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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 nominal ratio. 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 		11/8 105/64 is at 10 fifthspan -> 7edo [⟨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 it will have ⟨0 1 * * * * 7] in the 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.)
19/16 at -3(m3) 700.829 for regular temperament with nominal 19/16 at m3, see some 19-limit variation in Meantone family (search ⟨0 1 * * * * * -3] in mappings.)
45/34 at +11(A3) 698.661 for regular temperament with nominal 45/34 at A3, see some 17-limit variation in Meantone family (search ⟨0 1 4 * * * -5] in mappings.)
17/15 at -10(d3) 698.331 for regular temperament with nominal 17/15 at d3, see some 17-limit variation in Meantone family (search ⟨0 1 4 * * * -5] in mappings.)

"(14/11)*(13/11) is for Gentle region, then what is for meantone? 33/28, thrice, 99/84, flatly approx., 100/85, 20/17, (14/11)*(20/17)*(2/3)=560/561"

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