Kite's thoughts on pergens: Difference between revisions
→Addenda: work in progress, finally writing down cool stuff I thought up last summer |
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==Finding a notation for a pergen== | ==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. 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: | 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: | ||
<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> | <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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== Addendum (late 2023) == | == Addendum (late 2023) == | ||
WORK IN PROGRESS | WORK IN PROGRESS | ||
=== 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 === | ||
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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). | 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 conventional 2.3.^ monzos, the first two numbers are usually fairly small, and the second number is never negative. This facilitates one's search. | ||
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Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EI turns out to be. | Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EI turns out to be. | ||
The EI always equals the VC plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. | The EI always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EI would otherwise have a negative stepspan, or is a diminished unison. | ||
In our Triyo example, 250/243 plus 3 downed syntonic commas = v<sup>3</sup>A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EI always has a zero prime-5-count. | In our Triyo example, 250/243 plus 3 downed syntonic commas = v<sup>3</sup>A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EI always has a zero prime-5-count. | ||
Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EI. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so 135/128 or the schisma are possibilities. But either of these would lead | Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EI. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EI (v<sup>3</sup>AA1 and v<sup>3</sup>d<sup>4</sup>4 respectively), making a very awkward notation. | ||
Next let's specify the AC, experiment with the VC and see what the EI turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EI (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EI for (P8/2, P5). | Next let's specify the AC, experiment with the VC and see what the EI turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EI (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EI for (P8/2, P5). | ||
More examples: Laquinyo/Magic is (P8, P12/5) and has VC = [-10 -1 5 0] | More examples: Laquinyo/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^<sup>5</sup>dd2. Gugu/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EI. | ||
Again, this is the recommended EI. | |||
Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2. | Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2. | ||