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>2018-01-29 06:~~15:34~~ UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-29 11:06:20 UTC</tt>.

: The original revision id was <tt>~~625514983~~</tt>.

: The original revision id was <tt>625531813</tt>.

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

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* ∆w also must range between -(c+2) / (2b+2c) and (c-2) / (2b+2c) comma

If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.

If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.

Another example: for porcupine, the comma is 250/243 = 49.2¢, b = -5 and c = 3. Second range: 1/2 comma to -1/10 comma, again within the first range. Third range: 5/4 comma to -1/4 comma. Total range: -1/10 comma to 1/2 comma.

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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:

<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>

If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.

If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.

Another example: for porcupine, the comma is 250/243 = 49.2¢, b = -5 and c = 3. Second range: 1/2 comma to -1/10 comma, again within the first range. Third range: 5/4 comma to -1/4 comma. Total range: -1/10 comma to 1/2 comma.

@@ 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>2018-01-29 06:15:34 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-29 11:06:20 UTC</tt>.<br>
-: The original revision id was <tt>625514983</tt>.<br>
+: The original revision id was <tt>625531813</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 716: / Line 716: @@
 * ∆w also must range between -(c+2) / (2b+2c) and (c-2) / (2b+2c) comma
-If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.
+If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.
 Another example: for porcupine, the comma is 250/243 = 49.2¢, b = -5 and c = 3. Second range: 1/2 comma to -1/10 comma, again within the first range. Third range: 5/4 comma to -1/4 comma. Total range: -1/10 comma to 1/2 comma.
@@ Line 4,016: / Line 4,016: @@
 For any 5-limit temperament, the boundaries of the sweet spot can be reasonably defined such that going beyond them makes the error of 3/2, 5/4 or 6/5 more than half a comma. Assuming untempered octaves, let ∆w be the error of 3/2 from just, and ∆y be the error of 5/4 from just, with positive values meaning sharp. The error for 6/5 is ∆w - ∆y. For a comma (a,b,c) of cents K, we have b·∆w + c·∆y + K = 0¢. One pair of limits, or range, is simply that ∆w must be between -K/2 and +K/2. Substitute ±K/2 for ∆y and solve for ∆w to get another range for ∆w. The error of 6/5 = ∆w - ∆y = ±K/2, thus ∆y = ∆w ± K/2. Substitute this for ∆y to get a third range. The intersection of these three ranges is the sweet spot:&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;∆w must range between -1/2 comma and 1/2 comma&lt;/li&gt;&lt;li&gt;∆w also must range between -(c+2) / 2b comma and (c-2) / 2b comma&lt;/li&gt;&lt;li&gt;∆w also must range between -(c+2) / (2b+2c) and (c-2) / (2b+2c) comma&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.&lt;br /&gt;
+If b = -c, ignore the third range, to avoid dividing by zero. We'll use familiar meantone as an example, even though it doesn't require ups and downs, and can't tip over. The comma is (-4,4,-1), so b = 4 and c = -1. The second range is -1/8 comma to -3/8 comma, which falls entirely within the first range. The third range is -1/6 comma to -1/2 comma. The intersection of the three ranges is -1/6 to -3/8, thus the 5th must be flattened by 1/6 to 3/8 of a comma. At 1/6-comma, ∆y = This includes historical tunings of 1/4 comma, 1/5 comma, 2/7 comma and 1/3 comma.&lt;br /&gt;
 &lt;br /&gt;
 Another example: for porcupine, the comma is 250/243 = 49.2¢, b = -5 and c = 3. Second range: 1/2 comma to -1/10 comma, again within the first range. Third range: 5/4 comma to -1/4 comma. Total range: -1/10 comma to 1/2 comma.&lt;br /&gt;