Kite's thoughts on pergens: Difference between revisions

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==Finding a notation for a pergen==

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. ~~<s>~~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.~~)</s> (''Edited to add: not true, v3AA1, v5AA1, etc. are possible, as is vvA31.''~~) 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:

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 + xEI and M = nG + yEI', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EI, with EI occurring x times in one octave, and xEI is the octave's '''multi-enharmonic''', or '''multi-EI''' for short</li><li>y is the count for EI', with EI' occurring y times in one multigen, and yEI' is the multigen's multi-EI</li><li>For false doubles using single-pair notation, EI = EI', 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 EI" and new counts, P8 = mP + x'EI", and M' = n'G' + y'EI"</li></ul>

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WORK IN PROGRESS

=== New terminology ===

=== New terminology===

All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

=== The EI(s) define the pergen ===

===The EI(s) define the pergen===

The pergen can be derived directly from the EI(s). Thus the EI(s) define both the pergen and the notation. An EI can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EI is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike conventional 2.3.^ ~~monzos~~, the first two numbers are ~~usually~~ fairly small, and the second number is never negative. ~~This~~ facilitates one's search.

One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EI). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

=== Arrow commas ===

===Simplifying "doubled" EI's===

Consider an EI of v3AA1. AA1 is "doubled" in the sense that AA1 = A1 + A1. The EI's 2.3.^ monzo is [-22 14 -3]. The doubledness is apparent from the first two numbers both being even. The EI implies a mapping of [(1 2 2) (0 -3 -14)]. The pergen is (P8, P4/3).

We can derive P and G from this matrix. P = P8 = [1 0 0] and G = ^m2 = [8 -5 1]. We can make a 3x3 gencom matrix from P, G and EI.

Consider the twin squares

P8 (1 0 0)

vm3 (5 -3 -1)

v3AA1 (-22 14 -3)

Certain uninflected EI's naturally split into smaller pieces, because both numbers of the 2.3 monzo are even (or [[threeven]], fourven, etc.)

It is always possible to simplify a doubled EI.

===Arrow commas===

The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyo/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EI is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyo/Porcupine with an EI of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

=== The three commas ===

===The three commas ===

Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EI. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

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 ==Finding a notation for a pergen==
-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. <s>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.)</s> (''Edited to add: not true, v<sup>3</sup>AA1, v<sup>5</sup>AA1, etc. are possible, as is vvA<sup>3</sup>1.'') 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:
+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 + xEI and M = nG + yEI', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for EI, with EI occurring x times in one octave, and xEI is the octave's '''multi-enharmonic''', or '''multi-EI''' for short</li><li>y is the count for EI', with EI' occurring y times in one multigen, and yEI' is the multigen's multi-EI</li><li>For false doubles using single-pair notation, EI = EI', 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 EI" and new counts, P8 = mP + x'EI", and M' = n'G' + y'EI"</li></ul>
@@ Line 4,421: / Line 4,421: @@
 WORK IN PROGRESS
-=== New terminology ===
+=== New terminology===
 All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.
-=== The EI(s) define the pergen ===
+===The EI(s) define the pergen===
 The pergen can be derived directly from the EI(s). Thus the EI(s) define both the pergen and the notation. An EI can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.
 For example, if the EI is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).
-One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v<sup>3</sup>m2 = [1 1 -3], etc. Unlike conventional 2.3.^ monzos, the first two numbers are usually fairly small, and the second number is never negative. This facilitates one's search.
+One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v<sup>3</sup>m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EI). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.
-=== Arrow commas ===
+===Simplifying "doubled" EI's===
+Consider an EI of v<sup>3</sup>AA1. AA1 is "doubled" in the sense that AA1 = A1 + A1. The EI's 2.3.^ monzo is [-22 14 -3]. The doubledness is apparent from the first two numbers both being even. The EI implies a mapping of [(1 2 2) (0 -3 -14)]. The pergen is (P8, P4/3).
+We can derive P and G from this matrix. P = P8 = [1 0 0] and G = ^m2 = [8 -5 1]. We can make a 3x3 gencom matrix from P, G and EI.
+Consider the twin squares
+<nowiki><tt></nowiki>
+P8       (1  0  0)
+vm3    (5 -3 -1)
+v<sup>3</sup>AA1  (-22 14 -3)
+Certain uninflected EI's naturally split into smaller pieces, because both numbers of the 2.3 monzo are even (or [[threeven]], fourven, etc.)
+It is always possible to simplify a doubled EI.
+===Arrow commas===
 The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.
 Consider Triyo/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EI is v<sup>3</sup>A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyo/Porcupine with an EI of v<sup>3</sup>A1, the vAC is v(81/80) or [-4 4 -1 -1].
-=== The three commas ===
+===The three commas ===
 Thus when we consider a single-comma temperament along with its notation, there are <u>three</u> commas of interest, the VC, the vAC and the EI. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.