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===temperaments | ===temperaments spectrum=== | ||
{| class="wikitable" | {| class="wikitable" | ||
! [[Fifthspan]] | ! [[Fifthspan]] | ||
| Line 82: | Line 82: | ||
! 6 | ! 6 | ||
! Remarks | ! Remarks | ||
! Mapping development | |||
|- | |- | ||
! [[Meantone family#Flattone|Flattone]] | ! [[Meantone family#Flattone|Flattone]] | ||
| Line 89: | Line 90: | ||
| 11/8 | | 11/8 | ||
| | | | ||
| [{{val| 1 0 ... }}, {{val| 0 1 4 -9 }}] | |||
|- | |- | ||
! [[Septimal meantone]] | ! [[Septimal meantone]] | ||
| Line 96: | Line 98: | ||
| 7/5 | | 7/5 | ||
| Good 4:5:7 in 10 fifthspan<sub>p-p</sub> | | Good 4:5:7 in 10 fifthspan<sub>p-p</sub> | ||
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 }}] +19 | |||
|- | |- | ||
! [[Meantone family#Mohajira|Mohajira]] | ! [[Meantone family#Mohajira|Mohajira]] | ||
| Line 103: | Line 106: | ||
| | | | ||
| 7/5 is at -9.5 fifthspan | | 7/5 is at -9.5 fifthspan | ||
| [{{val| 1 0 ... }}, {{val| 0 2 8 -11 }}] *2-31 | |||
|- | |- | ||
! [[Meantone family#Unidecimal meantone aka Huygens|Undecimal meantone]] | ! [[Meantone family#Unidecimal meantone aka Huygens|Undecimal meantone]] | ||
| Line 110: | Line 114: | ||
| 7/5 | | 7/5 | ||
| Good 4:5:7 in 10 fifthspan<sub>p-p</sub> | | Good 4:5:7 in 10 fifthspan<sub>p-p</sub> | ||
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 18 }}] | |||
|- | |- | ||
! [[Meantone family#Dominant|Dominant]] | ! [[Meantone family#Dominant|Dominant]] | ||
| Line 117: | Line 122: | ||
| 10/7 | | 10/7 | ||
| inaccurate | | inaccurate | ||
| [{{val| 1 0 ... }}, {{val| 0 1 4 -2 }}] -12 | |||
|- | |- | ||
! [[Schismatic family#Schism|Schism]] | ! [[Schismatic family#Schism|Schism]] | ||
| Line 124: | Line 130: | ||
| 7/5 | | 7/5 | ||
| inaccurate | | inaccurate | ||
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -2 }}] -12 | |||
|- | |- | ||
! [[Schismatic family#Grackle|Grackle]] | ! [[Schismatic family#Grackle|Grackle]] | ||
| Line 131: | Line 138: | ||
| | | | ||
| 7/5 is at -18 fifthspan | | 7/5 is at -18 fifthspan | ||
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -26 }}] -24 | |||
|- | |- | ||
! [[Schismatic family#Garibaldi|Garibaldi]] | ! [[Schismatic family#Garibaldi|Garibaldi]] | ||
| Line 138: | Line 146: | ||
| 10/7 | | 10/7 | ||
| Good 4:5:6:7 in 15 fifthspan<sub>p-p</sub><br />Good 4:6 & 5:7 in 6 fifthspan<sub>p-p</sub> | | Good 4:5:6:7 in 15 fifthspan<sub>p-p</sub><br />Good 4:6 & 5:7 in 6 fifthspan<sub>p-p</sub> | ||
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 }}] +12 | |||
|- | |- | ||
! [[Schismatic family#Andromeda|Andromeda]] | ! [[Schismatic family#Andromeda|Andromeda]] | ||
| Line 145: | Line 154: | ||
| 10/7 | | 10/7 | ||
| 11/9 is at -20 fifthspan -> [[41edo]] | | 11/9 is at -20 fifthspan -> [[41edo]] | ||
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 -18 }}] | |||
|- | |- | ||
! [[Hemififths]] | ! [[Hemififths]] | ||
| Line 152: | Line 162: | ||
| 10/7 | | 10/7 | ||
| 5/4 is at 12.5 fifthspan | | 5/4 is at 12.5 fifthspan | ||
| [{{val| 1 0 ... }}, {{val| 0 2 25 13 5 }}] *2+41 | |||
|- | |- | ||
! [[Chromatic pairs#Edson|Edson]] | ! [[Chromatic pairs#Edson|Edson]] | ||
| Line 159: | Line 170: | ||
| 10/7 | | 10/7 | ||
| -> [[29edo]] | | -> [[29edo]] | ||
| [{{val| 1 0 ... }}, {{val| 0 1 no-five -14-(-8) -18-(-8) -21-(-8) }}] | |||
|- | |- | ||
! [[Gentle region]]<br />[[No-fives subgroup temperaments#Leapfrog|Leapfrog]] | ! [[Gentle region]]<br />[[No-fives subgroup temperaments#Leapfrog|Leapfrog]] | ||
| Line 166: | Line 178: | ||
| | | | ||
| | | | ||
| [{{val| 1 0 ... }}, {{val| 0 1 no-five 15 11 8 }}] +29 | |||
|- | |- | ||
! [[Archytas clan#Supra|Supra]] | ! [[Archytas clan#Supra|Supra]] | ||
| Line 173: | Line 186: | ||
| | | | ||
| | | | ||
| [{{val| 1 0 ... }}, {{val| 0 1 no-five -2 -6 }}] -17 | |||
|} | |} | ||
| Line 178: | Line 192: | ||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
! Eigenmonzo<br />(unchanged interval) !! data-sort-type="number"|at<br />(fifthspan) !! data-sort-type="number"|Generator<br />(cents) !! class="unsortable"| | ! Eigenmonzo<br />(unchanged interval) !! data-sort-type="number"|at<br />(fifthspan) !! data-sort-type="number"|Generator<br />(cents) !! class="unsortable"|in this temperament | ||
|- | |- | ||
| 11/8 || +6(A4) || 691.886 || | | 11/8 || +6(A4) || 691.886 || Flattone | ||
|- | |- | ||
| 6/5 || -3(m3) || 694.786 || 1/3 comma | | 6/5 || -3(m3) || 694.786 || Meantone (1/3 comma) | ||
|- | |- | ||
| 9/7 || -8(d4) || 695.614 || Septimal | | 9/7 || -8(d4) || 695.614 || Septimal meantone | ||
|- | |- | ||
| 5/4 || +4(M3) || 696.578 || 1/4 comma | | 5/4 || +4(M3) || 696.578 || Meantone (1/4 comma) | ||
|- | |- | ||
| 7/5 || +6(A4) || 697.085 || Septimal | | 7/5 || +6(A4) || 697.085 || Septimal meantone | ||
|- | |- | ||
| 14/11 || -8(d4) || 697.812 || Undecimal | | 14/11 || -8(d4) || 697.812 || Undecimal meantone | ||
|- | |- | ||
| || || || | | || || || | ||
| Line 206: | Line 220: | ||
| 7/5 || -6(d5) || 702.915 || Garibaldi | | 7/5 || -6(d5) || 702.915 || Garibaldi | ||
|- | |- | ||
| 13/11 || -3(m3) || 703.597 || | | 13/11 || -3(m3) || 703.597 || Leapfrog | ||
|- | |- | ||
| 14/11 || +4(M3) || 704.377 || | | 14/11 || +4(M3) || 704.377 || Leapfrog | ||
|- | |- | ||
| 27/22 || -8(d4) || 705.682 || | | 27/22 || -8(d4) || 705.682 || Leapfrog | ||
|- | |- | ||
| 11/9 || -8(d4) || 706.574 || Supra | | 11/9 || -8(d4) || 706.574 || Supra | ||
| Line 216: | Line 230: | ||
| 11/8 || -6(d5) || 708.114 || Supra | | 11/8 || -6(d5) || 708.114 || Supra | ||
|- | |- | ||
| 9/7 || +4(M3) || 708.771 || Archy 1/4 comma | | 9/7 || +4(M3) || 708.771 || Archy (1/4 comma) | ||
|- | |- | ||
| 5/4 || +9(A2) || 709.590 || Superpyth | | 5/4 || +9(A2) || 709.590 || Superpyth | ||
| Line 222: | Line 236: | ||
| 6/5 || -8(d4) || 710.545 || Superpyth | | 6/5 || -8(d4) || 710.545 || Superpyth | ||
|- | |- | ||
| 7/6 || -3(m3) || 711.043 || Archy 1/3 comma | | 7/6 || -3(m3) || 711.043 || Archy (1/3 comma) | ||
|} | |} | ||
Revision as of 14:37, 11 November 2021
- 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: Meanenneadecal?, 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 |
|---|---|---|---|---|---|---|
| Flattone | 21/16 | 16/11 | 5/4 | 11/8 | [⟨1 0 ...], ⟨0 1 4 -9]] | |
| Septimal meantone | 9/7 | 10/7 | 5/4 | 7/5 | Good 4:5:7 in 10 fifthspanp-p | [⟨1 0 ...], ⟨0 1 4 10]] +19 |
| 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]] |
| 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 | 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]] |
| 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 | 9/7 14/11 |
[⟨1 0 ...], ⟨0 1 no-five -2 -6]] -17 |
pan-5L2s tuning spectrum
| Eigenmonzo (unchanged interval) |
at (fifthspan) |
Generator (cents) |
in this temperament |
|---|---|---|---|
| 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 |
| 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) |