Topic: A cubic alternative to Canasta

1 scales

Thread (1 messages)

From: Gene Ward Smith (2004-02-15)
Subject: A cubic alternative to Canasta

If we take the 7-limit lattice of tetrads and considers shells of
tetrads around a major (or inverting, a minor) tetrad, we get shells
of size 1,6,12,8,6,24,24,0, ... (Note we have moats again, from the
well-known theorem that numbers of the form 8n-1 cannot be represented
as a sum of three squares.) If we take shells 0 to 2, we get 19
tetrads and 28 notes, adding shell 3 gives us four more notes, 32, but
eight more tetrads, for a total of 27 in a 3x3x3 cube. 

If we look at what ratios between scale degrees can approximate, we
find 2401/2400, 225/224, 1029/1024 as the three smallest commas giving
near 7-limit consonances, making these scales excellent candidates for
tempering by miracle. Doing so also exterminates an unwanted scale
interval of 2401/2400, so the 32 notes of the JI cube boil down to 31
for the miracle cube, the same number as Canasta. While I have seen no
great rush to compose in Canasta, this would be an alternative for
anyone thinking about it--me, for instance.

Here is the miracle version of the 19 tetrad scale, in terms of secors:

Shell 2 scale: [-20, -15, -14, -13, -12, -10, -9, -8, -7, -6, -5, -4,
-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 19]

Three more notes (not four, because of the tempering) gives us the
Miracle Cube Scale:

[-21, -20, -15, -14, -13, -12, -10, -9, -8, -7, -6, -5, -4, -3, -2,
-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 18, 19]

This doesn't beat Canasta in terms of quantity of chords, but it's
nearly as good and nicely arraged. It has an inverted version which
differs, so you get two versions of the scale. Here is the
major-tetrad centered version in TOP tuning:

! mircube.scl
Major harmonic cube of 27 tetrads in TOP miracle tuning
31
!
33.424588
66.849178
83.296052
116.720642
150.145230
200.016694
233.441286
266.865874
316.737338
350.161928
383.586516
433.457980
466.882568
500.307160
583.603212
617.027802
650.452390
700.323854
733.748442
767.173032
817.044498
850.469084
883.893676
900.340550
933.765140
967.189728
1000.614318
1017.061192
1083.910370
1150.759550
1200.631010