Diasem: Difference between revisions
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* < 4\7 if L < m + s | * < 4\7 if L < m + s | ||
(This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.) | (This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.) | ||
== Modes == | == Modes == | ||
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. To minimize bias (e.g. towards certain tunings), this article names the modes after the [[semiquartal]] mode that the diasem mode is based on (the semiquartal modes are given in [[UDP notation]]). We also have provided diatonic-based names, which are useful in tunings where S is relatively small, such as 26edo, and 31edo, and superdiatonic-based names, which may be useful in tunings where L is close to M (such as 28edo). | Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. To minimize bias (e.g. towards certain tunings), this article names the modes after the [[semiquartal]] mode that the diasem mode is based on (the semiquartal modes are given in [[UDP notation]]). We also have provided diatonic-based names, which are useful in tunings where S is relatively small, such as 26edo, and 31edo, and superdiatonic-based names, which may be useful in tunings where L is close to M (such as 28edo). | ||
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Like [[superpyth]], JI diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths. On the other hand, the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out [[81/80]] are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps. | Like [[superpyth]], JI diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths. On the other hand, the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out [[81/80]] are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps. | ||
=== As a Fokker block === | |||
[[File:Diasem as fokker block.png|600px|thumbnail|2.3.7 JI diasem as a Fokker block]] | |||
The 2.3.7 JI diasem scale can be viewed as a [[Fokker block]] living in the 2.3.7 octave-equivalent pitch class lattice. The x-axis goes along the 3 direction and the y-axis goes along the 7 direction. | |||
The diagram shows the LMLSLMLSL mode. All the notes of the mode are marked as solid purple dots. Notes of the lattice outside the mode are black hollow dots. The red dashed lines are separated by the chroma 49/48, and the blue dotted lines are separated by the chroma 567/512. Note that both 49/48 and 567/512 are tempered out by (the 2.3.7 [[patent val]] of) [[9edo]]. | |||
The notes of diasem form a Fokker block, which is a fundamental domain of the pitch class lattice; it is possible to tile the entire infinite lattice with copies of left-hand diasem translated by (49/48)<sup>''m''</sup>(567/512)<sup>''n''</sup> for integer ''m'' and ''n''. Including any one of the other three points on the boundary (28/27, 147/128, or 64/63) instead of 9/8 also yields Fokker blocks (more specifically, modes of the three other [[dome]]s of diasem). However, only one other choice, 28/27, yields a diasem scale, and it yields the left-handed diasem mode MLLSLMLSL. | |||
As a Fokker block, 2.3.7 JI diasem is also a product word scale, namely between semaphore[9] (LsLsLsLsL) and mavila[9] (LLLsLLLsL). | |||
== Tunings == | == Tunings == | ||
{{todo|cleanup}} <!-- the table is unreadable in wikitext --> | {{todo|cleanup}} <!-- the table is unreadable in wikitext --> | ||