><br /> | |||
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.<br /> | |||
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Another example: for porcupine, the comma is 250/243 = 49¢, b = -5 and c = 3. Second range: 1/2 comma to -1/10 comma, again within the first range. Third range: 5/4 comma to -1/4 comma. Total range: -1/10 comma to 1/2 comma.<br /> | |||
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The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). Let K' be the cents of this comma, which may be descending, hence K' may be negative. At exactly the tipping point, ∆w = -K'/(b+4c).<br /> | |||
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For porcupine, the 3-limit comma is (-11, 7) = A1 = 114¢. The tipping point is ∆w = -114/7 = -16¢. Expressed as a fraction of the vanishing comma 250/243, ∆w = -(114¢)/(7·49¢) comma = -1/3 comma. This falls outside of the sweet spot, and porcupine won't tip over.<br /> | |||
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