Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625617753 - Original comment: last one with tipping point / sweet spot math** |
Wikispaces>TallKite **Imported revision 625620919 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-30 17:05:46 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>625620919</tt>.<br> | ||
: The revision comment was: <tt> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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||= (P8/4, P5) ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished, | ||= (P8/4, P5) ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished, | ||
^1 = 81/80 || | ^1 = 81/80 || | ||
||= (P8, P4/4) ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= | ||= (P8, P4/4) ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= negri || | ||
||= (P8, P5/4) ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot || | ||= (P8, P5/4) ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot || | ||
||= (P8, P11/4) ||= v<span style="vertical-align: super;">4</span>dd3 ||= C^<span style="vertical-align: super;">4</span> ``=`` Eb<span style="vertical-align: super;">3</span> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F ||= || | ||= (P8, P11/4) ||= v<span style="vertical-align: super;">4</span>dd3 ||= C^<span style="vertical-align: super;">4</span> ``=`` Eb<span style="vertical-align: super;">3</span> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F ||= || | ||
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Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic. | Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic. | ||
==Tipping points== | |||
||||~ bare enharmonic | ||||~ bare enharmonic | ||
interval ||~ 3-exponent ||~ tipping | interval ||~ 3-exponent ||~ tipping | ||
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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;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | </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 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#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. | 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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==Tipping points and sweet spots== | ==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 | 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: | 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: | ||
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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. | 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 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. | ||
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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. | 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. | 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. | ||
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<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:52 --> </h1> | <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><!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:52 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:92:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 2em;"><a href="#toc13"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div> | ||
<!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div> | |||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --></div> | ||
<!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:54 --><u><strong>Definition</strong></u></h1> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
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<td style="text-align: center;">C Db^ Ebb^^=D#vv Ev F<br /> | <td style="text-align: center;">C Db^ Ebb^^=D#vv Ev F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">negri<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.<br /> | Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.<br /> | ||
<!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc4"><a name="Applications-Tipping points"></a><!-- ws:end:WikiTextHeadingRule:60 -->Tipping points</h2> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h1&gt; --><h1 id="toc5"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:62 --><u>Further Discussion</u></h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:64 -->Extremely large multigens</h2> | ||
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So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:66:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:66 -->Singles and doubles</h2> | ||
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If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.<br /> | If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.<br /> | ||
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A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br /> | A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:68:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Finding an example temperament"></a><!-- ws:end:WikiTextHeadingRule:68 -->Finding an example temperament</h2> | ||
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To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br /> | To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br /> | ||
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There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br /> | There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:70:&lt;h2&gt; --><h2 id="toc9"><a name="Further Discussion-Ratio and cents of the accidentals"></a><!-- ws:end:WikiTextHeadingRule:70 -->Ratio and cents of the accidentals</h2> | ||
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In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.<br /> | In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:72 -->Finding a notation for a pergen</h2> | ||
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There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br /> | There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br /> | ||
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This is a lot of math, but it only needs to be done once for each pergen!<br /> | This is a lot of math, but it only needs to be done once for each pergen!<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc11"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:74 -->Alternate enharmonics</h2> | ||
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Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br /> | Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br /> | ||
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</span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:050:``//`` -->//<!-- ws:end:WikiTextRawRule:050 -->d3=\\A2 -- \M3 -- P4<br /> | </span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:050:``//`` -->//<!-- ws:end:WikiTextRawRule:050 -->d3=\\A2 -- \M3 -- P4<br /> | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:051:``//`` -->//<!-- ws:end:WikiTextRawRule:051 -->=D#\\ -- E\ -- F</span><br /> | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:051:``//`` -->//<!-- ws:end:WikiTextRawRule:051 -->=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 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#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 /> | ||
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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 /> | 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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:76:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a><!-- ws:end:WikiTextHeadingRule:76 -->Chord names and scale names</h2> | ||
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Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | ||
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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.<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:80:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Pergens and MOS scales"></a><!-- ws:end:WikiTextHeadingRule:80 -->Pergens and MOS scales</h2> | ||
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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 /> | 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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:82 -->Pergens and EDOs</h2> | ||
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Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:84 -->Supplemental materials</h2> | ||
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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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:4963:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><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><!-- ws:end:WikiTextUrlRule:4963 --><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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:86 -->Various proofs (unfinished)</h2> | ||
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Given:<br /> | Given:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:88:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Tipping points and sweet spots"></a><!-- ws:end:WikiTextHeadingRule:88 -->Tipping points and sweet spots</h2> | ||
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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 | 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 /> | ||
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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 /> | |||
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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 <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 /> | |||
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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. <br /> | |||
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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<br /> | |||
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For multiple-comma temperaments, the tipping point can be derived from the pergen's enharmonic.<br /> | |||
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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:<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 /> | ||
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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.<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 /> | ||
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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.<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 /> | ||
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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 /> | 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 /> | ||
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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.<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:90 -->Miscellaneous Notes</h2> | ||
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<u><strong>Combining pergens</strong></u><br /> | <u><strong>Combining pergens</strong></u><br /> | ||