Diasem

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Diasem (also denoted 2s in groundfault's aberrismic theory) is a 9-note max-variety-3, generator-offset scale with step signature 5L 2m 2s, equivalent to the semiquartal (5L 4s) mos with two of the small steps made larger and the other two made smaller. Diasem is chiral, with two rotationally non-equivalent variants: right-handed (RH) diasem LmLsLmLsL and left-handed (LH) diasem LsLmLsLmL; these step patterns are mirror images. The fact that the small step of diatonic is made smaller results in 26edo and 31edo diasem having better melodic properties than the respective diatonic scales. 21edo is the smallest edo to support a non-degenerate diasem.

Diasem can be tuned as a 2.3.7 subgroup JI scale or a tempered version thereof, where L represents 9/8, m represents 28/27, and s represents 64/63. This interpretation, or more generally the series of generator sequence scales generated by GS(7/6, 8/7) or GS(8/7, 7/6), has been named Tas (from Archytas) by Scott Dakota.

"Diasem" is a name given by groundfault (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "Semaphore") since its step sizes are intermediate between that of diatonic (5L 2s) and semiquartal (5L 4s); it is also a pun based on the diesis, which appears as the small step in the scale in the 31edo and 36edo tunings.

Comparison of diasem with semiquartal and diatonic in 62edo
Name Structure Step Sizes Graphical Representation
Semiquartal 5L 4s 10\62, 3\62 ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤
Diasem 5L 2m 2s 10\62, 4\62, 2\62 ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤
Diatonic 5L 2s 10\62, 6\62 ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤

Intervals

The following is a table of diasem intervals and their abstract sizes in terms of L, m and s. Given concrete sizes of L, m and s in edo steps or cents, you can compute the concrete size of any interval in diasem using the following expressions.

Interval sizes in diasem
Interval class Sizes 2.3.7 JI 21edo (L:m:s = 3:2:1) 31edo (L:m:s = 5:2:1)
1-steps small s 64/63, 27.26¢ 1\21, 57.14¢ 1\31, 38.71¢
medium m 28/27, 62.96¢ 2\21, 114.29¢ 2\31, 77.42¢
large L 9/8, 203.91¢ 3\21, 171.43¢ 5\31, 193.55¢
2-steps small L + s 8/7, 231.17¢ 4\21, 228.57¢ 6\31, 232.26¢
medium L + m 7/6, 266.87¢ 5\21, 285.71¢ 7\31, 270.97¢
large 2L 81/64, 407.82¢ 6\21, 342.86¢ 10\31, 387.10¢
3-steps small L + m + s 32/27, 294.14¢ 6\21, 342.86¢ 8\31, 309.68¢
medium 2L + s 9/7, 435.08¢ 7\21, 400.00¢ 11\31, 425.81¢
large 2L + m 21/16, 470.78¢ 8\21, 457.14¢ 12\31, 464.52¢
4-steps small 2L + m + s 4/3, 498.04¢ 9\21, 514.29¢ 13\31, 503.23¢
medium 3L + s 81/56, 638.99¢ 10\21, 571.43¢ 16\31, 619.35¢
large 3L + m 189/128, 674.69¢ 11\21, 628.57¢ 17\31, 658.06¢
5-steps small 2L + m + 2s 256/189, 525.31¢ 10\21, 571.43¢ 14\31, 541.94¢
medium 2L + 2m + s 112/81, 561.01¢ 11\21, 628.57¢ 15\31, 580.65¢
large 3L + m + s 3/2, 701.96¢ 12\21, 685.71¢ 18\31, 696.77¢
6-steps small 3L + m + 2s 32/21, 729.22¢ 13\21, 742.86¢ 19\31, 735.48¢
medium 3L + 2m + s 14/9, 764.92¢ 14\21, 800.00¢ 20\31, 774.19¢
large 4L + m + s 27/16, 905.87¢ 15\21, 857.14¢ 23\31, 890.32¢
7-steps small 3L + 2m + 2s 128/81, 792.18¢ 15\21, 857.14¢ 21\31, 812.90¢
medium 4L + m + 2s 12/7, 933.13¢ 16\21, 914.29¢ 24\31, 929.03¢
large 4L + 2m + s 7/4, 968.83¢ 17\21, 971.43¢ 25\31, 967.74¢
8-steps small 4L + 2m + 2s 16/9, 996.09¢ 18\21, 1028.57¢ 26\31, 1006.45¢
medium 5L + m + 2s 54/28, 1137.04¢ 19\21, 1085.71¢ 29\31, 1122.58¢
large 5L + 2m + s 63/32, 1172.74¢ 20\21, 1142.86¢ 30\31, 1161.29¢

The octave can be called the "perfect 9-step" in TAMNAMS.

Properties

Any diasem scale with positive step sizes has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:

  • > 4\7 if L > m + s
  • = 4\7 if L = m + s
  • < 4\7 if L < m + s

(This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.)

The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset L + m or L + s (respectively a flat minor third or a sharp major second in tunings of diasem with "reasonable" fifths and small s steps).


Alterations

  • Diasem Melodic Minor LmLsLLsLm

In JI and similar tunings

Like Superpyth, JI diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths. On the other hand, the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out 81/80 are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what Superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps.

As a Fokker block

2.3.7 JI diasem as a Fokker block

The 2.3.7 JI diasem scale can be viewed as a Fokker block living in the 2.3.7 octave-equivalent pitch class lattice. The x-axis goes along the 3 direction and the y-axis goes along the 7 direction.

The diagram shows the LmLsLmLsL mode. Each dot represents a pitch class of a note in the 2.3.7 lattice. All the notes of the mode are marked as solid purple dots. Notes of the lattice outside the mode are black hollow dots. The red dashed lines are separated by the chroma 49/48, and the blue dotted lines are separated by the chroma 567/512. Note that both 49/48 and 567/512 are tempered out by (the 2.3.7 patent val of) 9edo.

The notes of diasem form the {49/48, 567/512} Fokker block, which is a fundamental domain of the 2.3.7 pitch class lattice; it is possible to tile the entire infinite lattice with copies of right-hand diasem transLated by (49/48)m(567/512)n for integer m and n. Including any one of the other three points on the boundary (28/27, 147/128, or 64/63) instead of 9/8 aLso yields Fokker blocks, more specifically, modes of three of the other domes of diasem, and transLates of the parallelogram that do not have lattice points on the boundary lead to other domes of this Fokker block. However, only one other choice, 28/27, yields a diasem scale, and it yields the left-handed diasem mode mLLsLmLsL.

As a Fokker block, 2.3.7 JI diasem is aLso a product word scale, a product of the tempered 2.3.7 mosses Semaphore[9] (LsLsLsLsL) and septimal Mavila[9] (LLLsLLLsL).

Tunings

Diasem tunings
Tuning L:m:s Good JI approximations other comments Degrees of the mode LmLsLmLsL
1 2 3 4 5 6 7 8
2.3.7 subgroup interpretation 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9
JI 7.479:2.309:1 Just 7/6, 8/7, and 3/2 203.910 266.871 470.781 498.045 701.955 764.916 968.826 996.090
21edo 3:2:1 16/15, 23/16 and 39/32 171.429 285.714 457.143 514.286 685.714 800 971.429 1028.571
26edo 4:2:1 14/11, 8/7 and 11/8 184.615 276.923 461.538 507.692 692.308 784.615 969.231 1015.385
28edo 4:3:1 5/4 and 13/7 171.429 300 471.429 514.286 685.714 814.286 985.714 1028.571
30edo 4:3:2 13/8 superdiatonic fifth 160 280 440 520 680 800 960 1040
31edo 5:2:1 Pental thirds and 7/5 193.548 270.968 464.516 503.226 696.774 774.194 967.742 1006.452
33edo 5:3:1 9/7, 13/11 and 10/9 181.818 290.909 472.727 509.091 690.909 763.636 981.818 1018.182
35edo 5:3:2 171.429 274.286 445.714 514.286 685.714 788.571 960 1028.571
35edo 5:4:1 171.429 308.571 480 514.286 685.714 822.857 994.286 1028.571
36edo 6:2:1 Septimal thirds and 3/2 200.000 266.667 466.667 500.000 700.000 766.667 966.667 1000.000
37edo 5:4:2 35/32 superdiatonic fifth 162.162 291.892 454.054 518.919 681.081 810.811 972.973 1037.838
38edo 6:3:1 189.474 284.2105 473.684 505.263 694.737 789.474 978.947 1010.526
39edo 5:4:3 superdiatonic fifth 153.846 276.923 430.769 523.077 676.923 800 953.846 1046.154
40edo 6:3:2 180 270 450 510 690 780 960 1020
40edo 6:4:1 180 300 480 510 690 810 990 1020
41edo 7:2:1 204.878 263.415 468.293 497.561 702.439 760.976 965.854 995.122
42edo 6:5:1 171.429 314.286 485.714 514.286 685.714 828.571 1000 1028.571
43edo 7:3:1 195.349 279.07 474.419 502.326 697.674 781.395 976.744 1004.651
44edo 6:4:3 11/10 (and 9/7) superdiatonic fifth 163.636 272.727 436.364 518.182 681.818 790.909 954.5455 1036.364
44edo 6:5:2 11/10 (and 9/7) superdiatonic fifth 163.636 300 463.636 518.182 681.818 818.182 981.818 1036.364
45edo 7:3:2 186.667 266.667 453.333 506.667 693.333 773.333 960 1013.333
45edo 7:4:1 186.667 293.333 480 506.667 693.333 800 986.667 1013.333
46edo 6:5:3 Neogothic thirds superdiatonic fifth 156.522 286.9565 443.478 521.739 678.231 808.696 965.218 1043.418
46edo 8:2:1 Neogothic thirds gentle fifth 208.696 260.87 469.565 495.652 704.348 756.522 965.218 991.314
47edo 7:4:2 178.723 280.851 459.578 510.638 689.362 791.489 970.212 1021.277
47edo 7:5:1 178.723 306.383 485.106 510.638 689.362 817.021 995.744 1021.277
48edo 6:5:4 superdiatonic fifth 150 275 425 525 675 800 950 1050
48edo 8:3:1 superdiatonic fifth 200 275 475 500 700 775 975 1000
49edo 7:4:3 171.429 269.388 440.817 514.286 685.714 783.6735 955.102 1028.571
49edo 7:5:2 171.429 293.878 465.756 514.286 685.714 808.163 979.592 1028.571
49edo 7:6:1 171.429 318.367 489.796 514.286 685.714 832.653 1004.082 1028.571
50edo 8:3:2 192 264 456 504 696 768 960 1008
50edo 8:4:1 192 288 480 504 696 792 984 1008
51edo 7:5:3 superdiatonic fifth 164.706 282.353 447.059 517.647 682.353 800 964.706 1035.294
51edo 7:6:2 superdiatonic fifth 164.706 305.882 470.588 517.647 682.353 823.529 988.235 1035.294
52edo 8:5:1 184.615 300 484.615 507.692 692.308 807.692 992.308 1015.385
53edo 7:5:4 27/20 superdiatonic fifth 158.491 271.698 429.189 520.755 679.245 792.453 950.944 1041.509
53edo 7:6:3 27/20 superdiatonic fifth 158.491 294.34 452.831 520.755 679.245 815.094 973.585 1041.509

Tuning examples

LsLLmLsLm

An example in the RH Diasem Lydian mode LsLLmLsLm. (score)

14edo, L:M:S = 2:1:1 (degenerate; this is basic semiquartal)

16edo, L:M:S = 2:2:1 (degenerate; this is basic superdiatonic)

19edo, L:M:S = 3:1:1 (degenerate; this is hard semiquartal)

21edo, L:M:S = 3:2:1

23edo, L:M:S = 3:2:2 (degenerate; this is soft semiquartal)

24edo, L:M:S = 4:1:1 (degenerate; this is superhard semiquartal)

26edo, L:M:S = 4:2:1

28edo, L:M:S = 4:3:1

31edo, L:M:S = 5:2:1

33edo, L:M:S = 5:3:1

33edo, L:M:S = 5:2:2 (degenerate; this is semihard semiquartal)

35edo, L:M:S = 5:4:1

35edo, L:M:S = 5:3:2

36edo, L:M:S = 6:2:1

38edo, L:M:S = 6:3:1

41edo, L:M:S = 7:2:1

mLsLmLLsL

An example in LH Diasem Locrian mode mLsLmLLsL. (score)

14edo, L:M:S = 2:1:1 (degenerate; this is basic semiquartal)

16edo, L:M:S = 2:2:1 (degenerate; this is basic superdiatonic)

19edo, L:M:S = 3:1:1 (degenerate; this is hard semiquartal)

21edo, L:M:S = 3:2:1

24edo, L:M:S = 4:1:1 (degenerate; this is superhard semiquartal)

26edo, L:M:S = 4:2:1

28edo, L:M:S = 4:3:1

31edo, L:M:S = 5:2:1

33edo, L:M:S = 5:3:1

35edo, L:M:S = 5:4:1

36edo, L:M:S = 6:2:1

38edo, L:M:S = 6:3:1

41edo, L:M:S = 7:2:1

mLLsLmLsL

21edo, L:M:S = 3:2:1

26edo, L:M:S = 4:2:1

31edo, L:M:S = 5:2:1

Supersets

The diasem scale extends to a 14-note generator-offset scale: LmLsLmLsL and LsLmLsLmL both extend to the scale mcmcmsmcmcmsmc (c = L − m), with two 7-note mosses generated by the diasem's fifths separated by m. This scale is not chiral. This scale extends diasem like how blackdye is a 10-note non-chiral generator-offset superset of the Zarlino scale's AG pattern 3L 2m 2s, LmsLmLs. The 14-note superset is one of:

  • 5L 7m 2s (if m < c < L)
  • a 2-step modmos of 12L 2s (if c = m)
  • 7L 5m 2s (if s < c < m)
  • 7L 7s (if c = s)
  • 7L 2m 5s (if c < s).

5L 7m 2s must have a diatonic fifth, since L > 2m > m + s. The 31edo tuning (c = 3\31, m = 2\31, s = 1\31) of the scale is ideal for the 81/80-tempering 2.3.5.7 interpretation.

Another superset is scscsmscscsmsc, with c = L - s (5L 2m 7s if c > m). 31edo diasem yields 5L 2m 7s with step ratio 4:2:1.

Both these tunings, 5L 7m 2s and 5L 2m 7s, have been named crossdye ("crossed eyes" referring to the two copies of 5L 2s diatonic + "blackdye", courtesy of cellularAutomaton). 5L 7m 2s has been called chromatic crossdye, and 5L 2m 7s has been called dietic crossdye and whitedye.

2.3.7 JI diasem also has the following generator-offset, SV3 supersets:

  • a 19-note superset: mLsmsLmsmLsmsLmsmLs (5L 7m 7s), with L = 2187/2048, m = 28/27, and s = 64/63,
  • a 29-note superset: mLsmLmLsmLmLmsLmLmLsmLmLmsLmL (12L 12m 5s), with L = 28/27, m = 64/63, and s = 531441/524188.

See also

  • Blackdye, a similar diatonic detempering but for 2.3.5

Links

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