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>2018-02-~~25 20~~:30:28 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-26 03:56:32 UTC</tt>.

: The original revision id was <tt>~~626878565~~</tt>.

: The original revision id was <tt>626886573</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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C/3 ``=`` C# ||= P8/2 = vA4 = ^d5

P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C

C - D\ - Eb/ - F ||= large sixfold red ||

C - D\ - Eb/ - F ||= large sixfold red

^1 = 1029/1024, /1 = 49/48 ||

||= 13 ||= (P8/2, P5/3)

half-8ve, third-5th ||= ^6d32 ||= C^6 ``=`` B#3 ||= P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3 ||= C - Fxv3=Gbb^3 C

C - D#vv - Fb^^ - G ||= large sixfold yellow ||

C - D#vv - Fb^^ - G ||= large sixfold yellow

^1 = 81/80 ||

||= ||= " ||= ^^d2,

\\\m2 ||= C^^ = B#

C``///`` = Db ||= P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C

C - /D - \F - G ||= lemba (50/49 & 1029/1024)

C - D/ - F\ - G ||= lemba (50/49 & 1029/1024)

^1 = (,-6,1,-1), /1 = 64/63 ||

^1 = (10,-6,1,-1), /1 = 64/63 ||

||= 14 ||= (P8/2, P11/3)

half-8ve, third-11th ||= v6M2 ||= C^6 ``=`` D ||= P8/2 = ^34 = v35

P11/3 = ^^4 = v45 ||= C - F^3=Gv3 - C

C - F^^ - Cvv - F ||= large sixfold jade ||

C - F^^ - Cvv - F ||= large sixfold jade, if 11/8 = P4

^1 = 33/32 ||

||= 15 ||= (P8/3, P4/3)

third-

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For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further%20Discussion-Various%20proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If ~~n = 1~~, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further%20Discussion-Various%20proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

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==Ratio and cents of the accidentals==

Looking at the table in the Applications section, the up symbol equals only a few ratios. 81/80, 64/63, 33/32 and 27/26 appear very often. These commas are used to map higher primes to 3-limit intervals, and are essential for notation. They are mapping commas or notational commas. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.

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. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.

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==Notating tunings with an arbitrary generator==

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it ~~barely~~ includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

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||= 560-585¢ ||= P11/3 ||= ||= ||= ||= ||

||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||= ||= ||

The total range of possible generators is fairly well covered, ~~but~~ there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.

The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. But there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.

Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.

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</td>

<td style="text-align: center;">large sixfold red

^1 = 1029/1024, /1 = 49/48

</td>

</tr>

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</td>

<td style="text-align: center;">large sixfold yellow

^1 = 81/80

</td>

</tr>

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</td>

<td style="text-align: center;">C - F#v=Gb^ - C

C - /D - \F - G

C - D/ - F\ - G

</td>

<td style="text-align: center;">lemba (50/49 &amp; 1029/1024)

^1 = (,-6,1,-1), /1 = 64/63

^1 = (10,-6,1,-1), /1 = 64/63

</td>

</tr>

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C - F^^ - Cvv - F

</td>

<td style="text-align: center;">large sixfold jade

<td style="text-align: center;">large sixfold jade, if 11/8 = P4

^1 = 33/32

</td>

</tr>

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For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof <a class="wiki_link" href="/pergen#Further%20Discussion-Various%20proofs">below</a>). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If ~~n = 1~~, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. &quot;Every&quot; means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof <a class="wiki_link" href="/pergen#Further%20Discussion-Various%20proofs">below</a>). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. &quot;Every&quot; means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under &quot;all pergens&quot;. (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

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<h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

Looking at the table in the Applications section, the up symbol equals only a few ratios. 81/80, 64/63, 33/32 and 27/26 appear very often. These commas are used to map higher primes to 3-limit intervals, and are essential for notation. They are mapping commas or notational commas. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.

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. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.

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<h2 id="toc18"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it ~~barely~~ includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G &gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

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</table>

The total range of possible generators is fairly well covered, ~~but~~ there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.

The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. But there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.

Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.

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finish proofs

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<h3 id="toc22"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.

<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>

<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>

(screenshot)

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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.

<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>

<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>

Screenshot of the first 38 pergens:

@@ 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>2018-02-25 20:30:28 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-26 03:56:32 UTC</tt>.<br>
-: The original revision id was <tt>626878565</tt>.<br>
+: The original revision id was <tt>626886573</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 242: / Line 242: @@
 C/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` C# ||= P8/2 = vA4 = ^d5
 P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C
-C - D\ - Eb/ - F ||= large sixfold red ||
+C - D\ - Eb/ - F ||= large sixfold red
+^1 = 1029/1024, /1 = 49/48 ||
 ||= 13 ||= (P8/2, P5/3)
 half-8ve, third-5th ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P8/2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5
 P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; C
-C - D#vv - Fb^^ - G ||= large sixfold yellow ||
+C - D#vv - Fb^^ - G ||= large sixfold yellow
+^1 = 81/80 ||
 ||=   ||= " ||= ^^d2,
 \\\m2 ||= C^^ = B#
 C``///`` = Db ||= P8/2 = vA4 = ^d5
 P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C
-C - /D - \F - G ||= lemba (50/49 &amp; 1029/1024)
+C - D/ - F\ - G ||= lemba (50/49 &amp; 1029/1024)
-^1 = (,-6,1,-1), /1 = 64/63 ||
+^1 = (10,-6,1,-1), /1 = 64/63 ||
 ||= 14 ||= (P8/2, P11/3)
 half-8ve, third-11th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D ||= P8/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;5
 P11/3 = ^^4 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;5 ||= C - F^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C
-C - F^^ - Cvv - F ||= large sixfold jade ||
+C - F^^ - Cvv - F ||= large sixfold jade, if 11/8 = P4
+^1 = 33/32 ||
 ||= 15 ||= (P8/3, P4/3)
 third-
@@ Line 345: / Line 348: @@
 For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.
-Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further%20Discussion-Various%20proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If n = 1, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.
+Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further%20Discussion-Various%20proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.
 The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.
@@ Line 400: / Line 403: @@
 ==Ratio and cents of the accidentals==
+Looking at the table in the Applications section, the up symbol equals only a few ratios. 81/80, 64/63, 33/32 and 27/26 appear very often. These commas are used to map higher primes to 3-limit intervals, and are essential for notation. They are mapping commas or notational commas. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.
 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. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.
@@ Line 621: / Line 626: @@
 ==Notating tunings with an arbitrary generator==
-Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it barely includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.
+Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.
 The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G &gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.
@@ Line 649: / Line 654: @@
 ||= 560-585¢ ||= P11/3 ||=   ||=   ||=   ||=   ||
 ||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||=   ||=   ||
-The total range of possible generators is fairly well covered, but there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.
+The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. But there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.
 Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.
@@ Line 2,043: / Line 2,048: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;large sixfold red&lt;br /&gt;
+^1 = 1029/1024, /1 = 49/48&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,062: / Line 2,068: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;large sixfold yellow&lt;br /&gt;
+^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,079: / Line 2,086: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C&lt;br /&gt;
-C - /D - \F - G&lt;br /&gt;
+C - D/ - F\ - G&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;lemba (50/49 &amp;amp; 1029/1024)&lt;br /&gt;
-^1 = (,-6,1,-1), /1 = 64/63&lt;br /&gt;
+^1 = (10,-6,1,-1), /1 = 64/63&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,101: / Line 2,108: @@
 C - F^^ - Cvv - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;large sixfold jade&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;large sixfold jade, if 11/8 = P4&lt;br /&gt;
+^1 = 33/32&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,568: / Line 2,576: @@
 For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.&lt;br /&gt;
 &lt;br /&gt;
-Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Various%20proofs"&gt;below&lt;/a&gt;). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If n = 1, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. &amp;quot;Every&amp;quot; means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.&lt;br /&gt;
+Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Various%20proofs"&gt;below&lt;/a&gt;). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. &amp;quot;Every&amp;quot; means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.&lt;br /&gt;
 &lt;br /&gt;
 The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under &amp;quot;all pergens&amp;quot;. (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.&lt;br /&gt;
@@ Line 2,747: / Line 2,755: @@
 &lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
   &lt;br /&gt;
+Looking at the table in the Applications section, the up symbol equals only a few ratios. 81/80, 64/63, 33/32 and 27/26 appear very often. These commas are used to map higher primes to 3-limit intervals, and are essential for notation. They are mapping commas or notational commas. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.&lt;br /&gt;
+&lt;br /&gt;
 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. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 3,704: / Line 3,714: @@
 &lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
   &lt;br /&gt;
-Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it barely includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;
+Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;
 &lt;br /&gt;
 The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G &amp;gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.&lt;br /&gt;
@@ Line 4,041: / Line 4,051: @@
 &lt;/table&gt;
-The total range of possible generators is fairly well covered, but there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.&lt;br /&gt;
+The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. But there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.&lt;br /&gt;
 &lt;br /&gt;
 Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.&lt;br /&gt;
@@ Line 5,673: / Line 5,683: @@
 finish proofs&lt;br /&gt;
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7097:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7097 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7102:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7102 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7098:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7098 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7103:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7103 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc22"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Notaion guide PDF&lt;/h3&gt;
   &lt;br /&gt;
 This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7099:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7099 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7104:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7104 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,689: / Line 5,699: @@
   &lt;br /&gt;
 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7100:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7100 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7105:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7105 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshot of the first 38 pergens:&lt;br /&gt;