Diasem: Difference between revisions
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Chain 2: '''Abv''' LS(L-S)SLM '''Ebv''' LSLML '''Bbv''' S(L-S)SLML '''Fv''' SLMLS(L-S) '''Cv''' | Chain 2: '''Abv''' LS(L-S)SLM '''Ebv''' LSLML '''Bbv''' S(L-S)SLML '''Fv''' SLMLS(L-S) '''Cv''' | ||
Notice that we have two generator chains of equal length. To give this scale a generator-offset structure we can treat the large 7-step as the offset of the 10-note scale. We treat L+S as one scale step and consider | Notice that we have two generator chains of equal length. To give this scale a generator-offset structure we can treat the large 7-step as the offset of the 10-note scale. We treat L+S as one scale step and consider the scale an interleaving of two pentatonic scales, using the notes of C-D-F-G-Bb for the even numbered notes and Cv-Ebv-Fv-Abv-Bbv for the odd ones. This gives the following ordering: C Cv D Ebv F Fv G Abv Bb Bbv C, or in step sizes, -s L+s M L+s -s L+s M L+s -s L+s. This is formally a [[blackdye]] (sL'ML's'L'ML's'L') pattern, albeit with a negative step size s' = -s! This scale has been called '''negative blackdye''' or '''negative-s blackdye'''. | ||
== In JI and similar tunings == | == In JI and similar tunings == | ||
Revision as of 00:01, 12 July 2022
Diasem is a 9-note max-variety-3, generator-offset scale with step pattern 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 diasem LMLSLMLSL and left-handed diasem LSLMLSLML; these 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. The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset a subminor third or a supermajor second. 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.
"Diasem" is a name given by ks26 (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.
| Name | Structure | Step Sizes | Graphical Representation |
|---|---|---|---|
| Semiquartal | 5L4s | 10\62, 3\62 | ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤ |
| Diasem | 5L2m2s | 10\62, 4\62, 2\62 | ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤ |
| Diatonic | 5L2s | 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 class | Sizes | 2.3.7 JI | 21edo (L:m:s = 3:2:1) | 31edo (L:m:s = 5:2:1) | |
|---|---|---|---|---|---|
| 1-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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.
Important 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.)
Modes
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. To minimize bias (e.g. towards certain tunings), this article names the modes after the semiquartal mode that the diasem mode is based on (the semiquartal modes are given in UDP notation). We also have provided diatonic-based names, which are useful in tunings where S is relatively small, such as 26edo, and 31edo, and superdiatonic-based names, which may be useful in tunings where L is close to M (such as 28edo).
Cyclic order
The modes arranged in cyclic order:
| Left-handed modes | Right-handed modes |
|---|---|
| LSLMLSLML LH Diasem 4|4 LH Diasem Mixo LH Diasem Superaeolian |
LMLSLMLSL RH Diasem 4|4 RH Diasem Aeolian RH Diasem Supermixo |
| SLMLSLMLL LH Diasem 0|8 LH Diasem Bright Aeolian LH Diasem Superlocrian |
MLSLMLSLL RH Diasem 0|8 RH Diasem Locrian RH Diasem Olympian |
| LMLSLMLLS LH Diasem 5|3 LH Diasem Dark Aeolian LH Diasem Superionian |
LSLMLSLLM RH Diasem 5|3 RH Diasem Ionian RH Diasem Superaeolian |
| MLSLMLLSL LH Diasem 1|7 LH Diasem Locrian LH Diasem Corinthian |
SLMLSLLML RH Diasem 1|7 RH Diasem Bright Dorian RH Diasem Superlocrian |
| LSLMLLSLM LH Diasem 6|2 LH Diasem Ionian LH Diasem Superdorian |
LMLSLLMLS RH Diasem 6|2 RH Diasem Dark Dorian RH Diasem Superionian |
| SLMLLSLML LH Diasem 2|6 LH Diasem Bright Dorian LH Diasem Superphrygian |
MLSLLMLSL RH Diasem 2|6 RH Diasem Phrygian RH Diasem Corinthian |
| LMLLSLMLS LH Diasem 7|1 LH Diasem Dark Dorian LH Diasem Superlydian |
LSLLMLSLM RH Diasem 7|1 RH Diasem Lydian RH Diasem Superdorian |
| MLLSLMLSL LH Diasem 3|5 LH Diasem Phrygian LH Diasem Supermixo |
SLLMLSLML RH Diasem 3|5 RH Diasem Bright Mixo RH Diasem Superphrygian |
| LLSLMLSLM LH Diasem 8|0 LH Diasem Lydian LH Diasem Olympian |
LLMLSLMLS RH Diasem 8|0 RH Diasem Dark Mixo RH Diasem Superlydian |
Arranged by generator chain
When we arrange the modes in the order given by rotating each mode by the generator (the perfect fifth) we obtain the following families of modes (">" roughly means 'brighter than'):
- RH
- LSLLMLSLM (RH Lydian) > LSLMLSLLM (RH Ionian) > SLLMLSLML (RH Bright Mixo) > SLMLSLLML (RH Bright Dorian)
- LLMLSLMLS (RH Dark Mixo) > LMLSLLMLS (RH Dark Dorian) > LMLSLMLSL (RH Aeolian) > MLSLLMLSL (RH Phrygian) > MLSLMLSLL (RH Locrian)
- LH
- LLSLMLSLM (LH Lydian) > LSLMLLSLM (LH Ionian) > LSLMLSLML (LH Mixo) > SLMLLSLML (LH Bright Dorian) > SLMLSLMLL (LH Bright Aeolian)
- LMLLSLMLS (LH Dark Dorian) > LMLSLMLLS (LH Dark Aeolian) > MLLSLMLSL (LH Phrygian) > MLSLMLLSL (LH Locrian)
This provides a clear motivation for the diatonic-based mode names.
Negative-s blackdye
Consider right-hand diasem, fixing a choice of positive step sizes. There exists a way of superimposing a left-hand diasem mode on the right hand diasem so that the right-hand diasem and the left-hand diasem overlap in 8 notes, yielding a scale of 10 notes, possibly after changing the mode of right-hand diasem. For example, superimposing LMLSLMLSL (RH Aeolian) and LMLSLMLLS (LH Dark Aeolian) gives LMLSLMLS(L-S)S (in fifth-based notation on C: C D Ebv Fv F G Abv Bbv Bb Cv C, where v denotes lowering by s). Note that the union is achiral. This new scale has two chains of perfect fifths each spanning 5 notes:
Chain 1: Bb (L-S)SLMLS F LMLS(L-S)S C LMLSL G MLS(L-S)SL D
Chain 2: Abv LS(L-S)SLM Ebv LSLML Bbv S(L-S)SLML Fv SLMLS(L-S) Cv
Notice that we have two generator chains of equal length. To give this scale a generator-offset structure we can treat the large 7-step as the offset of the 10-note scale. We treat L+S as one scale step and consider the scale an interleaving of two pentatonic scales, using the notes of C-D-F-G-Bb for the even numbered notes and Cv-Ebv-Fv-Abv-Bbv for the odd ones. This gives the following ordering: C Cv D Ebv F Fv G Abv Bb Bbv C, or in step sizes, -s L+s M L+s -s L+s M L+s -s L+s. This is formally a blackdye (sL'ML's'L'ML's'L') pattern, albeit with a negative step size s' = -s! This scale has been called negative blackdye or negative-s blackdye.
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.
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
Extensions
The diasem scale extends to a 14-note AG 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 AG extension of the Zarlino scale's AG pattern 3L 2m 2s, LmsLmLs. The 14-note extension 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 extension 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 could be called chromatic crossdye and 5L 2m 7s could be called dietic crossdye.
See also
- Blackdye, a similar diatonic detempering but for 2.3.5
Links
- Play JI diasem - Sevish Scale Workshop
- Play 26edo diasem - Sevish Scale Workshop