User:Dummy index

• 2-dimensional tuning spectrum
• 3rd-tritave temperaments
• About harmonics
• Bimetallic MOS
• Chromatic pairs and how we define haplotonic
• Heuristics for picking a nonstandard basis of JI subgroup
• Just odd intonation
• Layouts
• MOS tree with continued fraction
• No-1s odd-limit consistency
• Random lists of temperaments by generator size
• Related to 1729/1728
• SandBox
• Semitritave

memo

12ET-complementary comma pairs (e.g. syntonic-schismatic relation)

M3 or d4	A: 4*P5=M3+2*P8	B: 8*P5+d4=5*P8	Remarks
32/27	2187/2048=[-11 7⟩	256/243=[8 -5⟩	A/B=[-19 12⟩, A: (7edo), B: (5edo)
6/5	135/128=[-7 3 1⟩	(64/63)^2*(245/243)=[12 -9 1⟩	A/B=[-19 12⟩, A: Mavila, B: Superpyth
11/9	729/704=[-6 6 0 0 -1⟩	(64/63)^2/(99/98)=[13 -6 0 0 -1⟩	A/B=[-19 12⟩, A: Flattone, B: Supra
8192/6561	531441/524288=[-19 12⟩	1/1	A: (12edo)
5/4	81/80=[-4 4 -1⟩	32805/32768=[-15 8 1⟩	A*B=[-19 12⟩, A: Meantone, B: Schismatic
81/64	1/1	531441/524288=[-19 12⟩	B: (12edo)
9/7	64/63=[6 -2 0 -1⟩	59049/57344=[-13 10 0 -1⟩	B/A=[-19 12⟩, A: Archytas clan, B: Septimal meantone
4/3	256/243=[8 -5⟩	2187/2048=[-11 7⟩	B/A=[-19 12⟩, A: (5edo), B: (7edo)

Q: Mavila must have the fifth flatter than 7edo's, why be placed between 7edo and 5edo?

A: I wrote the 32/27 in this table as a monzo-ish value. 32/27 constructed of P5 & P8 will much sharper when flatter P5 situation.

(3/2)^(1/2)	2187/2048=[-11 7⟩	17-comma=[27 -17⟩	A/B=[-38 24⟩, A: (7edo), B: (17edo)
(3/2)^(4/7)	531441/524288=[-19 12⟩	531441/524288=[-19 12⟩	A*B=[-38 24⟩, A: (12edo), B: (12edo)
(3/2)^(2/3)	256/243=[8 -5⟩	[-41 26⟩	B/A=[-49 31⟩, A: (5edo), B: (26edo)

temperaments spectrum

Fifthspan	-8	-6	4	6	Remarks	Mapping development
Pelogic	25/18	14/9	6/5 8/7	9/7		[⟨1 0 ...], ⟨0 1 -3 -4 -1]]
Armodue	10/7	11/7	6/5 7/6	14/11		[⟨1 0 ...], ⟨0 1 -3 5 -1]] +9
Septimal mavila	7/5	25/16	6/5	32/25		[⟨1 0 ...], ⟨0 1 -3 -11 -1]] -16
Hornbostel	25/18 48/35	25/16	6/5	32/25		[⟨1 0 ...], ⟨0 1 -3 12]] +23
Plutus	32/25 48/35	16/11	5/4 7/6	11/8		[⟨1 0 ...], ⟨0 1 4 5 6]] +7-7
Flattone	21/16	16/11	5/4 11/9	11/8		[⟨1 0 ...], ⟨0 1 4 -9 6]] -14
Meanenneadecal	9/7	16/11	5/4 11/9	11/8		[⟨1 0 ...], ⟨0 1 4 10 6]] +19
Septimal meantone	9/7	10/7	5/4	7/5	Good 4:5:7 in 10 fifthspanp-p	[⟨1 0 ...], ⟨0 1 4 10]]
Mohajira	14/11		5/4		7/5 is at -9.5 fifthspan	[⟨1 0 ...], ⟨0 2 8 -11]] *2-31
Undecimal meantone	14/11	10/7	5/4	7/5	Good 4:5:7 in 10 fifthspanp-p	[⟨1 0 ...], ⟨0 1 4 10 18]] +12
Dominant	32/25	7/5	5/4	10/7	inaccurate	[⟨1 0 ...], ⟨0 1 4 -2]] -12
Schism	5/4	10/7	81/64	7/5	inaccurate	[⟨1 0 ...], ⟨0 1 -8 -2]] -12
Grackle	5/4		81/64		7/5 is at -18 fifthspan	[⟨1 0 ...], ⟨0 1 -8 -26]] -24
Garibaldi	5/4	7/5	81/64 80/63	10/7	Good 4:5:6:7 in 15 fifthspanp-p Good 4:6 & 5:7 in 6 fifthspanp-p	[⟨1 0 ...], ⟨0 1 -8 -14]] +12
Andromeda	5/4	7/5	14/11	10/7	11/9 is at -20 fifthspan -> 41edo	[⟨1 0 ...], ⟨0 1 -8 -14 -18 -21]]
Hemififths		7/5	14/11	10/7	5/4 is at 12.5 fifthspan	[⟨1 0 ...], ⟨0 2 25 13 5]] *2+41
Edson		7/5	14/11	10/7	-> 29edo	[⟨1 0 ...], ⟨0 1 no-five -14-(-8) -18-(-8) -21-(-8)]]
Gentle region Leapfrog	27/22		14/11			[⟨1 0 ...], ⟨0 1 no-five 15 11 8]] +29
Supra	11/9	11/8	9/7 14/11	16/11		[⟨1 0 ...], ⟨0 1 no-five -2 -6]] -17

pan-5L2s tuning spectrum

Eigenmonzo (unchanged interval)	at (fifthspan)	Generator (cents)	in this temperament (e.g.)
6/5	+4(M3)	678.910	Mavila
11/9	+4(M3)	686.852	Flattone
11/8	+6(A4)	691.886	Flattone
6/5	-3(m3)	694.786	Meantone (1/3 comma)
9/7	-8(d4)	695.614	Septimal meantone
5/4	+4(M3)	696.578	Meantone (1/4 comma)
7/5	+6(A4)	697.085	Septimal meantone
11/8	+18(AA3)	697.295	Undecimal meantone
14/11	-8(d4)	697.812	Undecimal meantone
7/5	-18(dd6)	700.972	Grackle
5/4	-8(d4)	701.711	Schismatic
6/5	+9(A2)	701.738	Schismatic
3/2	+1(P5)	701.955	Pythagorean
11/8	-18(dd6)	702.705	Andromeda
7/5	-6(d5)	702.915	Garibaldi
13/11	-3(m3)	703.597	Leapfrog
14/11	+4(M3)	704.377	Leapfrog
27/22	-8(d4)	705.682	Leapfrog
11/9	-8(d4)	706.574	Supra
11/8	-6(d5)	708.114	Supra
9/7	+4(M3)	708.771	Archy (1/4 comma)
5/4	+9(A2)	709.590	Superpyth
6/5	-8(d4)	710.545	Superpyth
7/6	-3(m3)	711.043	Archy (1/3 comma)

Clarify Meantone#Tuning spectrum

19/17	at +2(M2)	696.279	for regular temperament with nominal 19/17 at M2, see some 19-limit variation in Meantone family (search ⟨0 1 * * * * -5 -3] in mappings.
18/17	at +7(A1)	699.850	for regular temperament with nominal 18/17 at A1, see some 17-limit variation in Meantone family (search ⟨0 1 * * * * -5] in mappings.
18/17	at -5(m2)	700.209	don't know regular temperament with nominal 18/17 at m2, but theoretically it will have ⟨0 1 * * * * 7] in mapping.
17/16	at -5(m2)	699.009	for regular temperament with nominal 17/16 at m2, see some 17-limit variation in Meantone family (search ⟨0 1 * * * * -5] in mappings.