Pergen names: 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>2017-11-19 22:06:16 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-19 22:33:12 UTC</tt>.

: The original revision id was <tt>~~621986909~~</tt>.

: The original revision id was <tt>621987807</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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=__Applications__=

Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g or r/b) could be used.

Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this contrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.

Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.

Not all possible combinations of periods and generators are unique pergens. {P8, WWP5/2} is actually {P8, P5/2}. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.

The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... If the octave is split, the ~~first~~ genchain shows the octave: In C - F#v=Gb^ - C, the last C is an ~~8ve~~ above the first one.

The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

(table is under construction)

||~ pergen set ||~ valid range

of the 5th ||~ ~~equivalence(s)~~ ||~ enharmonic

of the 5th ||~ equivalences ||~ enharmonic

interval ||~ genchain(s) ||~ example

temperament ||~ ~~example~~ edos

temperament ||~ compatible edos

(12-31 only) ||

||= {P8, P5} ||= 600-720¢ ||= none ||= none ||= C - G ||= meantone ||= 12, 16, 19, 23, 26 ||

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||= ||= ||= ||= ||= ||= ||= ||

||= {P8, P12/2} ||= ||= P12/2 = ^M6 = vm7 ||= vvm2 ||= ||= magic ||= ||

||= ~~{P8, WWP4/2}~~ ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

|| ~~{P8, WWP5/2}~~ || || || || || || ||

|| || || || || || || ||

||= {P8/2, P4/2} ||= ||= P8/2 = vA4 = ^d5,

P4/2 = /M2 = \m3 ||= ^^d2,

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||= {P8/2, P12/2} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

~~||= {P8/2, WWP4/2| ||= ||= ||= ||= ||= ||= ||~~

||= ||= ||= ||= ||= ||= ||= ||

||= ~~{P8/2, WWP5/2}~~ ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

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<h1 id="toc2"><a name="Applications"></a>Applications</h1>

Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g or r/b) could be used.

Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this contrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.

Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.

Not all possible combinations of periods and generators are unique pergens. {P8, WWP5/2} is actually {P8, P5/2}. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.

The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... If the octave is split, the ~~first~~ genchain shows the octave: In C - F#v=Gb^ - C, the last C is an ~~8ve~~ above the first one.

The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

(table is under construction)

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of the 5th

</th>

<th>~~equivalence(s)~~

<th>equivalences

</th>

<th>enharmonic

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temperament

</th>

<th>~~example~~ edos

<th>compatible edos

(12-31 only)

</th>

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</tr>

<tr>

<td style="text-align: center;">~~{P8, WWP4/2}~~

</td>

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</tr>

<tr>

<td>~~{P8, WWP5/2}~~

<td>

</td>

<td>

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</tr>

<tr>

<td style="text-align: center;">~~{P8/2, WWP4/2|~~

</td>

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</tr>

<tr>

<td style="text-align: center;">~~{P8/2, WWP5/2}~~

</td>

@@ 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>2017-11-19 22:06:16 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-19 22:33:12 UTC</tt>.<br>
-: The original revision id was <tt>621986909</tt>.<br>
+: The original revision id was <tt>621987807</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 103: / Line 103: @@
 =__Applications__=
-Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g or r/b) could be used.
+Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this contrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.
-Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.
+Not all possible combinations of periods and generators are unique pergens. {P8, WWP5/2} is actually {P8, P5/2}. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.
-The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... If the octave is split, the first genchain shows the octave: In C - F#v=Gb^ - C, the last C is an 8ve above the first one.
+The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
 (table is under construction)
 ||~ pergen set ||~ valid range
-of the 5th ||~ equivalence(s) ||~ enharmonic
+of the 5th ||~ equivalences ||~ enharmonic
 interval ||~ genchain(s) ||~ example
-temperament ||~ example edos
+temperament ||~ compatible edos
 (12-31 only) ||
 ||= {P8, P5} ||= 600-720¢ ||= none ||= none ||= C - G ||= meantone ||= 12, 16, 19, 23, 26 ||
@@ Line 131: / Line 131: @@
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||= {P8, P12/2} ||=   ||= P12/2 = ^M6 = vm7 ||= vvm2 ||=   ||= magic ||=   ||
-||= {P8, WWP4/2} ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-|| {P8, WWP5/2} ||   ||   ||   ||   ||   ||   ||
+||   ||   ||   ||   ||   ||   ||   ||
 ||= {P8/2, P4/2} ||=   ||= P8/2 = vA4 = ^d5,
 P4/2 = /M2 = \m3 ||= ^^d2,
@@ Line 141: / Line 141: @@
 ||= {P8/2, P12/2} ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||= {P8/2, WWP4/2| ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||= {P8/2, WWP5/2} ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
@@ Line 673: / Line 673: @@
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g or r/b) could be used.&lt;br /&gt;
+Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this contrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.&lt;br /&gt;
 &lt;br /&gt;
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.&lt;br /&gt;
 &lt;br /&gt;
-Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.&lt;br /&gt;
+Not all possible combinations of periods and generators are unique pergens. {P8, WWP5/2} is actually {P8, P5/2}. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.&lt;br /&gt;
 &lt;br /&gt;
-The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... If the octave is split, the first genchain shows the octave: In C - F#v=Gb^ - C, the last C is an 8ve above the first one.&lt;br /&gt;
+The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.&lt;br /&gt;
 &lt;br /&gt;
 (table is under construction)&lt;br /&gt;
@@ Line 692: / Line 692: @@
 of the 5th&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;equivalence(s)&lt;br /&gt;
+         &lt;th&gt;equivalences&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;enharmonic&lt;br /&gt;
@@ Line 702: / Line 702: @@
 temperament&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;example edos &lt;br /&gt;
+         &lt;th&gt;compatible edos &lt;br /&gt;
 (12-31 only)&lt;br /&gt;
 &lt;/th&gt;
@@ Line 900: / Line 900: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;{P8, WWP4/2}&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 916: / Line 916: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;{P8, WWP5/2}&lt;br /&gt;
+         &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,015: / Line 1,015: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;{P8/2, WWP4/2|&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,047: / Line 1,047: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;{P8/2, WWP5/2}&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;