Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-28 03:53:47 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-28 04:21:22 UTC</tt>.

: The original revision id was <tt>~~625471835~~</tt>.

: The original revision id was <tt>625472215</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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 __very__ unlikely 5th, and tipping is impossible. As noted, 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 boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by "far worse" is meant an error twice as large, meantone's sweet spot is from

The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by "far worse" is meant an error twice as large, meantone's sweet spot is from 1/5-comma to 1/2-comma. This is assuming the JI ratios of interest are only 3/2, 5/4, 6/5 and their 8ve inverses. If 9/4 and 9/5 are included as well, the sweet spot is from

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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of ~~this~~ comma. At exactly the tipping point, the error of the 5th from just is C / (b+4c).

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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of the uninverted comma. At exactly the tipping point, the error of the 5th from just is -C / (b+4c). Let C' = the cents of the vanishing comma. The error of the 5th expressed as a fraction of the vanishing comma is (C / C') / (b+4c).

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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 very unlikely 5th, and tipping is impossible. As noted, 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 boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by &quot;far worse&quot; is meant an error twice as large, meantone's sweet spot is from

The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by &quot;far worse&quot; is meant an error twice as large, meantone's sweet spot is from 1/5-comma to 1/2-comma. This is assuming the JI ratios of interest are only 3/2, 5/4, 6/5 and their 8ve inverses. If 9/4 and 9/5 are included as well, the sweet spot is from

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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of ~~this~~ comma. At exactly the tipping point, the error of the 5th from just is C / (b+4c).

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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of the uninverted comma. At exactly the tipping point, the error of the 5th from just is -C / (b+4c). Let C' = the cents of the vanishing comma. The error of the 5th expressed as a fraction of the vanishing comma is (C / C') / (b+4c).

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2018-01-28 03:53:47 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-28 04:21:22 UTC</tt>.<br>
-: The original revision id was <tt>625471835</tt>.<br>
+: The original revision id was <tt>625472215</tt>.<br>
 : 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>
@@ Line 713: / Line 713: @@
 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. As noted, 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 boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by "far worse" is meant an error twice as large, meantone's sweet spot is from
+The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by "far worse" is meant an error twice as large, meantone's sweet spot is from 1/5-comma to 1/2-comma. This is assuming the JI ratios of interest are only 3/2, 5/4, 6/5 and their 8ve inverses. If 9/4 and 9/5 are included as well, the sweet spot is from
-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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of this comma. At exactly the tipping point, the error of the 5th from just is C / (b+4c).
+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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of the uninverted comma. At exactly the tipping point, the error of the 5th from just is -C / (b+4c). Let C' = the cents of the vanishing comma. The error of the 5th expressed as a fraction of the vanishing comma is (C / C') / (b+4c).
@@ Line 3,997: / Line 3,997: @@
 The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a &lt;u&gt;very&lt;/u&gt; unlikely 5th, and tipping is impossible. As noted, at least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.&lt;br /&gt;
 &lt;br /&gt;
-The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by &amp;quot;far worse&amp;quot; is meant an error twice as large, meantone's sweet spot is from&lt;br /&gt;
+The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by &amp;quot;far worse&amp;quot; is meant an error twice as large, meantone's sweet spot is from 1/5-comma to 1/2-comma. This is assuming the JI ratios of interest are only 3/2, 5/4, 6/5 and their 8ve inverses. If 9/4 and 9/5 are included as well, the sweet spot is from&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of this comma. At exactly the tipping point, the error of the 5th from just is C / (b+4c).&lt;br /&gt;
+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 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of the uninverted comma. At exactly the tipping point, the error of the 5th from just is -C / (b+4c). Let C' = the cents of the vanishing comma. The error of the 5th expressed as a fraction of the vanishing comma is (C / C') / (b+4c).&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;