7-limit signatures

Recall that cubic lattice coordinates for 7-limit tetrads associate the
3-tuple of integers [a,b,c] with the major triad with root

3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2)

if a+b+c is even, and the minor tetrad with root

3^((-a+b+c-1)/2) 5^((a-b+c+1)/2) 7^((a+b-c+1)/2)

if a+b+c is odd. This means that [2,0,0], [0,2,0], [0,0,2] represent the

major tetrads with roots 5*7/3, 3*7/5, 3*5/7 respectively; when octave
reduced these are 35/24, 21/20, and 15/14.

If L is a wedgie for a 7-limit linear temperament, we may define the
*signature* of L as S = [-L[1]+L[2]+L[3], L[1]-L[2]+L[3],
L[1]+L[2]-L[3]].
This is a 3-tuple representing the number of generator steps in the
octave
plus generator formulation of the temperament for 35/24, 21/20, 15/14
respectively, weighted by the number of periods to the octave. In the
case
where the octave is the period, it uniquely defines the tetrad in terms
of
steps by sending the tetrad [a,b,c] to S[1]*a + S[2]*b + S[3]*c steps.
For
example, taking the meantone wedgie of [1,4,10,12,-13,4] gives us a
signature of [13,-7,5], so the minor tonic tetrad [-1,0,0] is sent to -13
steps,
the dominant major tetrad [0,1,1] to -2 steps, and so forth; for major
tetrads these steps are twice the number of generator steps for the root
of
the tetrad, while the minor tetrads fill in the gaps in ways which depend

on the temperament--for instance, here we get [-1,1,-1] ~ [1,-2,0] at -1
step, equivalent under 126/125.

Just as temperaments with a generator which is a consonant interval are
of
particular interest, temperaments where one of the signature values is
+-1
are of interest, with miracle, whose signature is [-15,11,1] an example.

In this case the ordering of tetrads by steps corresponds to a chain of
adjacent tetrads in the lattice, so the step ordering is of particular
interest. Miracle now relates [-1,0,0] not just to 15 steps, but to the
tetrad [0,0,15], and [0,1,1] to [0,0,12], and so forth. This helps to
keep
track of the connectivity of the tetrads when using miracle. Moreover,
we
may define miracle MOS in terms of tetrads--Blackjack for instance can be

described as a chain of sixteen consecutive [0,0,n] tetrads, where n
starts
from an even number (representing a major tetrad) and runs up to an odd
number (minor tetrad.) For example, the chain from [0,0,0] (major tonic)

to [0,0,15] (minor tonic.)

Here is a list of temperaments with this unital signature property:

[[1, 1, 3, 3], [0, 6, -7, -2]] [6, -7, -2, 15, 20, -25] Miracle

generators [1200., 116.5729472] signatures [-15, 11, 1]

rms 1.637405196 comp 24.92662917 bad 1017.380173

ets [10, 21, 31, 41, 72, 103]

[[1, 0, -4, 6], [0, 1, 4, -2]] [1, 4, -2, -16, 6, 4] Dominant seventh

generators [1200., 1902.225977] signatures [1, -5, 7]

rms 20.16328150 comp 9.836559603 bad 1950.956872

ets [5, 7, 12]

[[1, 1, 2, 3], [0, 9, 5, -3]] [9, 5, -3, -21, 30, -13]
Quartaminorthirds

generators [1200., 77.70708739] signatures [-7, 1, 17]

rms 3.065961726 comp 27.04575317 bad 2242.667500

ets [15, 16, 31, 46]

[[1, 1, 1, 2], [0, 8, 18, 11]] [8, 18, 11, -25, 5, 10] Octafifths

generators [1200., 88.14540671] signatures [21, 1, 15]

rms 2.064339812 comp 34.23414357 bad 2419.357925

ets [27, 41, 68]

[[1, 2, 2, 3], [0, 4, -3, 2]] [4, -3, 2, 13, 8, -14] Tertiathirds

generators [1200., -125.4687958] signatures [-5, 9, -1]

rms 12.18857055 comp 14.72969740 bad 2644.480844

ets [1, 9, 10, 19, 29]

[[1, 0, 7, -5], [0, 1, -3, 5]] [1, -3, 5, 20, -5, -7] Hexadecimal

generators [1200., 1873.109081] signatures [1, 9, -7]

rms 18.58450012 comp 12.33750942 bad 2828.823679

ets [7, 9, 16]

[[1, 25, -31, -8], [0, 26, -37, -12]] [26, -37, -12, 76, 92, -119]

generators [1200., -1080.705187] signatures [-75, 51, 1]

rms .2219838332 comp 118.1864167 bad 3100.676640

ets [10, 171, 513]

[[1, 3, 6, 5], [0, 20, 52, 31]] [20, 52, 31, -74, 7, 36]

generators [1200., -84.87642563] signatures [63, -1, 41]

rms .3454637898 comp 96.52895120 bad 3218.975773

ets [99, 212, 311, 410]

[[1, 2, 2, 2], [0, 5, -4, -10]] [5, -4, -10, -12, 30, -18]

generators [1200., -97.68344522] signatures [-19, -1, 11]

rms 6.041345016 comp 24.27272426 bad 3559.349900

ets [12, 37]

[[1, 3, 0, 2], [0, 14, -23, -8]] [14, -23, -8, 46, 52, -69]

generators [1200., -121.1940013] signatures [-45, 29, -1]

rms .8353054234 comp 68.53846955 bad 3923.865443

ets [10, 99]

[[1, 12, 15, 1], [0, 23, 28, -4]] [23, 28, -4, -88, 71, -9]

generators [1200., -543.2692838] signatures [1, -9, 55]

rms .7218691130 comp 78.22290415 bad 4416.989140

ets [53]

[[1, 2, 3, 4], [0, 5, 8, 14]] [5, 8, 14, 10, -8, 1]

generators [1200., -102.3994286] signatures [17, 11, -1]

rms 8.609470174 comp 22.70605087 bad 4438.739304

ets [12]

[[1, 2, 1, 1], [0, 6, -19, -26]] [6, -19, -26, -7, 58, -44]

generators [1200., -83.37933102] signatures [-51, -1, 13]

rms 1.487254275 comp 55.50097036 bad 4581.275174

ets [29, 72]

[[1, 43, -58, -17], [0, 46, -67, -22]] [46, -67, -22, 137, 164, -213]

generators [1200., -1080.392876] signatures [-135, 91, 1]

rms .1267147296 comp 211.5126443 bad 5668.912722

ets [10, 301, 311, 612]

[[1, 2, 3, 3], [0, 6, 10, 3]] [6, 10, 3, -21, 12, 2]

generators [1200., -82.00647655] signatures [7, -1, 13]

rms 12.62928610 comp 21.39334917 bad 5780.113425

ets [15, 29]

[[1, 2, 1, 2], [0, 4, -13, -8]] [4, -13, -8, 18, 24, -30]

generators [1200., -122.3321832] signatures [-25, 9, -1]

rms 6.403982242 comp 31.21994593 bad 6241.865585

ets [10]

[[1, 1, 2, 2], [0, 4, 2, 5]] [4, 2, 5, 6, 3, -6]

generators [1200., 187.6316444] signatures [3, 7, 1]

rms 47.68000484 comp 11.69073209 bad 6516.579639

ets [6]

[[1, 0, -3, 6], [0, 3, 10, -6]] [3, 10, -6, -42, 18, 9]

generators [1200., 638.4642643] signatures [1, -13, 19]

rms 9.885351494 comp 25.98120378 bad 6672.839126

ets [15]

[[1, 2, 3, 3], [0, 5, 8, 2]] [5, 8, 2, -18, 11, 1]

generators [1200., -100.0317906] signatures [5, -1, 11]

rms 21.64417648 comp 17.58481613 bad 6692.936885

ets [12]

[[1, 2, 5, 6], [0, 4, 26, 31]] [4, 26, 31, -1, -38, 32]

generators [1200., -123.5352658] signatures [53, 9, -1]

rms 2.267858844 comp 56.46645397 bad 7230.978171

ets [29, 68]

[[1, 2, 2, 3], [0, 5, -4, 2]] [5, -4, 2, 16, 11, -18]

generators [1200., -99.19646785] signatures [-7, 11, -1]

rms 21.21541236 comp 18.58251802 bad 7325.893533

ets [1, 12]

[[1, 3, 2, 4], [0, 13, -3, 11]] [13, -3, 11, 34, 19, -35]

generators [1200., -130.2049690] signatures [-5, 27, -1]

rms 4.481233722 comp 41.46170034 bad 7703.566083

ets [9, 37, 46]

[[1, 12, 10, 5], [0, 19, 14, 4]] [19, 14, 4, -30, 47, -22]

generators [1200., -657.8863907] signatures [-1, 9, 29]

rms 3.032624788 comp 52.44877824 bad 8342.369709

ets [31]

[[1, 23, -56, 83], [0, 47, -128, 176]] [47, -128, 176, 768, -147, -312]

generators [1200., -546.7680257] signatures [1, 351, -257]

rms .3610890892e-1 comp 481.2637469 bad 8363.357505

ets [1578]

[[1, 13, 17, -1], [0, 21, 27, -7]] [21, 27, -7, -92, 70, -6]

generators [1200., -652.3887024] signatures [-1, -13, 55]

rms 1.469925034 comp 75.92946624 bad 8474.535049

ets [46, 57, 103]

[[1, 2, 4, 5], [0, 4, 16, 21]] [4, 16, 21, 4, -22, 16]

generators [1200., -125.5372720] signatures [33, 9, -1]

rms 6.562501740 comp 35.99263747 bad 8501.523814

ets [19]

[[1, 1, 3, 4], [0, 7, -8, -14]] [7, -8, -14, -10, 42, -29]

generators [1200., 101.5775171] signatures [-29, 1, 13]

rms 7.012328960 comp 35.52454740 bad 8849.513343

ets [12]

[[1, 2, -1, -1], [0, 6, -48, -55]] [6, -48, -55, 7, 104, -90]

generators [1200., -83.05774075] signatures [-109, -1, 13]

rms .6644554968 comp 115.7156146 bad 8897.127847

ets [29, 130]

[[1, 2, 3, 3], [0, 7, 11, 3]] [7, 11, 3, -24, 15, 1]

generators [1200., -73.16557361] signatures [7, -1, 15]

rms 16.40779159 comp 24.26315309 bad 9659.276719

ets [16]

[[1, 2, 3, 3], [0, 8, 13, 4]] [8, 13, 4, -27, 16, 2]

generators [1200., -63.00613990] signatures [9, -1, 17]

rms 12.64637740 comp 28.07029990 bad 9964.608569

ets [19]

It might be remarked that the signatures with the middle-sized (in
absolute value) components relatively small are an interesting subclass
of
these unital signature temperaments; they are associated with certain
planar temperaments of a kind not usually considered. Examples are
[-5,27,-1],
covered by 46, [-109,-1,13], covered by 130, and [1,-9,55], covered
(though
not very well) by 53.