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Encyclopedia of Microtonal Music Theory

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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.

. . . . . . . . .
[Gene Ward Smith, Yahoo tuning-math message 10877 (Sat Jul 17, 2004 1:13 am)]

family name schismic
period octave
generator fourth or fifth
5-limit

name schismic
comma 32805/32768
mapping [<1 2 1|, <0 -1 8|]
poptimal generator 120/289
MOS 12, 17, 29, 41, 53, 65, 118, 171
7-limit

name schismic, pontiac, 118&171
wedgie <<1 -8 39 -15 59 113||
mapping [<1 2 -1 19|, <0 -1 8 -39|]
7&9 limit copoptimal generator 732/1763
TM basis {4375/4374, 32805/32768}
MOS 12, 17, 29, 41, 53, 65, 118, 171

name garibaldi, 41&53
wedgie <<1 -8 -14 -15 -25 -10||
mapping [<1 2 -1 -3|, <0 -1 8 14|]
7&9 limit copoptimal generator 39/94
TOP period (cents): 1200.760624 TOP generator: 498.119330
TOP generator (cents): 498.119330
TM basis {225/224, 3125/3087}
MOS 12, 17, 29, 41, 53, 94

name schism, 12&17
wedgie <<1 -8 -2 -15 -6 18||
mapping [<1 2 -1 2|, <0 -1 8 2|]
7 limit poptimal generator 27/65
9 limit poptimal generator 22/53
TM basis {64/63, 360/343}
MOS 12, 17, 29, 41, 53

name grackle, 65&77
wedgie <<1 -8 -26 -15 -44 -38||
mapping [<1 2 -1 -8|, <0 -1 8 26|]
7 limit poptimal generator 170/409
9 limit poptimal generator 133/320
TM basis {126/125, 32805/32768}
MOS 12, 17, 29, 41, 53, 65, 77, 89
11 limit

name garibaldi, 41&53
wedgie <<1 -8 -14 23 -15 -25 33 -10 81 113||
mapping [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
poptimal generator 95/229
TM basis {225/224, 385/384, 2200/2187}
MOS 12, 17, 29, 41, 53, 94
name garybald, 29&41
wedgie <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
mapping [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
poptimal generator 63/152
TM basis {100/99, 225/224, 245/242}
MOS 12, 17, 29, 41, 70
. . . . . . . . .
[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]
                                                    |
                                                    |
           -----------------------------------------------------------------------------------------------
           |                         |                                           |                       |
           |                         |                                           |                       |
          (?)                    (mother?)                             (illegitimate uncle?)            (?)
        pontiac                   garibaldi                                   schism                  grackle
[32805/32768, 4375/4374]     [225/224, 3125/3087]                        [64/63, 360/343]      [32805/32768, 126/125]
                                     |
                                     |
               ----------------------------------
               |                                |
               |                                |
           (daughter)                       (daughter)
           garibaldi                         garybald
  [225/224, 385/384, 2200/2187]      [100/99, 225/224, 245/242]