Decic

Decic is a temperament for the 7, 11, 13, and 17 prime limits. It is a member of marvel temperaments, cloudy clan, and linus temperaments. It has a period of 1/10 octave and tempers out 225/224 and 16807/16384. The fifth of decic in size is a meantone fifth, but four of them are not used to reach the 5th harmonic. Instead, 14/13, 15/14 and 16/15 are equated to 1/10 of an octave, and from this it derives its name. Not only the meantone fifth (flat 3/2) or fourth (sharp 4/3), but also the magic major third (flat 5/4) can be used as a generator.

Temperament data

Decic temperament (10&50)

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨10 16 23 28 35 37 41], ⟨0 1 -1 0 2 0 1]]

POTE generators:

• 7-limit: ~3/2 = 698.69596
• 11-limit: ~3/2 = 696.79119
• 13-limit: ~3/2 = 696.99342
• 17-limit: ~3/2 = 697.08527

TOP generators:

• 7-limit: ~15/14 = 120.18411, ~3/2 = 699.76795
• 11-limit: ~15/14 = 120.14165, ~3/2 = 697.61366
• 13-limit: ~14/13 = 120.11775, ~3/2 = 697.67733
• 17-limit: ~14/13 = 120.12744, ~3/2 = 697.82559

Diamond monotone ranges:

• 7-odd-limit: ~3/2 = [680.00000, 720.00000] (34\60 to 6\10)
• 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
• 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)

Diamond tradeoff ranges:

• 7 and 9-odd-limit: ~3/2 = [693.12909, 702.51219]
• 11, 13, and 15-odd-limit: ~3/2 = [689.36294, 702.51219]
• 17-odd-limit: ~3/2 = [689.36294, 704.95541]

Diamond monotone and tradeoff ranges:

• 7-odd-limit: ~3/2 = [693.12909, 702.51219]
• 9-odd-limit: ~3/2 = [696.00000, 702.51219]
• 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]

Optimal GPV sequences:

• 7-limit: 10, 30b, 40, 50, 60, 110d, 170cdd
• 11 and 13-limit: 10, 30be, 40, 50
• 17-limit: 10, 30beg, 40, 50

Badness:

• 7-limit: 0.089135
• 11-limit: 0.063900
• 13-limit: 0.036880
• 17-limit: 0.025064

Interval chains

Intervals of decic

Generator		-3	-2	-1	0	1	2	3
Period 0	Cents*	1131.256	1154.171	1177.085	0.000	22.915	45.829	68.744
	Ratios				1/1		40/39	28/27, 25/24
Period 1	Cents*	51.256	74.171	97.085	120.000	142.915	165.829	188.744
	Ratios	36/35, 33/32	22/21, 21/20	18/17, 17/16	16/15, 15/14, 14/13	13/12, 12/11		10/9
Period 2	Cents*	171.256	194.171	217.085	240.000	262.915	285.829	308.744
	Ratios	11/10	9/8	17/15	8/7, 15/13	7/6	20/17, 13/11
Period 3	Cents*	291.256	314.171	337.085	360.000	382.915	405.829	428.744
	Ratios		6/5	17/14	11/9, 16/13, 21/17	26/21, 5/4	14/11
Period 4	Cents*	411.256	434.171	457.085	480.000	502.915	525.829	548.744
	Ratios		9/7	22/17, 13/10, 17/13, 21/16		4/3	15/11
Period 5	Cents*	531.256	554.171	577.085	600.000	622.915	645.829	668.744
	Ratios		11/8, 18/13	7/5	24/17, 17/12	10/7	13/9, 16/11

* in 17-limit POTE tuning

Scales

• Decic50