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Planar temperament tictactoe diagrams

🔗Gene Ward Smith <gwsmith@svpal.org>

6/13/2005 3:01:47 AM

If we have a planar temperament in the strict sense, meaning octaves
are the period, and a set of intervals in it, a tictactoe diagram is
one with an X where there is a note and an O where there is not, in a
grid arrangement defined by the generators. If the generators are the
ones given by Hermite reduction, and if the set of notes is the otonal
tetrad, you get a tictactoe diagram showing how the basic 7-limit
chords work, the utonal tetrad being the radial inverse. If we are
representing a linear temperament in a plane, more than one comma
gives the same generators, and so more than one configuration is
useful to know. For instance meantone in the 5-limit plane has both
the 225/224 and the 126/125 configuration, as well as others. Below
are tictactoe diagrams for some 7-limit commas leading to planar
temperaments. This sort of thing is particularly useful for Tonescape,
but I think is generally interesting. It should be clear looking at
this why 225/224 and 2401/2400 are so important as 7-limit planar
temperaments.

1029/1000

X X O O O
O X O O X

250/243

X O O O O O
X O O X O X

36/35

X O O
X X O
O O X

525/512

O X O
O X X
O O O
X O O

49/48

X O O
X X X

3125/3072

X O O O O O
X X O O O X

686/675

X O
O X
O O
X X

64/63

O O X O
X O X X

875/864

X O O O
X X O O
O O O O
O O O O
O O O X

81/80

X O O O O
X X O O X

3125/3087

O X
O O
O X
O O
O O
X X

2430/2401

O O O X
O O O O
O O O O
O O X O
X X O O

245/243

O X O
O O O
X X O
O O X

126/125

X O O O
O O O O
O O X O
O O X X

4000/3969

X O O O
O O X O
O O O O
O O X X

1728/1715

X O
O O
O O
X X
O X

1029/1024

O X O O O
X X O O X

15625/15552

X O O O O O O
X O O O O X X

225/224

O O X
X O O
X X O

19683/19600

X O O O O O O O O O
X O X O O O O O O O
O O O O O O O O O X

16875/16807

O O O X
O O O X
O O O O
O O O O
O O O O
X X O O

10976/10935

O O X O
O O O O
O O O X
X X O O

3136/3125

X O
O O
O O
X O
O O
X X

5120/5103

X O O O O O X O
O O O O O O X X

6144/6125

O O X
O O O
O X X
O O O
O O O
X O O

65625/65536

O X O
O X X
O O O
O O O
O O O
O O O
X O O

32805/32768

O O O O O O O O X O
X O O O O O O O X X

703125/702464

O O O X
O O O O
O O O O
O O O O
O X O O
O O O O
O O O O
X X O O

420175/419904

O O O O X
O O O O O
O O O O O
O O O O O
O O O O O
X X O O O
O O O O O
X O O O O

2401/2400

O X O
O X O
X O X

4375/4374

X O O O O O O O
X X O O O O O O
O O O O O O O O
O O O O O O O O
O O O O O O O O
O O O O O O O X

78125000/78121827

X O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O O
O O O O O O O O O O O O O X O O
O O O O O O O O O O O O O X O X