Kite's thoughts on pergens: Difference between revisions

Line 1:

<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>2018-02-22 14:23:44 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-22 14:51:46 UTC</tt>.

: The original revision id was <tt>~~626776603~~</tt>.

: The original revision id was <tt>626777991</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.

Line 337:

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 occuring 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 occuring 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.

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 occuring 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.

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 occuring splits are listed too, under "all pergens".

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 occuring 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.

||||~ pergen ||~ secondary splits of a 12th or less ||

||||~ pergen ||~ secondary splits ~~<=~~ 12th ||

||||= all pergens ||= M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2 ||

||||~ half-splits ||~ ||

Line 353:

Line 352:

||= (P8, P5/3) ||= third-5th ||= m2/3, m6/3, M9/6, A8/3, A12/3 ||

||= (P8, P11/3) ||= third-11th ||= M2/3, M3/6, A4/9, A5/12, m9/3, P12/3 ||

||= (P8/3, P4/2) ||= third-8ve half-4th ||= half-4th ~~+ third-8ve~~ + M6/6, m10/6, A11/12 ||

||= (P8/3, P4/2) ||= third-8ve half-4th ||= third-8ve + half-4th + M6/6, m10/6, A11/12 ||

||= (P8/3, P5/2) ||= third-8ve half-5th ||= half-5th ~~+ third-8ve~~ + m3/6, d5/12 ||

||= (P8/3, P5/2) ||= third-8ve half-5th ||= third-8ve + half-5th + m3/6, d5/12 ||

||= (P8/2, P4/3) ||= half-8ve third-4th ||= half-8ve + third-4th + A4/6, M10/6 ||

||= (P8/2, P5/3) ||= half-8ve third-5th ||= half-8ve + third-5th + m6/6, M9/6, A12/6 ||

||= (P8/2, P11/3) ||= half-8ve third-11th ||= half-8ve + third-11th + M2/6, M3/12, A4/18, A5/24 ||

||= (P83, P4/3) ||= third-everything ||= (every 3-limit interval)/3 ||

==Singles and doubles==

Line 379:

Line 377:

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)/nm). The new multigen M' is the product of the original pergen's 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.

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)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's 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.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6⋅G - m3. The comma splits both the octave and the fifth.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus __true doubles require commas of at least 7-limit__, whereas false doubles require only 5-limit.

Line 425:

Line 423:

There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

* For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2

* x is the count for E, with E ~~occuring~~ x times in one octave, and xE is the octave's **multi-enharmonic**, or **multi-E** for short

* x is the count for E, with E occurring x times in one octave, and xE is the octave's **multi-enharmonic**, or **multi-E** for short

* y is the count for E', with E' ~~occuring~~ y times in one multigen, and yE' is the multigen's multi-E

* y is the count for E', with E' occurring y times in one multigen, and yE' is the multigen's multi-E

* For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics

* The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + ~~zE", and P8 = mP + xE~~"

* The unreduced pergen is (P8/m, M'/n'), with a new enharmonic E" and new counts, P8 = mP + x'E", and M' = n'G' + y'E"

The **keyspan** of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The **stepspan** of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Line 2,540:

Line 2,538:

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 occuring 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 occuring 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.~~ ~~

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 occuring 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.

~~ ~~

~~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 occuring splits are listed too, under~~ &quot;~~all pergens~~&quot;.

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 occuring 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.

Line 2,550:

Line 2,547:

<th colspan="2">pergen

</th>

<th>secondary splits ~~&lt;=~~ 12th

<th>secondary splits of a 12th or less

</th>

</tr>

Line 2,640:

Line 2,637:

<td style="text-align: center;">third-8ve half-4th

</td>

<td style="text-align: center;">half-4th ~~+ third-8ve~~ + M6/6, m10/6, A11/12

<td style="text-align: center;">third-8ve + half-4th + M6/6, m10/6, A11/12

</td>

</tr>

Line 2,648:

Line 2,645:

<td style="text-align: center;">third-8ve half-5th

</td>

<td style="text-align: center;">half-5th ~~+ third-8ve~~ + m3/6, d5/12

<td style="text-align: center;">third-8ve + half-5th + m3/6, d5/12

</td>

</tr>

Line 2,685:

Line 2,682:

</table>

~~ ~~

<h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

Line 2,705:

Line 2,701:

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)/nm). The new multigen M' is the product of the original pergen's 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.

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)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's 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.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6⋅G - m3. The comma splits both the octave and the fifth.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.

Line 2,750:

Line 2,746:

There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

<ul><li>For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for E, with E ~~occuring~~ x times in one octave, and xE is the octave's multi-enharmonic, or multi-E for short</li><li>y is the count for E', with E' ~~occuring~~ y times in one multigen, and yE' is the multigen's multi-E</li><li>For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics</li><li>The unreduced pergen is (P8/m, M'/n'), with M' = n'~~G' + zE~~&quot;, and P8 = mP + xE&quot;</li></ul>

<ul><li>For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for E, with E occurring x times in one octave, and xE is the octave's multi-enharmonic, or multi-E for short</li><li>y is the count for E', with E' occurring y times in one multigen, and yE' is the multigen's multi-E</li><li>For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic E&quot; and new counts, P8 = mP + x'E&quot;, and M' = n'G' + y'E&quot;</li></ul>

The keyspan of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The stepspan of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Line 5,605:

Line 5,601:

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)

Line 5,615:

Line 5,611:

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>2018-02-22 14:23:44 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-22 14:51:46 UTC</tt>.<br>
-: The original revision id was <tt>626776603</tt>.<br>
+: The original revision id was <tt>626777991</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 337: / Line 337: @@
 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 occuring 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 occuring 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.
+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 occuring 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.
-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 occuring splits are listed too, under "all pergens".
+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 occuring 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.
+||||~ pergen ||~ secondary splits of a 12th or less ||
-||||~ pergen ||~ secondary splits &lt;= 12th ||
 ||||= all pergens ||= M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2 ||
 ||||~ half-splits ||~   ||
@@ Line 353: / Line 352: @@
 ||= (P8, P5/3) ||= third-5th ||= m2/3, m6/3, M9/6, A8/3, A12/3 ||
 ||= (P8, P11/3) ||= third-11th ||= M2/3, M3/6, A4/9, A5/12, m9/3, P12/3 ||
-||= (P8/3, P4/2) ||= third-8ve half-4th ||= half-4th + third-8ve + M6/6, m10/6, A11/12 ||
+||= (P8/3, P4/2) ||= third-8ve half-4th ||= third-8ve + half-4th + M6/6, m10/6, A11/12 ||
-||= (P8/3, P5/2) ||= third-8ve half-5th ||= half-5th + third-8ve + m3/6, d5/12 ||
+||= (P8/3, P5/2) ||= third-8ve half-5th ||= third-8ve + half-5th + m3/6, d5/12 ||
 ||= (P8/2, P4/3) ||= half-8ve third-4th ||= half-8ve + third-4th + A4/6, M10/6 ||
 ||= (P8/2, P5/3) ||= half-8ve third-5th ||= half-8ve + third-5th + m6/6, M9/6, A12/6 ||
 ||= (P8/2, P11/3) ||= half-8ve third-11th ||= half-8ve + third-11th + M2/6, M3/12, A4/18, A5/24 ||
 ||= (P83, P4/3) ||= third-everything ||= (every 3-limit interval)/3 ||
 ==Singles and doubles==
@@ Line 379: / Line 377: @@
 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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - m&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M)/nm). The new multigen M' is the product of the original pergen's 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.
+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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - m&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's 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.
-For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P5) / (3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - m3. The comma splits both the octave and the fifth.
+For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P5) / (3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G' - m3. The comma splits both the octave and the fifth.
 This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P4) / (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;4) = (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus __true doubles require commas of at least 7-limit__, whereas false doubles require only 5-limit.
@@ Line 425: / Line 423: @@
 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:
 * For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2
-* x is the count for E, with E occuring x times in one octave, and xE is the octave's **multi-enharmonic**, or **multi-E** for short
+* x is the count for E, with E occurring x times in one octave, and xE is the octave's **multi-enharmonic**, or **multi-E** for short
-* y is the count for E', with E' occuring y times in one multigen, and yE' is the multigen's multi-E
+* y is the count for E', with E' occurring y times in one multigen, and yE' is the multigen's multi-E
 * For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics
-* The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE", and P8 = mP + xE"
+* The unreduced pergen is (P8/m, M'/n'), with a new enharmonic E" and new counts, P8 = mP + x'E", and M' = n'G' + y'E"
 The **keyspan** of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The **stepspan** of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.
@@ Line 2,540: / Line 2,538: @@
 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 occuring 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 occuring 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.&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 occuring 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;
-&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 occuring splits are listed too, under &amp;quot;all pergens&amp;quot;.&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 occuring 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 2,550: / Line 2,547: @@
          &lt;th colspan="2"&gt;pergen&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;secondary splits &amp;lt;= 12th&lt;br /&gt;
+         &lt;th&gt;secondary splits of a 12th or less&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
@@ Line 2,640: / Line 2,637: @@
          &lt;td style="text-align: center;"&gt;third-8ve half-4th&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-4th + third-8ve + M6/6, m10/6, A11/12&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;third-8ve + half-4th + M6/6, m10/6, A11/12&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,648: / Line 2,645: @@
          &lt;td style="text-align: center;"&gt;third-8ve half-5th&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-5th + third-8ve + m3/6, d5/12&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;third-8ve + half-5th + m3/6, d5/12&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,685: / Line 2,682: @@
 &lt;/table&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt;Singles and doubles&lt;/h2&gt;
@@ Line 2,705: / Line 2,701: @@
 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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - m&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M)/nm). The new multigen M' is the product of the original pergen's 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;
+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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - m&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's 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;
 &lt;br /&gt;
-For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P5) / (3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This &lt;u&gt;is&lt;/u&gt; explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - m3. The comma splits both the octave and the fifth.&lt;br /&gt;
+For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P5) / (3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This &lt;u&gt;is&lt;/u&gt; explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G' - m3. The comma splits both the octave and the fifth.&lt;br /&gt;
 &lt;br /&gt;
 This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P4) / (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;4) = (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus &lt;u&gt;true doubles require commas of at least 7-limit&lt;/u&gt;, whereas false doubles require only 5-limit.&lt;br /&gt;
@@ Line 2,750: / Line 2,746: @@
   &lt;br /&gt;
 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &amp;lt; |x| &amp;lt;= m/2 and 0 &amp;lt; |y| &amp;lt;= n/2&lt;/li&gt;&lt;li&gt;x is the count for E, with E occuring x times in one octave, and xE is the octave's &lt;strong&gt;multi-enharmonic&lt;/strong&gt;, or &lt;strong&gt;multi-E&lt;/strong&gt; for short&lt;/li&gt;&lt;li&gt;y is the count for E', with E' occuring y times in one multigen, and yE' is the multigen's multi-E&lt;/li&gt;&lt;li&gt;For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics&lt;/li&gt;&lt;li&gt;The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE&amp;quot;, and P8 = mP + xE&amp;quot;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &amp;lt; |x| &amp;lt;= m/2 and 0 &amp;lt; |y| &amp;lt;= n/2&lt;/li&gt;&lt;li&gt;x is the count for E, with E occurring x times in one octave, and xE is the octave's &lt;strong&gt;multi-enharmonic&lt;/strong&gt;, or &lt;strong&gt;multi-E&lt;/strong&gt; for short&lt;/li&gt;&lt;li&gt;y is the count for E', with E' occurring y times in one multigen, and yE' is the multigen's multi-E&lt;/li&gt;&lt;li&gt;For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics&lt;/li&gt;&lt;li&gt;The unreduced pergen is (P8/m, M'/n'), with a new enharmonic E&amp;quot; and new counts, P8 = mP + x'E&amp;quot;, and M' = n'G' + y'E&amp;quot;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 The &lt;strong&gt;keyspan&lt;/strong&gt; of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The &lt;strong&gt;stepspan&lt;/strong&gt; of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,605: / Line 5,601: @@
   &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:7026: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:7026 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7024: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:7024 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,615: / Line 5,611: @@
   &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:7027: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:7027 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7025: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:7025 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshot of the first 38 pergens:&lt;br /&gt;