User:Kaiveran/Nupelog

Nupelog is a 7-note generator-offset scale with step pattern 2L 2M 3s (sMLssML), equivalent to the 2L 5s MOS with two of the small steps made larger. It occupies a broad triangle on the chart of MV3 scales, whose vertices are generators 0\1,1\2 (degenerate to 2edo),

Unanswered Questions:

• Is it chiral?
• What are the just intervals strongly implied by certain tunings?

"Nupelog" is a name given to this region by Kaiveran, reflecting both the pelogic family of temperaments, and the fact that several measured tunings of the Indonesian pelog have a very similar step pattern.

Intervals

The following is a table of blackdye 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 blackdye using the following expressions.

Interval sizes in blackdye

Interval class	Sizes	5-limit JI	32edo (L:m:s = 5:2:1)	34edo (L:m:s = 5:3:1)
1-step	s m L	81/80, 21.51¢ 16/15, 111.73¢ 10/9, 182.40¢	1\32, 37.50¢ 2\32, 75.00¢ 5\32, 187.50¢	1\34, 35.29¢ 3\34, 105.88¢ 5\34, 176.47¢
2-step	L + s L + m	9/8, 203.91¢ 32/27, 294.14¢	6\32, 225.00¢ 7\32, 262.50¢	6\34, 211.77¢ 8\34, 282.35¢
3-step	L + 2s L + m + s 2L + s 2L + m	729/640‚ 225.42¢ 6/5, 315.64¢ 5/4, 386.31¢ 320/243, 476.54¢	7\32, 262.50¢ 8\32, 300.00¢ 11\32, 412.50¢ 12\32, 450.00¢	7\34, 247.06¢ 9\34, 317.65¢ 11\34, 388.24¢ 13\34, 458.82¢
4-step	2L + 2s 2L + m + s	81/64, 407.82¢ 4/3, 498.04¢	12\32, 450.00¢ 13\32, 487.50¢	12\34, 423.53¢ 14\34, 494.12¢
5-step	2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s	27/20, 519.55¢ 64/45, 609.78¢ 45/32, 590.22¢ 40/27, 680.45¢	14\32, 525.00¢ 15\32, 562.50¢ 17\32, 637.50¢ 18\32, 675.00¢	15\34, 529.41¢ 17\34, 600.00¢ 17\34, 600.00¢ 19\34, 670.59¢
6-step	3L + m + 2s 3L + 2m + s	3/2, 701.96¢ 128/81, 792.18¢	19\32, 712.50¢ 20\32, 750.00¢	20\34, 705.88¢ 22\34, 776.47¢
7-step	3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s	243/160, 723.46¢ 8/5, 813.69¢ 5/3, 884.36¢ 1280/729, 974.58¢	20\32, 750.00¢ 21\32, 787.50¢ 24\32, 900.00¢ 25\32, 937.50¢	21\34, 741.18¢ 23\34, 811.77¢ 25\34, 882.35¢ 27\34, 952.94¢
8-step	4L + m + 3s 4L + 2m + 2s	27/16, 905.87¢ 16/9, 996.09¢	25\32, 937.50¢ 26\32, 975.00¢	26\34, 917.65¢ 28\34, 988.24¢
9-step	5L + 2m + s 5L + m + 2s 4L + 2m + 2s	9/5, 1017.60¢ 15/8, 1088.27¢ 160/81, 1178.49¢	27\32, 1012.50¢ 30\32, 1125.00¢ 31\32, 1162.50¢	29\34, 1023.53¢ 31\34, 1094.12¢ 33\34, 1164.71¢

The octave in blackdye can be called the "perfect 10-step" in TAMNAMS.

Modes

Cyclic order

The modes arranged in cyclic order: (Note: The mode names are based on the 5-limit JI interpretation; modes in a less JI-like tuning may differ greatly from what these names suggest.)

Pattern	Name
SLMLSLMLSL	Blackdye Bright Aeolian
LMLSLMLSLS	Blackdye Dark Aeolian
MLSLMLSLSL	Blackdye Locrian
LSLMLSLSLM	Blackdye Ionian
SLMLSLSLML	Blackdye Bright Dorian
LMLSLSLMLS	Blackdye Dark Dorian
MLSLSLMLSL	Blackdye Phrygian
LSLSLMLSLM	Blackdye Lydian
SLSLMLSLML	Blackdye Bright Mixo
LSLMLSLMLS	Blackdye Dark Mixo

By generator chain

Because blackdye has two chains of 5 fifth generators, it has two chains of 5 modes each separated by one generator:

1. LSLSLMLSLM (Lydian) > LSLMLSLSLM (Ionian) > LSLMLSLMLS (Dark Mixo) > LMLSLSLMLS (Dark Dorian) > LMLSLMLSLS (Dark Aeolian)
2. SLSLMLSLML (Bright Mixo) > SLMLSLSLML (Bright Dorian) > SLMLSLMLSL (Bright Aeolian) > MLSLSLMLSL (Phrygian) > MLSLMLSLSL (Locrian)

Properties

1. The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.
2. The fifth is:
   • sharper than 700¢, if L + 3S > 2M
   • equal to 700¢, if L + 3S = 2M
   • flatter than 700¢, if L + 3S < 2M

Tunings

Blackdye tunings

Tuning	L:M:S	Degrees of the mode SLMLSLMLSL
		1	2	3	4	5	6	7	8	9
5-Limit Interpretation		81/80	9/8	6/5	4/3	27/20	3/2	8/5	16/9	9/5
JI	8.481:5.195:1	21.506	203.910	315.641	498.045	519.551	701.955	813.686	996.090	1017.596
22edo	3:2:1	54.545	218.182	327.273	490.909	545.455	709.090	818.182	981.818	1036.364
27edo	4:2:1	44.444	222.222	311.111	488.889	533.333	711.111	800.000	977.778	1022.222
29edo	4:3:1	41.379	206.897	331.034	496.551	537.931	703.448	827.586	993.103	1034.483
32edo	4:3:2	75.000	225.000	337.500	487.500	562.500	712.500	825.000	975.000	1050.000
32edo	5:2:1	37.500	225.000	300.000	487.500	525.000	712.500	787.500	975.000	1012.500
34edo	5:3:1	35.294	211.765	317.647	494.117	529.412	705.882	811.764	988.235	1023.529
37edo	5:3:2	64.865	227.027	324.324	486.486	551.351	713.514	810.810	972.973	1037.838
36edo	5:4:1	33.333	200.000	333.333	500.000	533.333	700.000	833.333	1000.000	1033.333
39edo	5:4:2	61.538	215.385	338.462	492.308	553.846	707.692	830.769	984.615	1046.154
42edo	5:4:3	85.714	228.571	342.857	485.714	571.429	714.286	828.571	971.429	1057.143
39edo	5:4:2	61.538	215.385	338.462	492.308	553.846	707.692	830.769	984.615	1046.154
37edo	6:2:1	32.432	227.027	291.892	486.486	518.919	713.514	778.378	972.973	1005.405
39edo	6:3:1	30.769	215.385	307.692	492.308	523.077	707.692	800.000	984.615	1015.384
42edo	6:3:2	57.143	228.571	314.286	485.714	542.857	714.286	800.000	971.429	1028.571
41edo	6:4:1	29.268	204.878	321.951	497.561	526.829	702.439	819.512	995.122	1024.390
47edo	6:4:3	76.596	229.787	331.915	485.106	561.702	702.439	714.894	970.213	1046.809
42edo	7:2:1	28.571	228.571	285.714	485.714	514.286	714.286	771.429	971.429	1000.000
44edo	7:3:1	27.273	218.182	300.000	490.909	518.182	709.091	790.901	981.818	1009.091
46edo	7:4:1	26.087	208.696	313.043	495.652	521.739	704.348	808.696	991.304	1017.391
53edo	8:5:1	22.642	203.773	316.981	498.113	520.755	701.887	815.094	996.226	1018.868

Example

An example in Blackdye Bright Aeolian, SLMLSLMLSL (score)

22edo (L:M:S = 3:2:1)

27edo (L:M:S = 4:2:1)

29edo (L:M:S = 4:3:1)

32edo (L:M:S = 5:2:1)

34edo (L:M:S = 5:3:1)

36edo (L:M:S = 5:4:1)

Scala files

• SNS ((2/1, 3/2)-5, 10/9)-10

See also

• Diasem, a similar diatonic-like mos detempering but for 2.3.7