schismic
[Joe Monzo]
A qualifier used to describe temperaments which are based on the distribution of the schisma. For a detailed explanation, see Graham Breed's "Schismic Temperaments" page.
Pythagorean tuning can be considered one instance of a schismic temperament.
The standard 12-edo tuning is unique in that it is both a schismic (-1 schisma) and a meantone (-1/11-comma) temperament, meaning that it equates 5:4 [== 51] with both 81:64 [== 34] and 8192:6561 [== 3-8].
Some other well-known schismic temperaments are those of Helmholtz, Groven, and the "dinarra" tuning of Eduard Sabat-Garibaldi.
Not to be confused with schismic tuning, which see.
. . . . . . . . .
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family name
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schismic
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period
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octave
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generator
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fourth or fifth
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5-limit
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name
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schismic
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comma
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32805/32768
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mapping
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[<1 2 1|, <0 -1 8|]
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poptimal generator
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120/289
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MOS
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12, 17, 29, 41, 53, 65, 118, 171
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7-limit
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name
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schismic, pontiac, 118&171
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wedgie
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<<1 -8 39 -15 59 113||
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mapping
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[<1 2 -1 19|, <0 -1 8 -39|]
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7&9 limit copoptimal generator
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732/1763
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TM basis
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{4375/4374, 32805/32768}
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MOS
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12, 17, 29, 41, 53, 65, 118, 171
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name
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garibaldi, 41&53
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wedgie
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<<1 -8 -14 -15 -25 -10||
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mapping
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[<1 2 -1 -3|, <0 -1 8 14|]
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7&9 limit copoptimal generator
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39/94
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TOP period (cents):
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1200.760624 TOP generator: 498.119330
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TOP generator (cents):
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498.119330
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TM basis
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{225/224, 3125/3087}
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MOS
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12, 17, 29, 41, 53, 94
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|
name
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schism, 12&17
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wedgie
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<<1 -8 -2 -15 -6 18||
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mapping
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[<1 2 -1 2|, <0 -1 8 2|]
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7 limit poptimal generator
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27/65
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9 limit poptimal generator
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22/53
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TM basis
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{64/63, 360/343}
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MOS
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12, 17, 29, 41, 53
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|
name
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grackle, 65&77
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|
wedgie
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<<1 -8 -26 -15 -44 -38||
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mapping
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[<1 2 -1 -8|, <0 -1 8 26|]
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|
7 limit poptimal generator
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170/409
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9 limit poptimal generator
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133/320
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TM basis
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{126/125, 32805/32768}
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MOS
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12, 17, 29, 41, 53, 65, 77, 89
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11 limit
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name
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garibaldi, 41&53
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|
wedgie
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<<1 -8 -14 23 -15 -25 33 -10 81 113||
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mapping
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[<1 2 -1 -3 13|, <0 -1 8 14 -23|]
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poptimal generator
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95/229
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TM basis
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{225/224, 385/384, 2200/2187}
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MOS
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12, 17, 29, 41, 53, 94
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|
name
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garybald, 29&41
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|
wedgie
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<<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
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mapping
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[<1 2 -1 -3 -4|, <0 -1 8 14 18|]
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poptimal generator
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63/152
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TM basis
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{100/99, 225/224, 245/242}
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MOS
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12, 17, 29, 41, 70
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. . . . . . . . .
[Joe Monzo]
The relationships of the schismic family are shown below on a family-tree, where the top generation is 5-limit and each succeeding generation is the next higher prime-limit:
(grandmother)
helmholtz
[32805/32768]
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|
-----------------------------------------------------------------------------------------------
| | | |
| | | |
(?) (mother?) (illegitimate uncle?) (?)
pontiac garibaldi schism grackle
[32805/32768, 4375/4374] [225/224, 3125/3087] [64/63, 360/343] [32805/32768, 126/125]
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|
----------------------------------
| |
| |
(daughter) (daughter)
garibaldi garybald
[225/224, 385/384, 2200/2187] [100/99, 225/224, 245/242]
. . . . . . . . .
