Kite's thoughts on pergens: Difference between revisions

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| | 81/80, 50/49

| | injera

| | biruyo ~~and gu~~

| | gu and biruyo

| | ~~rryy~~&~~amp;gT~~

| | g&rryyT

|-

| | (P8, P5/2)

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|}

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

zeros below the diagonal, and no zeros on the diagonal:

{| class="wikitable" style="text-align:center;"

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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. 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. To summarize:

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>'.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

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==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonics are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:

{| class="wikitable" style="text-align:center;"

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'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? ~~The~~ pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

@@ Line 60: / Line 60: @@
 | | 81/80, 50/49
 | | injera
-| | biruyo and gu
+| | gu and biruyo
-| | rryy&amp;gT
+| | g&rryyT
 |-
 | | (P8, P5/2)
@@ Line 196: / Line 196: @@
 |}
-Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with
+Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:
-zeros below the diagonal, and no zeros on the diagonal:
 {| class="wikitable" style="text-align:center;"
@@ Line 1,129: / Line 1,128: @@
 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. 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 <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize:
-<ul><li>'''A'''<nowiki/> '''double-split pergen is <u>explicitly false</u> if m = |b|, and not explicitly false if m &gt; |b|.'''</li><li>'''A double-split pergen is a <u>true double</u> if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>'.'''</li><li>'''A double-split pergen is a <u>true double</u> if''' '''GCD (m, n) &gt; |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>
+<ul><li>'''A'''<nowiki/> '''double-split pergen is <u>explicitly false</u> if m = |b|, and not explicitly false if m &gt; |b|.'''</li><li>'''A double-split pergen is a <u>true double</u> if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a <u>true double</u> if''' '''GCD (m, n) &gt; |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>
 A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.
@@ Line 1,406: / Line 1,405: @@
 ==Notating rank-3 pergens==
-Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:
+Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonics are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:
 {| class="wikitable" style="text-align:center;"
@@ Line 3,023: / Line 3,022: @@
 '''<u>Notating a pergen tuned to an EDO</u>'''
-If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? The pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.
+If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.
 When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.