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Kite's thoughts on pergens: Difference between revisions

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added an addenda for late 2024
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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.  
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.  


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.{{todo|expand|comment=apply to multi-comma temperaments|inline=1}}


== Addenda (late 2024) ==


=== Frequency of imperfect pergens ===
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depend on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings  [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%.


{{todo|expand|comment=apply to multi-comma temperaments|inline=1}}
This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect.
{| class="wikitable"
|+Pergens of the form (P8, x), showing generator and mapping (a = 1)
!unsplit
!half-splits
!third-splits
!quarter-splits
!fifth-splits
!sixth-splits
|-
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (1 1 1)
|P4/2 (1 2 -2)
|P4/3 (1 2 -3)
|P4/4 (1 2 -4)
|P4/5 (1 2 -5)
|P4/6 (1 2 -6)
|-
|
|P5/2 (1 1 2)
|P5/3 (1 1 3)
|P5/4 (1 1 4)
|P5/5 (1 1 5)
|P5/6 (1 1 6)
|-
|
|
|P11/3 (1 3 -3)
|P11/4 (1 3 -4)
|P11/5 (1 3 -5)
|P11/6 (1 3 -6)
|-
|
|
|
|P12/4 (1 0 4)
|P12/5 (1 0 5)
|P12/6 (1 0 6)
|-
|
|
|
|
|ccP4/5 (1 4 -5)
|ccP4/6 (1 4 -6)
|-
|
|
|
|
|
|ccP5/6 (1 -1 6)
|}
Of all the half-octave pergens, half of every other column (i.e. 25%) are imperfect. Imperfect pergens occur whenever b is not a multiple of a.
{| class="wikitable"
|+Pergens of the form (P8/2, x), showing generator and mapping (a = 2)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (2 2 1)
|'''M2/4 (2 3 2)'''
|P4/3 (2 4 -3)
|'''M2/8 (2 3 4)'''
|P4/5 (2 4 -5)
|'''M2/12 (2 3 6)'''
|-
|
|P4/2 (2 4 -2)
|P5/3 (2 2 3)
|P4/4 (2 4 -4)
|P5/5 (2 2 5)
|P4/6 (2 4 -6)
|-
|
|
|P11/3 (2 6 -3)
= '''M2/6'''
|P5/4 (2 2 4)
|P11/5 (2 6 -5)
|P5/6 (2 2 6)
|-
|
|
|
|'''cm7/8 (2 5 -4)'''
|P12/5 (2 0 5)
|P11/6 (2 6 -6)
|-
|
|
|
|
|ccP4/5 (2 8 -5)
|'''cm7/12 (2 5 -6)'''
|-
|
|
|
|
|
|'''cM9/12 (2 1 6)'''
|}
Note that some of these pergens, when put in mingen form, become imperfect. For example, (P8/2, P11/3) becomes (P8/2, M2/6). Also note that for many of these pergens, the generators are comma-sized, and MOS scales will either be very "hard" (L/s very large) or else will contain very many notes per octave. For example, to bring the L/s ratio down to about 5, (P8/2, M2/4) needs a 16 note scale, and (P8/2, P11/3) needs a 28 note scale!
 
Of all the third-octave pergens, two-thirds of every third column (2/9 or 22%) are imperfect:
{| class="wikitable"
|+Pergens of the form (P8/3, x), showing generator and mapping (a = 3)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (3 3 1)
|P4/2 (3 6 -2)
|'''m3/9 (3 5 -3)'''
|P4/4 (3 6 -4)
|P4/5 (3 6 -5)
|'''m3/18 (3 5 -6)'''
|-
|
|P5/2 (3 3 2)
|'''M6/9 (3 4 3)'''
|P5/4 (3 3 4)
|P5/5 (3 3 5)
|'''M6/18 (3 4 6)'''
|-
|
|
|P4/3 (3 6 -3)
|P11/4 (3 9 -4)
|P11/5 (3 9 -5)
|P4/6 (3 6 -6)
|-
|
|
|
|P12/4 (3 0 4)
|P12/5 (5 0 5)
|P5/6 (3 3 6)
|-
|
|
|
|
|ccP4/5 (3 12 -5)
|'''ccm3/18 (3 7 -6)'''
|-
|
|
|
|
|
|'''ccM6/18 (3 2 6)'''
|}
Of all the quarter-octave pergens, imperfection occurs in half of every 4th column and 3/4 of every 4th column (5/16 or 31.25%).
{| class="wikitable"
|+Pergens of the form (P8/4, x), showing generator and mapping (a = 4)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
!c = ±7
!c = ±8
|-
|P5 (4 4 1)
|'''m6/8 (4 7 -2)'''
|P4/3 (4 8 -3)
|'''M2/8 (4 6 4)'''
|P4/5 (4 8 -5)
|P4/6 (4 8 -6)
|P4/7 (4 8 -7)
|'''M2/16 (4 6 8)'''
|-
|
|P4/2 (4 8 -2)
'''M2/4'''
|P5/3 (4 4 3)
|'''m6/16 (4 7 -4)'''
|P5/5 (4 4 5)
|P5/6 (4 4 6)
|P5/7 (4 4 7)
|'''m6/32 (4 7 -8)'''
|-
|
|
|P11/3 (4 12 -3)
|'''M10/16 (4 5 4)'''
|P11/5 (4 12 -5)
'''m6/20'''
|P11/6 (4 12 -6)
|P11/7 (4 12 -7)
|'''M10/32 (4 5 8)'''
|-
|
|
|
|P4/4 (4 8 -4)
|P12/5 (4 0 5)
|'''m6/24 (4 7 -6)'''
|P12/7 (4 0 7)
|'''c<sup>3</sup>M3/32 (4 3 8)'''
|-
|
|
|
|
|ccP4/5 (4 16 -5)
|'''M10/24 (4 5 6)'''
|ccP4/7 (4 16 -7)
|'''ccm6/32 (4 9 -8)'''
|-
|
|
|
|
|
|'''ccm6/24 (4 9 -6)'''
|ccP5/7 (4 -4 7)
|'''cm7/16 (4 10 -8)'''
|-
|
|
|
|
|
|
|c<sup>3</sup>P4/7 (4 20 -7)
|P4/8 (4 8 -8)
|-
|
|
|
|
|
|
|
|P5/8 (4 4 8)
|}
Percentage of imperfect pergens in each category:
{| class="wikitable"
|+
!(P8, x)
!(P8/2, x)
!(P8/3, x)
!(P8/4, x)
!(P8/5, x)
!(P8/6, x)
!(P8/7, x)
|-
|none
|1/4
|2/9
|5/16
|4/25
|5/12
|6/49
|-
|0%
|25%
|22.22%
|31.25%
|16%
|41.67%
|12.24%
|}
[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
[[Category:Notation]]
[[Category:Notation]]
[[Category:Pages with proofs]]
[[Category:Pages with proofs]]
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