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>2017-12-~~20 19~~:25:03 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-22 17:09:05 UTC</tt>.

: The original revision id was <tt>~~624132641~~</tt>.

: The original revision id was <tt>624192833</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 **genchain** (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...

C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...

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 table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.

The table shows compatible edos. An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.

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||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^P4 = vP5 ||= C - F^=Gv - C ||= 128/121,

^1 = 33/32 ||= " ||

||= " ||= ^^d32 ||= C^^ = B#3 ||= P8/2 = vAA4 = ^dd5 ||= C - F##v=Gbb^ - C ||= ||= " ||

||= " ||= ^^d32 ||= C^^ = B#3 ||= P8/2 = vAA4 = ^dd5 ||= C - F##v=Gbb^ - C ||= ? ||= " ||

||= (P8, P4/2) ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore

^1 = 64/63 ||= 14, 15*, 18b*, 19, 20*,

23, 24, 25*, 28*, 29,

30* ||

||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= ||= " ||

||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= ? ||= " ||

||= " ||= ^^d42 ||= C^^ = B#4 ||= P4/2 = vAA2 = ^dd3 ||= C - D##v=Eb3^ - F ||= ||= " ||

||= " ||= ^^d42 ||= C^^ = B#4 ||= P4/2 = vAA2 = ^dd3 ||= C - D##v=Eb3^ - F ||= ? ||= " ||

||= (P8, P5/2) ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira

^1 = 33/32 ||= 14*, 17, 18b, 20*, 21*,

24, 27, 28*, 30*, 31 ||

||= " ||= vvdd3 ||= C^^ = Eb3 ||= P5/2 = ^A2 = vd4 ||= ||= ||= ||

||= " ||= vvdd3 ||= C^^ = Eb3 ||= P5/2 = ^A2 = vd4 ||= C - D#^=Fbv - G ||= ? ||= " ||

||= (P8/2, P4/2) ||= \\m2,

vvA1,

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P4/2 = v3m2 ||= C - Eb^^ - Avv - C

C - Dbv3=E^3 - F ||= ||= 15, 18b*, 24, 30 ||

||= (P8/3, P5/2) ||= ||= ||= ||= ||= ||= 18b, 24, 30 ||

||= (P8/3, P5/2) ||= v6m3 ||= ||= ||= ||= ||= 18b, 24, 30 ||

||= (P8/2, P4/3) ||= ||= ||= ||= ||= ||= 14, 22, 28*, 30* ||

||= (P8/2, P4/3) ||= v6d4 ||= ||= ||= ||= ||= 14, 22, 28*, 30* ||

||= " ||= v6d34 ||= ||= ||= ||= ||= ||

||= (P8/2, P5/3) ||= ^6d32 ||= C^6 ``=`` B#3 ||= P8/2 = v3AA4 = ^3dd5

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

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==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, ~~simply~~ find a ratio for P or G that contains only one higher prime, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's ~~multi-gen~~ is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen M' is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

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__**Extra paragraphs:**__

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the notational comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c) we get [k,s,y,b]:

k = 12a + 19b + 4c + 10d

s = 7a + 11b + 2c + 6d

y =

Chord names: All rank-2 chords can be named using ups and downs, as if they were edos. For example, in half-octave, a 4:5:6 chord is C Ev G = C.v. Whenever the enharmonic isn't an A1, there are multiple spellings for many chords. It would be possible to spell the chord C Fb^ G, but there's no reason to. But in certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, as they are in 24-edo. Even in 12-edo, there are chords with ambiguous spellings. C Eb Gb Bbb = Cdim7, and C Eb G A = Cmin6. But without the 5th, the chord could be C Eb Bbb or C Eb A.

Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).

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The genchain (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...

C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...

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 table has a perchain (&quot;peer-chain&quot;, chain of periods) that shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

If the octave is split, the table has a perchain (&quot;peer-chain&quot;, chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.

The table shows compatible edos. An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.

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

</td>

<td style="text-align: center;">?

</td>

<td style="text-align: center;">&quot;

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

</td>

<td style="text-align: center;">?

</td>

<td style="text-align: center;">&quot;

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<td style="text-align: center;">C - D##v=Eb3^ - F

</td>

<td style="text-align: center;">?

</td>

<td style="text-align: center;">&quot;

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<td style="text-align: center;">P5/2 = ^A2 = vd4

</td>

<td style="text-align: center;">C - D#^=Fbv - G

</td>

<td style="text-align: center;">?

</td>

<td style="text-align: center;">&quot;

</td>

</tr>

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<td style="text-align: center;">(P8/3, P5/2)

</td>

<td style="text-align: center;">v6m3

</td>

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<tr>

<td style="text-align: center;">(P8/2, P4/3)

</td>

<td style="text-align: center;">v6d4

</td>

</td>

</td>

</td>

</td>

<td style="text-align: center;">14, 22, 28*, 30*

</td>

</tr>

<tr>

<td style="text-align: center;">&quot;

</td>

<td style="text-align: center;">v6d34

</td>

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</td>

<td style="text-align: center;">14, 22, 28*, 30*

</td>

</tr>

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<h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, ~~simply~~ find a ratio for P or G that contains only one higher prime, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's ~~multi-gen~~ is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is explicitly false. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is explicitly false. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its unreduced form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen M' is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

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Extra paragraphs:

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the notational comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c) we get [k,s,y,b]:

k = 12a + 19b + 4c + 10d

s = 7a + 11b + 2c + 6d

y =

Chord names: All rank-2 chords can be named using ups and downs, as if they were edos. For example, in half-octave, a 4:5:6 chord is C Ev G = C.v. Whenever the enharmonic isn't an A1, there are multiple spellings for many chords. It would be possible to spell the chord C Fb^ G, but there's no reason to. But in certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, as they are in 24-edo. Even in 12-edo, there are chords with ambiguous spellings. C Eb Gb Bbb = Cdim7, and C Eb G A = Cmin6. But without the 5th, the chord could be C Eb Bbb or C Eb A.

Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).

@@ 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>2017-12-20 19:25:03 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-22 17:09:05 UTC</tt>.<br>
-: The original revision id was <tt>624132641</tt>.<br>
+: The original revision id was <tt>624192833</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 128: / Line 128: @@
 The **genchain** (chain of generators) in the table is only a short section of the full genchain.
 C - G implies ...Eb Bb F C G D A E B F# C#...
-C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...
+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 table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
+If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.
 The table shows compatible edos. An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.
@@ Line 157: / Line 157: @@
 ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^P4 = vP5 ||= C - F^=Gv - C ||= 128/121,
 ^1 = 33/32 ||= " ||
-||= " ||= ^^d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P8/2 = vAA4 = ^dd5 ||= C - F##v=Gbb^ - C ||=   ||= " ||
+||= " ||= ^^d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P8/2 = vAA4 = ^dd5 ||= C - F##v=Gbb^ - C ||= ? ||= " ||
 ||= (P8, P4/2) ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore
 ^1 = 64/63 ||= 14, 15*, 18b*, 19, 20*,
 , 24, 25*, 28*, 29,
 * ||
-||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||=   ||= " ||
+||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= ? ||= " ||
-||= " ||= ^^d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= P4/2 = vAA2 = ^dd3 ||= C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F ||=   ||= " ||
+||= " ||= ^^d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= P4/2 = vAA2 = ^dd3 ||= C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F ||= ? ||= " ||
 ||= (P8, P5/2) ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira
 ^1 = 33/32 ||= 14*, 17, 18b, 20*, 21*,
 , 27, 28*, 30*, 31 ||
-||= " ||= vvdd3 ||= C^^ = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P5/2 = ^A2 = vd4 ||=   ||=   ||=   ||
+||= " ||= vvdd3 ||= C^^ = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P5/2 = ^A2 = vd4 ||= C - D#^=Fbv - G ||= ? ||= " ||
 ||= (P8/2, P4/2) ||= \\m2,
 vvA1,
@@ Line 217: / Line 217: @@
 P4/2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C - Eb^^ - Avv - C
 C - Dbv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=E^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F ||=   ||= 15, 18b*, 24, 30 ||
-||= (P8/3, P5/2) ||=   ||=   ||=   ||=   ||=   ||= 18b, 24, 30 ||
+||= (P8/3, P5/2) ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;m3 ||=   ||=   ||=   ||=   ||= 18b, 24, 30 ||
-||= (P8/2, P4/3) ||=   ||=   ||=   ||=   ||=   ||= 14, 22, 28*, 30* ||
+||= (P8/2, P4/3) ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d4 ||=   ||=   ||=   ||=   ||= 14, 22, 28*, 30* ||
+||= " ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 ||=   ||=   ||=   ||=   ||=   ||
 ||= (P8/2, P5/3) ||= ^&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
@@ Line 285: / Line 286: @@
 ==Finding an example temperament==
-To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, simply find a ratio for P or G that contains only one higher prime, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.
+To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.
-Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
+Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
 If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen M' is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.
@@ Line 362: / Line 363: @@
 __**Extra paragraphs:**__
+Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the notational comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c) we get [k,s,y,b]:
+k = 12a + 19b + 4c + 10d
+s = 7a + 11b + 2c + 6d
+y =
+Chord names: All rank-2 chords can be named using ups and downs, as if they were edos. For example, in half-octave, a 4:5:6 chord is C Ev G = C.v. Whenever the enharmonic isn't an A1, there are multiple spellings for many chords. It would be possible to spell the chord C Fb^ G, but there's no reason to. But in certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, as they are in 24-edo. Even in 12-edo, there are chords with ambiguous spellings. C Eb Gb Bbb = Cdim7, and C Eb G A = Cmin6. But without the 5th, the chord could be C Eb Bbb or C Eb A.
 Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).
@@ Line 988: / Line 998: @@
 The &lt;strong&gt;genchain&lt;/strong&gt; (chain of generators) in the table is only a short section of the full genchain.&lt;br /&gt;
 C - G implies ...Eb Bb F C G D A E B F# C#...&lt;br /&gt;
-C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...&lt;br /&gt;
+C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...&lt;br /&gt;
-If the octave is split, the table has a &lt;strong&gt;perchain&lt;/strong&gt; (&amp;quot;peer-chain&amp;quot;, chain of periods) that shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.&lt;br /&gt;
+If the octave is split, the table has a &lt;strong&gt;perchain&lt;/strong&gt; (&amp;quot;peer-chain&amp;quot;, chain of periods) that 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;
 The table shows compatible edos. An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.&lt;br /&gt;
@@ Line 1,121: / Line 1,131: @@
          &lt;td style="text-align: center;"&gt;C - F##v=Gbb^ - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;?&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
@@ Line 1,156: / Line 1,166: @@
          &lt;td style="text-align: center;"&gt;C - D#v=Ebb^ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;?&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
@@ Line 1,172: / Line 1,182: @@
          &lt;td style="text-align: center;"&gt;C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;?&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
@@ Line 1,204: / Line 1,214: @@
          &lt;td style="text-align: center;"&gt;P5/2 = ^A2 = vd4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C - D#^=Fbv - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&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;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,413: / Line 1,423: @@
          &lt;td style="text-align: center;"&gt;(P8/3, P5/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;m3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,428: / Line 1,438: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, P4/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;14, 22, 28*, 30*&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,439: / Line 1,465: @@
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;14, 22, 28*, 30*&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,049: / Line 2,075: @@
 &lt;!-- ws:start:WikiTextHeadingRule:35:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:35 --&gt;Finding an example temperament&lt;/h2&gt;
   &lt;br /&gt;
-To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, simply find a ratio for P or G that contains only one higher prime, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.&lt;br /&gt;
+To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.&lt;br /&gt;
 &lt;br /&gt;
-Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is &lt;strong&gt;explicitly false&lt;/strong&gt;. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).&lt;br /&gt;
+Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is &lt;strong&gt;explicitly false&lt;/strong&gt;. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).&lt;br /&gt;
 &lt;br /&gt;
 If the pergen is not explicitly false, put the pergen in its &lt;strong&gt;unreduced&lt;/strong&gt; form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen M' is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &amp;lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.&lt;br /&gt;
@@ Line 2,126: / Line 2,152: @@
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Extra paragraphs:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the notational comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c) we get [k,s,y,b]:&lt;br /&gt;
+k = 12a + 19b + 4c + 10d&lt;br /&gt;
+s = 7a + 11b + 2c + 6d&lt;br /&gt;
+y =&lt;br /&gt;
+&lt;br /&gt;
+Chord names: All rank-2 chords can be named using ups and downs, as if they were edos. For example, in half-octave, a 4:5:6 chord is C Ev G = C.v. Whenever the enharmonic isn't an A1, there are multiple spellings for many chords. It would be possible to spell the chord C Fb^ G, but there's no reason to. But in certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, as they are in 24-edo. Even in 12-edo, there are chords with ambiguous spellings. C Eb Gb Bbb = Cdim7, and C Eb G A = Cmin6. But without the 5th, the chord could be C Eb Bbb or C Eb A.&lt;br /&gt;
+&lt;br /&gt;
 &lt;br /&gt;
 Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).&lt;br /&gt;