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autopatrolled, Sysops
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TallKite (talk | contribs)
autopatrolled, Sysops
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Wrote down some insights on using pergens to find canonical mappings
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::: Explanation of M2, m7, etc.: https://en.wikipedia.org/wiki/Interval_(music)#Main_intervals. Knowing these terms really helps with chord names like CM6 and Dm7.
::: Explanation of M2, m7, etc.: https://en.wikipedia.org/wiki/Interval_(music)#Main_intervals. Knowing these terms really helps with chord names like CM6 and Dm7.
::: N doesn't stand for anything. It's like x or y in algebra. But N isn't really important. The important thing is the pergen. I only wrote "Both fractions are always of the form 1/N" because I wanted to make clear that there is no "two-thirds-of-a-fifth" pergen. IOW the fraction always has a numerator of 1. If N still confuses you, I suggest you just ignore it. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 05:39, 16 April 2021 (UTC)
::: N doesn't stand for anything. It's like x or y in algebra. But N isn't really important. The important thing is the pergen. I only wrote "Both fractions are always of the form 1/N" because I wanted to make clear that there is no "two-thirds-of-a-fifth" pergen. IOW the fraction always has a numerator of 1. If N still confuses you, I suggest you just ignore it. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 05:39, 16 April 2021 (UTC)
== Kite's thoughts on canonical generators, which imply canonical mappings ==
The canonical generator of a rank-2 pergen is called the best-multigen generator. This is found by:
# minimizing the absolute value of the 2nd number in a multigen's monzo (e.g. P5/2 over M2/4 because P5 = (-1 1) and M2 = (-3 2)). I suspect this also minimizes the multigen's fraction.
# next, choosing the least-cents voicing of the multigen (e.g. P4/2 over P12/2 and P11/3 over P12/3)
# and finally, if the generator is P4, using P5 instead, to follow historical practice.
But a very popular canonical generator is the one that has the least cents. (x31eq.com's mappings always imply this generator.) If the least-cents generator differs from the best-multigen generator, it's always of the form G - nP or nP - G, where P = period, G = best-multigen generator, and n = round ((cents of G)/(cents of P)). A pergen that uses this alternate generator is called a least-cents pergen.
The best-multigen generator works better harmonically, and the least-cents generator works better melodically. For example with (P8/2, P5), MOS scales can be thought of as two familiar chains of 5ths, offset by a half-octave. Whereas with the corresponding least-cents pergen (P8/2, M2/2), MOS scales can be thought of as each half of the octave being filled in by a stack of semitones. But the best-multigen generator has the advantage that it simplifies the process of finding an ideal notation. This is why the best-multigen pergen is the canonical pergen.
The least-cents generator differs from the best-multigen generator in two cases. In the trivial case, the unsplit pergen becomes (P8, P4). The other case occurs sometimes but not always when the period is not an octave. The least-cents multigen in this case is always imperfect, because step #1 is skipped. PergenLister displays the least-cents generator's cents in the 3rd column. It usually displays this generator as a fraction of an imperfect multigen in the "Unreduced Pergen" column. However unreduced pergens #16 and #27 are not least-cents. This is because the purpose of this column is to find an alternate notation, not to find the least-cents generator.
'''Thus for many temperaments there are two canonical mappings''', one for each type of canonical generator. Except for the trivial case, both are useful and IMO the xenwiki should list both. For example, Sagugu aka Srutal is (P8/2, P5), which implies the mapping [(2 2 7) (0 1 -2)]. But the least-cents pergen (P8/2, M2/2) implies the mapping [(2 3 5) (0 1 -2)]. Another example: Trigu aka Augmented is (P8/3, P5) which implies [(3 3 4) (0 1 0)]. But the least-cents pergen (P8/3, m3/3) implies [(3 5 0) (0 -1 0)]. However any temperament in which the least-cents generator is the same as the best-multigen generator has only one canonical mapping, e.g. Zozo aka Semaphore which is (P8, P4/2).
This table shows the least-cents generator for pergens 1-32, if it differs from the best-multigen generator (asterisk indicates a true double):
{| class="wikitable" style="text-align:center;"
|+
! colspan="3" |best-multigen generator
! colspan="3" |least-cents generator
|-
! colspan="3" |unsplit pergen
! colspan="2" |generator
!cents
|-
!1
|(P8, P5)
|700 + c
|P - G
|P4
|500 - c
|-
! colspan="3" |half-split pergens
! colspan="3" |
|-
!2
|(P8/2, P5)
|700 + c
|G - P
|M2/2
|100 + c
|-
!3
|(P8, P4/2)
|250 - c/2
|
|
|
|-
!4
|(P8, P5/2)
|350 + c/2
|
|
|
|-
!5
|(P8/2, P4/2) *
|250 - c/2
|
|
|
|-
! colspan="3" |third-split pergens
! colspan="3" |
|-
!6
|(P8/3, P5)
|700 + c
|2P - G
|m3/3
|100 - c
|-
!7
|(P8, P4/3)
|167 - c/3
|
|
|
|-
!8
|(P8, P5/3)
|233 + c/3
|
|
|
|-
!9
|(P8, P11/3)
|567 - c/3
|
|
|
|-
!10
|(P8/3, P4/2)
|250 - c/2
|P - G
|M6/6
|150 + c/2
|-
!11
|(P8/3, P5/2)
|350 + c/2
|P - G
|m3/6
|50 - c/2
|-
!12
|(P8/2, P4/3)
|167 - c/3
|
|
|
|-
!13
|(P8/2, P5/3)
|233 + c/3
|
|
|
|-
!14
|(P8/2, P11/3)
|567 - c/3
|P - G
|M2/6
|33 + c/6
|-
!15
|(P8/3, P4/3) *
|167 - c/3
|
|
|
|-
! colspan="3" |quarter-split pergens
! colspan="3" |
|-
!16
|(P8/4, P5)
|700 + c
|G - 2P
|M3/4
|100 + c
|-
!17
|(P8, P4/4)
|125 - c/4
|
|
|
|-
!18
|(P8, P5/4)
|175 + c/4
|
|
|
|-
!19
|(P8, P11/4)
|425 - c/4
|
|
|
|-
!20
|(P8, P12/4)
|475 + c/4
|
|
|
|-
!21
|(P8/4, P4/2) *
|250 - c/2
|P - G
|M2/4
|50 + c/2
|-
!22
|(P8/2, M2/4)
|50 + c/2
|
|
|
|-
!23
|(P8/2, P4/4) *
|125 - c/4
|
|
|
|-
!24
|(P8/2, P5/4) *
|175 + c/4
|
|
|
|-
!25
|(P8/4, P4/3)
|167 - c/3
|P - G
|M10/12
|133 + c/3
|-
!26
|(P8/4, P5/3)
|233 + c/3
|P - G
|m6/12
|67 - c/3
|-
!27
|(P8/4, P11/3)
|567 - c/3
|2P - G
|M2/6
|33 + c/3
|-
!28
|(P8/3, P4/4)
|125 - c/4
|
|
|
|-
!29
|(P8/3, P5/4)
|175 + c/4
|
|
|
|-
!30
|(P8/3, P11/4)
|425 - c/4
|G - P
|m3/12
|25 - c/4
|-
!31
|(P8/3, P12/4)
|475 + c/4
|G - P
|M6/12
|75 + c/4
|-
!32
|(P8/4, P4/4) *
|125 - c/4
|
|
|
|}
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