Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-30 17:08:45 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-30 18:09:27 UTC</tt>.<br>

: The original revision id was <tt>~~625621057~~</tt>.<br>

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</span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span>

To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Tipping%20points|tipping points]] above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).

To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping%20points|tipping points]] above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-1,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3dd2 = P5. Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, so ^ = 67¢ + 8.67·c, about a third-tone.

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Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.

== ==

==Tipping points and sweet spots==

As noted above, 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 if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).

For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated.

For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. A d2 comma implies 12-edo and 700¢. See [[xenharmonic/pergen#Applications-Tipping%20points|tipping points]] above for a more complete list.

For multiple-comma temperaments, use the pergen's enharmonic. For example, third-5th has E = v<span style="vertical-align: super;">3</span>m2, and tips at 720¢. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.

An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80.

==Pergens and MOS scales==

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Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n')

[//unfinished proof//]

~~==Tipping points and sweet spots==~~

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 if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.

~~Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point, although it can happen.~~

The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).

For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated.

For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and 720¢. A d2 implies 12-edo and a 700¢ tipping point. See

~~For multiple-comma temperaments, the tipping point can be derived from the pergen's enharmonic.~~

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 these three ranges is the sweet spot:

* ∆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) and (c-2) / (2b+2c) comma

If b = -c, ignore the third range, to avoid dividing by zero. 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. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.

Another example: for porcupine, the comma is 250/243 = 49.2¢, 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.

For porcupine, the 3-limit comma is (-11, 7) = 113.7¢, which implies 7-edo. At the tipping point, ∆w = -113.7¢ / 7 = -16.2¢, which is the distance from a just 3/2 to 7-edo's 5th. Expressed as a fraction of the vanishing comma 250/243, ∆w = -113.7¢ / (7·49.2¢) comma = -0.33 comma. This falls outside of the sweet spot, and porcupine won't tip over.

For any 2.3.7 comma (a,b,0,c), the three JI ratios of interest are 3/2, 7/4 and 7/6. The sweet spot results from the exact same three ranges of ∆w. However, the tipping point formula changes, to reflect the new mapping comma 64/63. The 3-limit comma is (a,b,0,c) + c·(6,-2,0,-1) = (a+6c, b-2c). At the tipping point, ∆w = -K' / (b-2c).

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. Even relaxing the half-comma restriction to two-thirds-comma makes the ranges be -1/3 to 1/3, and ∆w = ±17.9¢, and the tipping point is just outside the sweet spot.

In theory, a few temperaments could tip over. Diminished (2.3.5 and (8,4,-4) = 648/625) will tip at 700¢, where ∆w = -2.0¢ and ∆y = 13.7¢, nearly a quarter comma. 5/4 becomes 400¢, and 6/5 is locked at 300¢. But in practice, there's no reason for ∆y to be that high. It should be at most only 2 or 3 times greater than ∆w. Another tipping temperament is quadruple red, 2.3.7 with (5,4,0,-4). It has the same tipping point, at 700¢, where ∆w = -2¢ and 7/4's error = 31¢, again far greater than ∆w. 7/4 is 1000¢, far sharper than it need be. For both these temperaments, the half-comma restriction creates an overly broad range for ∆w.

~~However, triple red (2.3.7 and 729/686)...~~

~~Negri 2.3.5 and (-14,3,4) tips!~~

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<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#~~toc13"&gt~~; </a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-~~Tipping points and sweet spots~~">~~Tipping points and sweet spots~~</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>

<div ~~style="margin-left: 2em;"~~>~~<a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>~~

</div>

~~</div>~~

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<br />

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</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see <a class="wiki_link" href="/pergen#Tipping%20points">tipping points</a> above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).<br />

To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see <a class="wiki_link" href="/pergen#Applications-Tipping%20points">tipping points</a> above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).<br />

<br />

For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-1,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3dd2 = P5. Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, so ^ = 67¢ + 8.67·c, about a third-tone.<br />

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<br />

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.<br />

<h2 id="toc13"> </h2>

<br />

<h2 id="toc14"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<br />

<h2 id="toc13"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<br />

As noted above, 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 if a single temperament &quot;tips over&quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.<br />

<br />

The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).<br />

<br />

For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated.<br />

<br />

For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. A d2 comma implies 12-edo and 700¢. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/pergen#Applications-Tipping%20points">tipping points</a> above for a more complete list.<br />

<br />

For multiple-comma temperaments, use the pergen's enharmonic. For example, third-5th has E = v<span style="vertical-align: super;">3</span>m2, and tips at 720¢. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.<br />

<br />

An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80.<br />

<br />

<h2 id="toc14"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<br />

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br />

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<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 />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><br />

<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 />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><br />

<br />

Screenshot of the first 38 pergens:<br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<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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<br />

<h2 id="toc18~~"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>~~

<h2 id="toc18"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

~~<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 if a single temperament &quot;tips over&quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.<br />

~~<br />~~

Fortunately, the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point, although it can happen.<br />

~~<br />~~

The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).<br />

~~<br />~~

For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated. <br />

~~<br />~~

For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and 720¢. A d2 implies 12-edo and a 700¢ tipping point. See<br />

~~<br />~~

~~For multiple-comma temperaments, the tipping point can be derived from the pergen's enharmonic.<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 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 these three ranges is the sweet spot:<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) and (c-2) / (2b+2c) comma</li></ul><br />

If b = -c, ignore the third range, to avoid dividing by zero. 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. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.<br />

~~<br />~~

Another example: for porcupine, the comma is 250/243 = 49.2¢, 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 />

~~<br />~~

For porcupine, the 3-limit comma is (-11, 7) = 113.7¢, which implies 7-edo. At the tipping point, ∆w = -113.7¢ / 7 = -16.2¢, which is the distance from a just 3/2 to 7-edo's 5th. Expressed as a fraction of the vanishing comma 250/243, ∆w = -113.7¢ / (7·49.2¢) comma = -0.33 comma. This falls outside of the sweet spot, and porcupine won't tip over.<br />

~~<br />~~

For any 2.3.7 comma (a,b,0,c), the three JI ratios of interest are 3/2, 7/4 and 7/6. The sweet spot results from the exact same three ranges of ∆w. However, the tipping point formula changes, to reflect the new mapping comma 64/63. The 3-limit comma is (a,b,0,c) + c·(6,-2,0,-1) = (a+6c, b-2c). At the tipping point, ∆w = -K' / (b-2c).<br />

~~<br />~~

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. Even relaxing the half-comma restriction to two-thirds-comma makes the ranges be -1/3 to 1/3, and ∆w = ±17.9¢, and the tipping point is just outside the sweet spot.<br />

~~<br />~~

In theory, a few temperaments could tip over. Diminished (2.3.5 and (8,4,-4) = 648/625) will tip at 700¢, where ∆w = -2.0¢ and ∆y = 13.7¢, nearly a quarter comma. 5/4 becomes 400¢, and 6/5 is locked at 300¢. But in practice, there's no reason for ∆y to be that high. It should be at most only 2 or 3 times greater than ∆w. Another tipping temperament is quadruple red, 2.3.7 with (5,4,0,-4). It has the same tipping point, at 700¢, where ∆w = -2¢ and 7/4's error = 31¢, again far greater than ∆w. 7/4 is 1000¢, far sharper than it need be. For both these temperaments, the half-comma restriction creates an overly broad range for ∆w.<br />

~~<br />~~

~~However, triple red (2.3.7 and 729/686)...<br />~~

~~<br />~~

~~Negri 2.3.5 and (-14,3,4) tips!<br />~~

~~<br />~~

~~<h2 id="toc19~~"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<br />

<u><strong>Combining pergens</strong></u><br />

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 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-30 17:08:45 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-30 18:09:27 UTC</tt>.<br>
-: The original revision id was <tt>625621057</tt>.<br>
+: The original revision id was <tt>625624145</tt>.<br>
 : The revision comment was: <tt></tt><br>
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@@ Line 440: / Line 440: @@
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F&lt;/span&gt;
-To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Tipping%20points|tipping points]] above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).
+To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping%20points|tipping points]] above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).
 For example, (P8, P5/3) has n = 3, G = ^M2, and E = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Assuming a reasonably just 5th, E needs to be upped, so E = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Add the multi-E ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-1,1] to the multigen P5 = [7,4] to get ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-2,1] = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3dd2 = P5. Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 = -200¢ - 26·c, so ^ = 67¢ + 8.67·c, about a third-tone.
@@ Line 466: / Line 466: @@
 Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.
-== ==
+==Tipping points and sweet spots==
+As noted above, 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 if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.
+The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).
+For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated.
+For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. A d2 comma implies 12-edo and 700¢. See [[xenharmonic/pergen#Applications-Tipping%20points|tipping points]] above for a more complete list.
+For multiple-comma temperaments, use the pergen's enharmonic. For example, third-5th has E = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2, and tips at 720¢. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.
+An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80.
 ==Pergens and MOS scales==
@@ Line 701: / Line 717: @@
 Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n')
 [//unfinished proof//]
-==Tipping points and sweet spots==
-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 if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.
-Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point, although it can happen.
-The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).
-For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated.
-For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and 720¢. A d2 implies 12-edo and a 700¢ tipping point. See
-For multiple-comma temperaments, the tipping point can be derived from the pergen's enharmonic.
-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 these three ranges is the sweet spot:
-* ∆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) and (c-2) / (2b+2c) comma
-If b = -c, ignore the third range, to avoid dividing by zero. 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. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.
-Another example: for porcupine, the comma is 250/243 = 49.2¢, 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.
-For porcupine, the 3-limit comma is (-11, 7) = 113.7¢, which implies 7-edo. At the tipping point, ∆w = -113.7¢ / 7 = -16.2¢, which is the distance from a just 3/2 to 7-edo's 5th. Expressed as a fraction of the vanishing comma 250/243, ∆w = -113.7¢ / (7·49.2¢) comma = -0.33 comma. This falls outside of the sweet spot, and porcupine won't tip over.
-For any 2.3.7 comma (a,b,0,c), the three JI ratios of interest are 3/2, 7/4 and 7/6. The sweet spot results from the exact same three ranges of ∆w. However, the tipping point formula changes, to reflect the new mapping comma 64/63. The 3-limit comma is (a,b,0,c) + c·(6,-2,0,-1) = (a+6c, b-2c). At the tipping point, ∆w = -K' / (b-2c).
--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. Even relaxing the half-comma restriction to two-thirds-comma makes the ranges be -1/3 to 1/3, and ∆w = ±17.9¢, and the tipping point is just outside the sweet spot.
-In theory, a few temperaments could tip over. Diminished (2.3.5 and (8,4,-4) = 648/625) will tip at 700¢, where ∆w = -2.0¢ and ∆y = 13.7¢, nearly a quarter comma. 5/4 becomes 400¢, and 6/5 is locked at 300¢. But in practice, there's no reason for ∆y to be that high. It should be at most only 2 or 3 times greater than ∆w. Another tipping temperament is quadruple red, 2.3.7 with (5,4,0,-4). It has the same tipping point, at 700¢, where ∆w = -2¢ and 7/4's error = 31¢, again far greater than ∆w. 7/4 is 1000¢, far sharper than it need be. For both these temperaments, the half-comma restriction creates an overly broad range for ∆w.
-However, triple red (2.3.7 and 729/686)...
-Negri 2.3.5 and (-14,3,4) tips!
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 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt; &lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextTocRule:92:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#toc0"&gt; &lt;/a&gt;&lt;/div&gt;
+  &lt;!-- ws:start:WikiTextTocRule:90:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#toc0"&gt; &lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Extremely large multigens"&gt;Extremely large multigens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Extremely large multigens"&gt;Extremely large multigens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#toc13"&gt; &lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials"&gt;Supplemental materials&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials"&gt;Supplemental materials&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
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@@ Line 2,488: / Line 2,463: @@
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:050:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:050 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:051:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:051 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
-To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see &lt;a class="wiki_link" href="/pergen#Tipping%20points"&gt;tipping points&lt;/a&gt; above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).&lt;br /&gt;
+To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see &lt;a class="wiki_link" href="/pergen#Applications-Tipping%20points"&gt;tipping points&lt;/a&gt; above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).&lt;br /&gt;
 &lt;br /&gt;
 For example, (P8, P5/3) has n = 3, G = ^M2, and E = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Assuming a reasonably just 5th, E needs to be upped, so E = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Add the multi-E ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-1,1] to the multigen P5 = [7,4] to get ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-2,1] = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3dd2 = P5. Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 = -200¢ - 26·c, so ^ = 67¢ + 8.67·c, about a third-tone.&lt;br /&gt;
@@ Line 2,514: / Line 2,489: @@
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 Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.&lt;br /&gt;
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+&lt;br /&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Pergens and MOS scales&lt;/h2&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;Tipping points and sweet spots&lt;/h2&gt;
+  &lt;br /&gt;
+As noted above, 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 if a single temperament &amp;quot;tips over&amp;quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &amp;quot;sweet spot&amp;quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.&lt;br /&gt;
+&lt;br /&gt;
+The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a &lt;u&gt;very&lt;/u&gt; unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).&lt;br /&gt;
+&lt;br /&gt;
+For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated.&lt;br /&gt;
+&lt;br /&gt;
+For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. A d2 comma implies 12-edo and 700¢. See &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/pergen#Applications-Tipping%20points"&gt;tipping points&lt;/a&gt; above for a more complete list.&lt;br /&gt;
+&lt;br /&gt;
+For multiple-comma temperaments, use the pergen's enharmonic. For example, third-5th has E = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2, and tips at 720¢. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.&lt;br /&gt;
+&lt;br /&gt;
+An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Pergens and MOS scales&lt;/h2&gt;
   &lt;br /&gt;
 Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;
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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.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextUrlRule:4933:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:4933 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;
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 Screenshot of the first 38 pergens:&lt;br /&gt;
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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:&lt;br /&gt;
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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 if a single temperament &amp;quot;tips over&amp;quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.&lt;br /&gt;
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-Fortunately, the temperament's &amp;quot;sweet spot&amp;quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point, although it can happen.&lt;br /&gt;
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-The tipping point depends on the choice of enharmonic. Half-8ve can be notated with an E of vvM2. The tipping point becomes 600¢, a &lt;u&gt;very&lt;/u&gt; unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals 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. 11's mapping comma can be either 33/32 (11/8 = P4) or 729/704 (11/8 = A4). 13's mapping comma can be either 27/26 (13/8 = M6) or 1053/1024 (13/8 = m6).&lt;br /&gt;
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-For single-comma temperaments, 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. The tipping point is that edo's 5th. Since the mapping comma is by definition a P1, the tipping point can be inferred from the way the comma is notated. &lt;br /&gt;
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-For example, porcupine's 250/243 comma is an A1 = (-11,7), which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and 720¢. A d2 implies 12-edo and a 700¢ tipping point. See&lt;br /&gt;
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-For multiple-comma temperaments, the tipping point can be derived from the pergen's enharmonic.&lt;br /&gt;
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-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 these three ranges is the sweet spot:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;∆w must range between -1/2 comma and 1/2 comma&lt;/li&gt;&lt;li&gt;∆w also must range between -(c+2) / 2b comma and (c-2) / 2b comma&lt;/li&gt;&lt;li&gt;∆w also must range between -(c+2) / (2b+2c) and (c-2) / (2b+2c) comma&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-If b = -c, ignore the third range, to avoid dividing by zero. 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. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.&lt;br /&gt;
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-Another example: for porcupine, the comma is 250/243 = 49.2¢, 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.&lt;br /&gt;
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-For porcupine, the 3-limit comma is (-11, 7) = 113.7¢, which implies 7-edo. At the tipping point, ∆w = -113.7¢ / 7 = -16.2¢, which is the distance from a just 3/2 to 7-edo's 5th. Expressed as a fraction of the vanishing comma 250/243, ∆w = -113.7¢ / (7·49.2¢) comma = -0.33 comma. This falls outside of the sweet spot, and porcupine won't tip over.&lt;br /&gt;
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-For any 2.3.7 comma (a,b,0,c), the three JI ratios of interest are 3/2, 7/4 and 7/6. The sweet spot results from the exact same three ranges of ∆w. However, the tipping point formula changes, to reflect the new mapping comma 64/63. The 3-limit comma is (a,b,0,c) + c·(6,-2,0,-1) = (a+6c, b-2c). At the tipping point, ∆w = -K' / (b-2c).&lt;br /&gt;
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--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. Even relaxing the half-comma restriction to two-thirds-comma makes the ranges be -1/3 to 1/3, and ∆w = ±17.9¢, and the tipping point is just outside the sweet spot.&lt;br /&gt;
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-In theory, a few temperaments could tip over. Diminished (2.3.5 and (8,4,-4) = 648/625) will tip at 700¢, where ∆w = -2.0¢ and ∆y = 13.7¢, nearly a quarter comma. 5/4 becomes 400¢, and 6/5 is locked at 300¢. But in practice, there's no reason for ∆y to be that high. It should be at most only 2 or 3 times greater than ∆w. Another tipping temperament is quadruple red, 2.3.7 with (5,4,0,-4). It has the same tipping point, at 700¢, where ∆w = -2¢ and 7/4's error = 31¢, again far greater than ∆w. 7/4 is 1000¢, far sharper than it need be. For both these temperaments, the half-comma restriction creates an overly broad range for ∆w.&lt;br /&gt;
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-However, triple red (2.3.7 and 729/686)...&lt;br /&gt;
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-Negri 2.3.5 and (-14,3,4) tips!&lt;br /&gt;
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 &lt;u&gt;&lt;strong&gt;Combining pergens&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;