Kite's thoughts on pergens: Difference between revisions
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== | ==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 it isn't acceptable for a single temperament to "tip over". That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. | As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to "tip over". That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. | ||
Does the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which | Does the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which does, but it remains an open question whether one exists. | ||
The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. At least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on. | The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. At least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on. | ||
For any 5-limit temperament, the boundaries of the sweet spot can be reasonably defined such that going beyond them makes the error of 3/2, 5/4 or 6/5 more than half a comma. Assuming untempered octaves, let ∆w be the error of | For any 5-limit temperament, the boundaries of the sweet spot can be reasonably defined such that going beyond them makes the error of 3/2, 5/4 or 6/5 more than half a comma. Assuming untempered octaves, let ∆w be the error of 3/2 from just, and ∆y be the error of 5/4 from just, with positive values meaning sharp. The error for 6/5 is ∆w - ∆y. For a comma (a,b,c) of cents K, we have b·∆w + c·∆y + K = 0¢. One pair of limits, or range, is simply that ∆w must be between -K/2 and +K/2. Substitute ±K/2 for ∆y and solve for ∆w to get another range for ∆w. The error of 6/5 = ∆w - ∆y = ±K/2, thus ∆y = ∆w ± K/2. Substitute this for ∆y to get a third range. The intersection of the three ranges is the sweet spot. This gives us: | ||
* ∆w must range between -1/2 comma and 1/2 comma | * ∆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 comma and (c-2) / 2b comma | ||
* ∆w also must range between -(c+2)/(2b+2c) | * ∆w also must range between -(c+2) / (2b+2c) and (c-2) / (2b+2c) comma | ||
We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma. | We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma. | ||
Another example: for porcupine, the comma is 250/243 = | 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. | ||
The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). Let K' be the cents of this comma, which may be descending, hence K' may be negative. At exactly the tipping point, ∆w = -K' / (b+4c). | |||
For porcupine, the 3-limit comma is (-11, 7) = 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 ratios 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), where K' is the 3-limit comma's cents. | |||
7-limit example: semaphore, 2.3.7 with 49/48 = 35.7¢, a = -4, b = -1, and c = 2. The sweet spot ranges are -1/2 to 1/2, 2 to 0, and -2 to 0. The sweet spot's range is very narrow, only one tuning: zero-comma, with a just fifth, and ∆w = 0¢. The 3-limit comma is (8,-5), and the tipping point is 5-edo's 5th = 720¢. At the tipping point, ∆w = 90.2¢ / 5 = 18.0¢. Semaphore won't tip over. | |||
It may seem odd to have the sweet range so narrow, and it's possible that the sweet spot may not exist for certain commas. Relaxing the half-comma restriction to two-thirds-comma | |||
The tipping point can be | The tipping point can be inferred directly from the way the comma is notated. Semaphore's 49/48 comma is a minor 2nd, and any m2 comma implies 5-edo. Since 720¢ is the sharpest possible 5th that heptatonic notation can support, no m2 comma will tip over. Porcupine's 250/243 comma is an A1, which always implies 7-edo, and a 685.7¢ tipping point. Any d2 comma implies 12-edo, with a 700¢ tipping point. | ||
==Miscellaneous Notes== | |||
__**Combining pergens**__ | __**Combining pergens**__ | ||
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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"><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:90:&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:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><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:98 --><!-- ws:start:WikiTextTocRule:99: --><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:99 --><!-- ws:start:WikiTextTocRule:100: --><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:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><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:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#toc12"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><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:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><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:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div> | ||
< | <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --></div> | ||
<!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:54 --><u><strong>Definition</strong></u></h1> | |||
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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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:4948:http://www.tallkite.com/misc_files/pergens.pdf --><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><!-- ws:end:WikiTextUrlRule:4948 --><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:4949: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:4949 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:2879:&lt;img src=&quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 460px; width: 704px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextLocalImageRule:2879 --><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:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion- | <!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Tipping points and sweet spots"></a><!-- ws:end:WikiTextHeadingRule:86 -->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 it isn't acceptable for a single temperament to &quot;tip over&quot;. That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.<br /> | As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to &quot;tip over&quot;. That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.<br /> | ||
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Does the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which | Does the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which does, but it remains an open question whether one exists.<br /> | ||
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The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. At least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.<br /> | The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. At least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.<br /> | ||
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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 | For any 5-limit temperament, the boundaries of the sweet spot can be reasonably defined such that going beyond them makes the error of 3/2, 5/4 or 6/5 more than half a comma. Assuming untempered octaves, let ∆w be the error of 3/2 from just, and ∆y be the error of 5/4 from just, with positive values meaning sharp. The error for 6/5 is ∆w - ∆y. For a comma (a,b,c) of cents K, we have b·∆w + c·∆y + K = 0¢. One pair of limits, or range, is simply that ∆w must be between -K/2 and +K/2. Substitute ±K/2 for ∆y and solve for ∆w to get another range for ∆w. The error of 6/5 = ∆w - ∆y = ±K/2, thus ∆y = ∆w ± K/2. Substitute this for ∆y to get a third range. The intersection of the three ranges is the sweet spot. This gives us:<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) | <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 /> | ||
We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.<br /> | We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.<br /> | ||
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Another example: for porcupine, the comma is 250/243 = | 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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The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). Let K' be the cents of this comma, which may be descending, hence K' may be negative. At exactly the tipping point, ∆w = -K' / (b+4c).<br /> | |||
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For porcupine, the 3-limit comma is (-11, 7) = 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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For any 2.3.7 comma (a,b,0,c), the three ratios 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), where K' is the 3-limit comma's cents.<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.<br /> | |||
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It may seem odd to have the sweet range so narrow, and it's possible that the sweet spot may not exist for certain commas. Relaxing the half-comma restriction to two-thirds-comma<br /> | |||
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The tipping point can be inferred directly from the way the comma is notated. Semaphore's 49/48 comma is a minor 2nd, and any m2 comma implies 5-edo. Since 720¢ is the sharpest possible 5th that heptatonic notation can support, no m2 comma will tip over. Porcupine's 250/243 comma is an A1, which always implies 7-edo, and a 685.7¢ tipping point. Any d2 comma implies 12-edo, with a 700¢ tipping point.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:88:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:88 -->Miscellaneous Notes</h2> | |||
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<u><strong>Combining pergens</strong></u><br /> | <u><strong>Combining pergens</strong></u><br /> | ||
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