Shells of 7-limit note-classes

If we say a note-class belongs to shell n if its symmetrical lattice
distance from the unison is sqrt(n), then the first nine shells are
listed below. At shell nine, a new phenomenon emerges--there are two
geometrically distinct (and algebraically distinguishable, using
invariants) kinds of notes in shell nine. At shell 14, we get another
interesting phenomenon--there are no notes in shell 14. In fact, it is
a moat! We also get moats at shell 46, shell 56 etc. If we regard what
shell a note belongs in as a measure of its complexity, note that
126/125 and 225/224 (which together generate meantone) are less
complex by this measure than 81/80--and in fact, 2401/2400, lying in
shell 11, is less complex than 81/80 in shell 13.

Shell 2 is interesting because of the special intervals 36/24, 21/20
and 15/14 which are so important when looking at the lattice of chords.
Shell 7 has both 64/63 and 126/125 in it, a fact I exploited in a
piece which progressed from 126/125-planar (Starling) to 64/63-planar
and back again.

Shell 1: 7-limit consonances
{12/7, 3/2, 5/3, 7/5, 8/7, 7/6, 6/5, 5/4, 4/3, 7/4, 8/5, 10/7}

Shell 2: tetrad lattice generators
{48/35, 21/20, 28/15, 40/21, 35/24, 15/14}

Shell 3: conjugates of 9/7, as well as 10/9, 16/15, 25/24, 36/35 and 49/48
{32/21, 28/25, 42/25, 48/25, 64/35, 49/40, 80/49, 35/32, 25/24, 16/15,
36/35, 49/48, 9/5, 9/7, 10/9, 25/14, 96/49, 21/16, 14/9, 60/49, 35/18,
15/8, 25/21, 49/30}

Shell 4: squares of consonances
{32/25, 64/49, 49/32, 49/25, 25/16, 25/18, 50/49, 9/8, 72/49, 16/9,
49/36, 36/25}

Shell 5: conjugates of 9/8 and 50/49
{147/80, 200/147, 75/56, 288/175, 63/50, 288/245, 63/40, 56/45,
175/96, 45/28, 90/49, 192/175, 80/63, 75/49, 245/144, 147/100, 100/63,
175/144, 49/45, 112/75, 98/75, 384/245, 160/147, 245/192}

Shell 6 conjugates of 105=3*5*7
{125/84, 480/343, 105/64, 168/125, 54/35, 35/27, 128/105, 343/240}

Shell 7: conjugates of 28/27, 64/63, 126/125 and 27/25
{384/343, 125/72, 54/49, 27/14, 147/128, 28/27, 256/245, 224/125,
196/125, 343/288, 640/343, 126/125, 216/175, 245/128, 64/63, 576/343,
360/343, 63/32, 147/125, 256/175, 400/343, 343/200, 175/128, 27/25,
40/27, 245/216, 144/125, 250/147, 125/96, 432/245, 45/32, 27/20,
125/98, 128/75, 49/27, 125/112, 343/300, 50/27, 343/320, 175/108,
125/63, 64/45, 192/125, 343/180, 600/343, 75/64, 256/147, 343/192}

Shell 8: squares of tetrad lattice generators
{2304/1225, 441/400, 800/441, 1225/1152, 225/196, 392/225}

Shell 9: there are two varities of shell-nine note-classes

9a: cubes of consonances, including 128/125
{432/343, 27/16, 128/125, 216/125, 32/27, 343/216, 343/250, 512/343,
343/256, 125/108, 500/343, 125/64}

9b: conjugates of 225/224
s92 := {1152/875, 441/250, 225/224, 1600/1029, 1225/768, 375/196,
1728/1225, 189/100, 392/375, 1225/864, 200/189, 343/225, 135/98,
500/441, 875/576, 640/441, 1715/1152, 196/135, 2304/1715, 441/320,
1536/1225, 1029/800, 450/343, 448/225}

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Shell 2 is interesting because of the special intervals 36/24, 21/20
> and 15/14 which are so important when looking at the lattice of chords.

35/24, 21/20 and 15/14--5*7/3, 3*7/5 and 3*5/7.