User:Moremajorthanmajor/2L 1s (perfect fourth-equivalent): Difference between revisions

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Line 40: Line 40:
!
!
|-
|-
|Do#, Sol#
|F/C/G ut#
Do#, Sol#
د#,
د#,


ص#
ص#
|Sol#
|F/C/G ut#
Sol#
ص#
ص#
|Re#
|G/D/A re#
Re#
ر#
ر#
|1\11, 46.154
|1\11, 46.154
Line 56: Line 59:
|3\9, 163.636
|3\9, 163.636
|-
|-
| Reb, Lab
| G/D/A reb
Reb, Lab
رb, لb
رb, لb
|Lab
|G/D/A reb
Lab
لb
لb
|Mib
|A/E/B mib
Mib
م
م
|3\11, 138.462
|3\11, 138.462
Line 69: Line 75:
|1\9, 54.545
|1\9, 54.545
|-
|-
|'''Re, La'''
|'''G/D/A re'''
'''Re, La'''
'''ر, ل'''
'''ر, ل'''
|'''La'''
|'''G/D/A re'''
'''La'''
ل
ل
|'''Mi'''
|'''A/E/B mi'''
'''Mi'''
م
م
|'''4\11,''' '''184.615'''
|'''4\11,''' '''184.615'''
Line 83: Line 92:
|'''4\9,''' '''218.182'''
|'''4\9,''' '''218.182'''
|-
|-
|Re#, La#
|G/D/A re#
Re#, La#
ر,# ل#
ر,# ل#
|La#
|G/D/A re#
La#
ل#
ل#
|Mi#
|A/E/B mi#
Mi#
م#
م#
|5\11, 230.769
|5\11, 230.769
|4\8, 252.632
| rowspan="2" |4\8, 252.632
|7\13, 270.967
|7\13, 270.967
| rowspan="2" |'''3\5,''' '''300'''
|3\5, 300
| 8\12, 331.034
| 8\12, 331.034
|5\7, 352.941
|5\7, 352.941
|7\9, 381.818
|7\9, 381.818
|-
|-
|'''Mib, Sib'''
|A/E/B mibb
Mibb, Sibb
مbb,تbb
|A/E/B mibb
Sibb
تbb
|B/F/C fab
Fab
فb
|6\11, 276.923
|6\13, 232.258
|2\5, 200
|4\12, 165.517
|2\7, 141.176
|2\9, 109.091
|-
|'''A/E/B mib'''
'''Mib, Sib'''
'''مb,تb'''
'''مb,تb'''
|'''Sib'''
|'''A/E/B mib'''
'''Sib'''
تb
تb
|'''F'''
|'''B/F/C fa'''
'''Fa'''
'''ف'''
'''ف'''
|'''7\11,''' '''323.077'''
|'''7\11,''' '''323.077'''
|'''5\8,''' '''315.789'''
|'''5\8,''' '''315.789'''
|'''8\13,''' '''309.677'''
|'''8\13,''' '''309.677'''
|'''3\5,''' '''300'''
|'''7\12,''' '''289.655'''
|'''7\12,''' '''289.655'''
|'''4\7,''' '''282.353'''
|'''4\7,''' '''282.353'''
|'''5\9,''' '''272.727'''
|'''5\9,''' '''272.727'''
|-
|-
|Mi, Si
|A/E/B mi
Mi, Si
م, ت
م, ت
| Si
| A/E/B mi
Si
ت
ت
|Fa#
|B/F/C fa#
Fa#
ف#
ف#
|8\11, 369.231
|8\11, 369.231
Line 124: Line 159:
|8\9, 436.364
|8\9, 436.364
|-
|-
|Mi#, Si#
|A/E/B mi#
Mi#, Si#
م,#ت#
م,#ت#
|Si#
|A/E/B mi#
Si#
ت#
ت#
|Fax
|B/F/C fax
Fax
فx
فx
|9\11, 415.385
|9\11, 415.385
Line 138: Line 176:
|11\9, 600
|11\9, 600
|-
|-
|Dob, Solb
|F/C/G utb
Dob, Solb
دb,
دb,


صb
صb
| Dob
| B/F/C fab
Dob
دb
دb
|Solb
|C/G/D solb
Solb
صb
صb
|10\11, 461.538
|10\11, 461.538
Line 153: Line 194:
|6\9, 327.273
|6\9, 327.273
|-
|-
!Do, Sol
!F/C/G ut
Do, Sol
د, ص
د, ص
!Do
!B/F/C fa
Do
د
د
!Sol
!C/G/D sol
Sol
ص
ص
!'''11\11,''' '''507.692'''
!'''11\11,''' '''507.692'''
Line 167: Line 211:
!'''9\9,''' '''490.909'''
!'''9\9,''' '''490.909'''
|-
|-
|Do#, Sol#
|F/C/G ut#
Do#, Sol#
د#,
د#,


ص#
ص#
|Do#
|B/F/C fa#
Do#
د#
د#
|Sol#
|C/G/D sol#
Sol#
ص#
ص#
|12\11, 553.846
|12\11, 553.846
Line 183: Line 230:
|12\9, 654.545
|12\9, 654.545
|-
|-
|Reb, Lab
|G/D/A reb
Reb, Lab
رb, لb
رb, لb
|Reb
|C/G/D solb
Reb
رb
رb
|Lab
|D/A/E lab
Lab
لb
لb
|14\11, 646.154
|14\11, 646.154
Line 196: Line 246:
|10\9, 545.455
|10\9, 545.455
|-
|-
|'''Re, La'''
|'''G/D/A re'''
'''Re, La'''
'''ر, ل'''
'''ر, ل'''
|'''Re'''
|'''C/G/D sol'''
'''Re'''
ر
ر
|'''La'''
|'''D/A/E la'''
'''La'''
ل
ل
|'''15\11,''' '''692.308'''
|'''15\11,''' '''692.308'''
Line 210: Line 263:
|'''13\9,''' '''709.091'''
|'''13\9,''' '''709.091'''
|-
|-
|Re#, La#
|G/D/A re#
Re#, La#
ر,# ل#
ر,# ل#
|Re#
|C/G/D sol#
Re#
د#
د#
|La#
|D/A/E la#
La#
ل#
ل#
|16\11, 738.462
|16\11, 738.462
Line 224: Line 280:
|16\9, 872.727
|16\9, 872.727
|-
|-
|'''Mib, Sib'''
|'''A/E/B mib'''
'''Mib, Sib'''
'''مb,تb'''
'''مb,تb'''
|'''Mib'''
|'''D/A/E lab'''
'''Mib'''
مb
مb
|'''Sib'''
|'''E/B/F síb'''
'''Sib'''
تb
تb
|'''18\11,''' '''830.769'''
|'''18\11,''' '''830.769'''
Line 237: Line 296:
|'''14\9,''' '''763.636'''
|'''14\9,''' '''763.636'''
|-
|-
|Mi, Si
|A/E/B mi
Mi, Si
م, ت
م, ت
|Mi
|D/A/E la
Mi
م
م
|Si
|E/B/F sí
Si
ت
ت
|19\11, 876.923
|19\11, 876.923
Line 251: Line 313:
|17\9, 927.727
|17\9, 927.727
|-
|-
|Mi#, Si#
|A/E/B mi#
Mi#, Si#
م,#ت#
م,#ت#
| Mi#
| D/A/E la#
Mi#
م#
م#
|Si#
|E/B/F sí#
Si#
ت#
ت#
|20\11, 923.077
|20\11, 923.077
Line 265: Line 330:
|20\9, 1090.909
|20\9, 1090.909
|-
|-
|Dob, Solb
|F/C/G utb
Dob, Solb
دb,
دb,


صb
صb
|Solb
|F/C/G utb
Solb
صb
صb
|Reb
|G/D/A reb
Reb
رb
رb
|21\11, 969.231
|21\11, 969.231
Line 280: Line 348:
|15\9, 818.182
|15\9, 818.182
|-
|-
!Do, Sol
!F/C/G ut
Do, Sol
د, ص
د, ص
!Sol
!F/C/G ut
Sol
ص
ص
!Re
!G/D/A re
Re
ر
ر
!22\11, 1015.385
!22\11, 1015.385
Line 1,216: Line 1,287:
|-
|-
|0
|0
|Do, Sol
|F/C/G ut
Do, Sol
د, ص
د, ص
|perfect unison
|perfect unison
|0
|0
|Do, Sol
|F/C/G ut
Do, Sol
د, ص
د, ص
|perfect fourth
|perfect fourth
|-
|-
| 1
| 1
|Mib, Sib
|A/E/B mib
Mib, Sib
صb, مb
صb, مb
|diminished third
|diminished third
| -1
| -1
|Re, La
|G/D/A re
Re, La
ر, ل
ر, ل
|perfect second
|perfect second
|-
|-
|2
|2
|Reb, Lab
|G/D/A reb
Reb, Lab
رb, لb
رb, لb
|diminished second
|diminished second
| -2
| -2
|Mi, Si
|A/E/B mi
Mi, Si
ص, م
ص, م
|perfect third
|perfect third
Line 1,245: Line 1,322:
|-
|-
|3
|3
|Do, Sol
|F/C/G utb
Dob, Solb
دb, صb
دb, صb
| diminished fourth
| diminished fourth
| -3
| -3
|Do#, Sol#
|F/C/G ut#
Do#, Sol#
د, #ص#
د, #ص#
|augmented unison (chroma)
|augmented unison (chroma)
|-
|-
|4
|4
|Mibb, Sibb
|A/E/B mibb
Mibb, Sibb
مbb, صbb
مbb, صbb
|doubly diminished third
|doubly diminished third
| -4
| -4
|Re#, La#
|G/D/A re#
Re#, La#
ر ,# ل#
ر ,# ل#
|augmented second
|augmented second
Line 1,265: Line 1,346:
The generator chain for this scale is as follows:
The generator chain for this scale is as follows:
{| class="wikitable"
{| class="wikitable"
|A/E/B mibb
|F/C/G utb
|G/D/A reb
|A/E/B mib
|F/C/G ut
|G/D/A re
|A/E/B mi
|F/C/G ut#
|G/D/A re#
|A/E/B mi#
|-
|Mibb
|Mibb
Sibb
Sibb

Revision as of 04:26, 24 November 2024

2L 1s<perfect fourth>, is a perfect fourth-repeating MOS scale. The notation "<perfect fourth>" means the period of the MOS is a perfect fourth, disambiguating it from octave-repeating 2L 1s.

The generator range is 171.4 to 240 cents, placing it near the diatonic major second, usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fourth complement (240 to 342.9 cents).

In the fourth-repeating version of the diatonic scale, each tone has a perfect fourth above it. The scale has one major chord and two minor chords.

Basic diatonic is in 5ed4/3, which is a very good fourth-based equal tuning similar to 12edo.

Notation

There are 6 main ways to notate this scale. One method uses a simple fourth repeating notation consisting of 3 naturals (eg. Do Re Mi, Sol La Si). Given that 1-5/4-3/2 is fourth-equivalent to a tone cluster of 1-9/8-5/4 and a fourth has too few notes for a structure analogous to the major scale, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple, quintuple or sextuple fourth (minor seventh, tenth, thirteenth or sixteenth or diminished nineteenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 9/8. Notating this way produces a minor tenth which is the Dorian mode of Middletown[6L 3s], also known as the Mahur scale in Persian/Arabic music, a minor thirteenth which is the Aeolian mode of Bijou[8L 4s]; the bastonic chromatic scale, a minor sixteenth which is the Phrygian mode of Hyperionic[10L 5s] or a diminished nineteenth which is the Locrian mode of Subsextal[12L 6s]. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth, 15 in quintuple fourth and 18 in sextuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal, hex or duohex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 or 0123456789XɜABCDEF0 with flats written F molle) may be used.

Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Fourth Seventh ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
Mixolydian Dorian
F/C/G ut#

Do#, Sol# د#,

ص#

F/C/G ut#

Sol# ص#

G/D/A re#

Re# ر#

1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
G/D/A reb

Reb, Lab رb, لb

G/D/A reb

Lab لb

A/E/B mib

Mib م

3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
G/D/A re

Re, La ر, ل

G/D/A re

La ل

A/E/B mi

Mi م

4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
G/D/A re#

Re#, La# ر,# ل#

G/D/A re#

La# ل#

A/E/B mi#

Mi# م#

5\11, 230.769 4\8, 252.632 7\13, 270.967 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
A/E/B mibb

Mibb, Sibb مbb,تbb

A/E/B mibb

Sibb تbb

B/F/C fab

Fab فb

6\11, 276.923 6\13, 232.258 2\5, 200 4\12, 165.517 2\7, 141.176 2\9, 109.091
A/E/B mib

Mib, Sib مb,تb

A/E/B mib

Sib تb

B/F/C fa

Fa ف

7\11, 323.077 5\8, 315.789 8\13, 309.677 3\5, 300 7\12, 289.655 4\7, 282.353 5\9, 272.727
A/E/B mi

Mi, Si م, ت

A/E/B mi

Si ت

B/F/C fa#

Fa# ف#

8\11, 369.231 6\8, 378.947 10\13, 387.097 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
A/E/B mi#

Mi#, Si# م,#ت#

A/E/B mi#

Si# ت#

B/F/C fax

Fax فx

9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
F/C/G utb

Dob, Solb دb,

صb

B/F/C fab

Dob دb

C/G/D solb

Solb صb

10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
F/C/G ut

Do, Sol د, ص

B/F/C fa

Do د

C/G/D sol

Sol ص

11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
F/C/G ut#

Do#, Sol# د#,

ص#

B/F/C fa#

Do# د#

C/G/D sol#

Sol# ص#

12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
G/D/A reb

Reb, Lab رb, لb

C/G/D solb

Reb رb

D/A/E lab

Lab لb

14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
G/D/A re

Re, La ر, ل

C/G/D sol

Re ر

D/A/E la

La ل

15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
G/D/A re#

Re#, La# ر,# ل#

C/G/D sol#

Re# د#

D/A/E la#

La# ل#

16\11, 738.462 12\8, 757.895 20\13, 774.294 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
A/E/B mib

Mib, Sib مb,تb

D/A/E lab

Mib مb

E/B/F síb

Sib تb

18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
A/E/B mi

Mi, Si م, ت

D/A/E la

Mi م

E/B/F sí

Si ت

19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
A/E/B mi#

Mi#, Si# م,#ت#

D/A/E la#

Mi# م#

E/B/F sí#

Si# ت#

20\11, 923.077 15\8, 947.378 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
F/C/G utb

Dob, Solb دb,

صb

F/C/G utb

Solb صb

G/D/A reb

Reb رb

21\11, 969.231 24\13, 929.033 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
F/C/G ut

Do, Sol د, ص

F/C/G ut

Sol ص

G/D/A re

Re ر

22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Mahur Bijou ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
G# 0#, E# 1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
Jf, Af 1b, 1d 3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
J, A 1 4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
J#, A# 1# 5\11, 230.769 4\8, 252.632 7\13, 270.968 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
Af, Bf 2b, 2d 7\11, 323.077 5\8, 315.789 8\13, 309.677 7\12, 289.655 4\7, 282.353 5\9, 272.727
A, B 2 8\11, 369.231 6\8, 378.947 10\13, 387.097 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
A#, B# 2# 9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
Bb, Cf 3b, 3d 10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
B, C 3 11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
B#, C# 3# 12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
Cf, Qf 4b, 4d 14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
C, Q 4 15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
C#, Q# 4# 16\11, 738.462 12\8, 757.895 20\13, 774.194 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
Qf, Df 5b, 5d 18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
Q, D 5 19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
Q#, D# 5# 20\11, 923.077 15\8, 947.368 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
Df, Sf 6b, 6d 21\11, 969.231 24\13, 929.033 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
D, S 6 22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
D#, S# 6# 23\11, 1061.538 17\8, 1073.684 28\13, 1083.871 11\5, 1100 27\12, 1117.241 16\7, 1129.412 21\9, 1145.455
Ef 7b, 7d 25\11, 1153.846 18\8, 1136.842 29\13, 1122.581 26\12, 1075.862 15\7, 1058.824 19\9, 1036.364
E 7 26\11, 1200 19\8, 1200 31\13, 1200 12\5, 1200 29\12, 1200 17\7, 1200 22\9, 1200
E# 7# 27\11, 1246.154 20\8, 1263.158 33\13, 1277.419 13\5, 1300 32\12, 1324.138 19\7, 1341.176 25\9, 1363.636
Ff 8b, Gd 29\11, 1338.462 21\8, 1326.316 34\13, 1316.129 31\12, 1282.759 18\7, 1270.588 23\9, 1254.545
F 8, G 30\11, 1384.615 22\8, 1389.474 36\13, 1393.548 14\5, 1400 34\12, 1406.897 20\7, 1411.765 26\9, 1418.182
F# 8#, G# 31\11, 1430.769 23\8, 1452.632 38\13, 1470.968 15\5, 1500 37\12, 1531.034 22\7, 1552.941 29\9, 1581.818
Gf 9b, Ad 32\11, 1476.923 37\13, 1432.258 14\5, 1400 33\12, 1365.517 19\7, 1341.176 24\9, 1309.091
G 9, A 33\11, 1523.077 24\8, 1515.789 39\13, 1509.677 15\5, 1500 36\12, 1489.655 21\7, 1482.353 27\9, 1472.727
G# 9#, A# 34\11, 1569.231 25\8, 1578.947 41\13, 1587.097 16\5, 1600 39\12, 1613.793 23\7, 1623.529 30\9, 1636.364
Jf, Af Xb, Bd 36\11, 1661.538 26\8, 1642.105 42\13, 1625.806 38\12, 1572.034 22\7, 1552.941 28\9, 1527.27
J, A X, B 37\11, 1707.692 27\8, 1705.263 44\13, 1703.226 17\5, 1700 41\12, 1696.552 24\7, 1694.118 31\9, 1690.909
J#, A# X#, B# 38\11, 1753.846 28\8, 1768.421 46\13, 1780.645 18\5, 1800 44\12, 1820.690 26\7, 1835.294 34\9, 1854.545
Af, Bf Eb, Dd 40\11, 1846.154 29\8, 1831.579 47\13, 1819.355 43\12, 1779.310 25\7, 1764.706 32\9, 1745.455
A, B E, D 41\11, 1892.308 30\8, 1894.737 49\13, 1896.774 19\5, 1900 46\12, 1903.448 27\7, 1905.882 35\9, 1909.090
A#, B# E#, D# 42\11, 1938.462 31\8, 1957.895 51\13, 1974.194 20\5, 2000 49\12, 2027.586 29\7, 2047.059 38\9, 2072.727
Bb, Cf 0b, Ed 43\11, 1984.615 50\13, 1935.484 19\5, 1900 45\12, 1862.069 26\7, 1835.294 33\9, 1800
B, C 0, E 44\11, 2030.769 32\8, 2021.053 52\13, 2012.903 20\5, 2000 48\12, 1986.207 28\7, 1976.471 36\9, 1963.636
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Hyperionic Subsextal ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
1# 0# 1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
2f 1f 3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
2 1 4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
2# 1# 5\11, 230.769 4\8, 252.632 7\13, 270.967 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
3f 2f 7\11, 323.077 5\8, 315.789 8\13, 309.677 7\12, 289.655 4\7, 282.353 5\9, 272.727
3 2 8\11, 369.231 6\8, 378.947 10\13, 387.098 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
3# 2# 9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
4f 3f 10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
4 3 11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
4# 3# 12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
5f 4f 14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
5 4 15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
5# 4# 16\11, 738.462 12\8, 757.895 20\13, 774.194 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
6f 5f 18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
6 5 19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
6# 5# 20\11, 923.077 15\8, 947.368 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
7f 6f 21\11, 969.231 24\13, 929.032 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
7 6 22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
7# 6# 23\11, 1061.538 17\8, 1073.684 28\13, 1083.871 11\5, 1100 27\12, 1117.241 16\7, 1129.412 21\9, 1145.455
8f 7f 25\11, 1153.846 18\8, 1136.842 29\13, 1122.581 26\12, 1075.862 15\7, 1058.824 19\9, 1036.364
8 7 26\11, 1200 19\8, 1200 31\13, 1200 12\5, 1200 29\12, 1200 17\7, 1200 22\9, 1200
8# 7# 27\11, 1246.154 20\8, 1263.158 33\13, 1277.419 13\5, 1300 32\12, 1324.138 19\7, 1341.176 25\9, 1363.636
9f 8f 29\11, 1338.462 21\8, 1326.316 34\13, 1316.129 31\12, 1282.759 18\7, 1270.588 23\9, 1254.545
9 8 30\11, 1384.615 22\8, 1389.474 36\13, 1393.548 14\5, 1400 34\12, 1406.897 20\7, 1411.765 26\9, 1418.182
9# 8# 31\11, 1430.769 23\8, 1452.632 38\13, 1470.968 15\5, 1500 37\12, 1531.034 22\7, 1552.941 29\9, 1581.818
Af 9f 32\11, 1476.923 37\13, 1432.258 14\5, 1400 33\12, 1365.517 19\7, 1341.176 24\9, 1309.091
A 9 33\11, 1523.077 24\8, 1515.789 39\13, 1509.677 15\5, 1500 36\12, 1489.655 21\7, 1482.353 27\9, 1472.727
A# 9# 34\11, 1569.231 25\8, 1578.947 41\13, 1587.097 16\5, 1600 39\12, 1613.793 23\7, 1623.529 30\9, 1636.364
Bf Xb 36\11, 1661.538 26\8, 1642.105 42\13, 1625.806 38\12, 1572.034 22\7, 1552.941 28\9, 1527.27
B X 37\11, 1707.692 27\8, 1705.263 44\13, 1703.226 17\5, 1700 41\12, 1696.552 24\7, 1694.118 31\9, 1690.909
B# X# 38\11, 1753.846 28\8, 1768.421 46\13, 1780.645 18\5, 1800 44\12, 1820.690 26\7, 1835.294 34\9, 1854.545
Cf ɛf 40\11, 1846.154 29\8, 1831.579 47\13, 1819.355 43\12, 1779.310 25\7, 1764.706 32\9, 1745.455
C ɛ 41\11, 1892.308 30\8, 1894.737 49\13, 1896.774 19\5, 1900 46\12, 1903.448 27\7, 1905.882 35\9, 1909.090
C# ɛ# 42\11, 1938.462 31\8, 1957.895 51\13, 1974.194 20\5, 2000 49\12, 2027.586 29\7, 2047.059 38\9, 2072.727
Df Af 43\11, 1984.615 50\13, 1935.484 19\5, 1900 45\12, 1862.069 26\7, 1835.294 33\9, 1800
D A 44\11, 2030.769 32\8, 2021.053 52\13, 2012.903 20\5, 2000 48\12, 1986.207 28\7, 1976.471 36\9, 1963.636
D# A# 45\11, 2076.923 33\8, 2084.211 54\13, 2090.323 21\5, 2100 51\12, 2110.345 30\7, 2117.647 39\9, 2127.273
Ef Bf 47\11, 2169.231 34\8, 2147.368 55\13, 2129.032 50\12, 2068.966 29\7, 2047.059 37\9, 2018.182
E B 48\11, 2215.385 35\8, 2210.526 57\13, 2206.452 22\5, 2200 53\12, 2193.103 31\7, 2188.235 40\9, 2181.818
E# B# 49\11, 2261.538 36\8, 2273.684 59\13, 2283.871 23\5, 2300 56\12, 2317.241 33\7, 2329.412 43\9, 2345.455
Ff Cf 51\11, 2353.846 37\8, 2336.842 61\13, 2322.581 55\12, 2275.864 32\7, 2258.824 41\9, 2236.364
F C 52\11, 2400 38\8, 2400 62\13, 2400 24\5, 2400 58\12, 2400 34\7, 2400 44\9, 2400
F# C# 53\11, 2446.154 39\8, 2463.158 64\13, 2477.419 25\5, 2500 61\12, 2524.138 36\7, 2541.176 47/9, 2563.636
1f Df 54\11, 2492.308 63\13, 2438.710 24\5, 2400 57\12, 2358.621 33\7, 2329.412 42\9, 2390.909
1 D 55\11, 2538.462 40\8, 2526.316 65\13, 2516.129 25\5, 2500 60\12, 2482.759 35\7, 2470.588 45\9, 2454.545
1# D# 56\11, 2584.615 41\8, 2589.474 67\13, 2593.548 26\5, 2600 63\12, 2606.897 37\7, 2611.765 48\9, 2618.182
2f Ef 58\11, 2676.923 42\8, 2652.632 69\13, 2670.968 62\12, 2565.517 36\7, 2541.176 46\9, 2509.091
2 E 59\11, 2723.077 43\8, 2715.789 70\13, 2709.677 27\5, 2700 65\12, 2689.655 38\7, 2682.353 49\9, 2672.727
2# E# 60\11, 2769.231 44\8, 2778.947 72\13, 2787.097 28\5, 2800 68\12, 2813.793 40\7, 2823.529 52\9, 2836.364
3f Ff 62\11, 2861.538 45\8, 2842.105 73\13, 2825.806 67\12, 2772.034 39\7, 2752.941 50\9, 2727.273
3 F 63\11, 2907.692 46\8, 2905.263 75\13, 2903.226 29\5, 2900 70\12, 2896.552 41\7, 2894.118 53\9, 2890.909
3# F# 64\11, 2953.846 47\8, 2968.421 77\13, 2980.645 30\5, 3000 73\12, 3020.690 43\7, 3035.294 55\9, 3000
4f 0f 65\11, 3000 76\13, 2941.935 29\5, 2900 69\29, 2855.172 40\7, 2823.529 52\9, 2836.364
4 0 66\11, 3046.154 48\8, 3031.579 78\13, 3019.355 30\5, 3000 72\12, 2979.310 42\7, 2964.706 54\9, 2945.455

Intervals

Generators Fourth notation Interval category name Generators Notation of 4/3 inverse Interval category name
The 3-note MOS has the following intervals (from some root):
0 F/C/G ut

Do, Sol د, ص

perfect unison 0 F/C/G ut

Do, Sol د, ص

perfect fourth
1 A/E/B mib

Mib, Sib صb, مb

diminished third -1 G/D/A re

Re, La ر, ل

perfect second
2 G/D/A reb

Reb, Lab رb, لb

diminished second -2 A/E/B mi

Mi, Si ص, م

perfect third
The chromatic 5-note MOS also has the following intervals (from some root):
3 F/C/G utb

Dob, Solb دb, صb

diminished fourth -3 F/C/G ut#

Do#, Sol# د, #ص#

augmented unison (chroma)
4 A/E/B mibb

Mibb, Sibb مbb, صbb

doubly diminished third -4 G/D/A re#

Re#, La# ر ,# ل#

augmented second

Genchain

The generator chain for this scale is as follows:

A/E/B mibb F/C/G utb G/D/A reb A/E/B mib F/C/G ut G/D/A re A/E/B mi F/C/G ut# G/D/A re# A/E/B mi#
Mibb

Sibb

Dob

Solb

Reb

Lab

Mib

Sib

Do

Sol

Re

La

Mi

Si

Do#

Sol#

Re#

La#

Mi#

Si#

مbb

تbb

دb

صb

رb

لb

مb

تb

د

ص

ر

ل

م

ت

د#

ص#

ر#

ل#

م#

ت#

dd3 d4 d2 d3 P1 P2 P3 A1 A2 A3

Modes

The mode names are based on the species of fourth:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd
Major LLs 2|0 P P
Minor LsL 1|1 P d
Phrygian sLL 0|2 d d

Temperaments

The most basic rank-2 temperament interpretation of diatonic is Mahuric. The name "Mahuric" comes from the “Mahur” scale in Persian and Arabic music. The major triad is spelled root-2g-(p+g) (p = 4/3, g = the whole tone) and approximates 4:5:6 in pental interpretations or 14:18:21 in septimal ones. Basic ~5ed4/3 fits both interpretations.

Mahuric-Meantone

Subgroup: 4/3.5/4.3/2

Comma list: 81/80

POL2 generator: ~9/8 = 193.6725¢

Mapping: [⟨1 0 1], ⟨0 2 1]]

Optimal ET sequence: ~(5ed4/3, 8ed4/3, 13ed4/3)

Mahuric-Superpyth

Subgroup: 4/3.9/7.3/2

Comma list: 64/63

POL2 generator: ~8/7 = 216.7325¢

Mapping: [⟨1 0 1], ⟨0 2 1]]

Optimal ET sequence: ~(5ed4/3, 7ed4/3, 9ed4/3, 11ed4/3)

Scale tree

The spectrum looks like this:

Generator

(bright)

Cents L s L/s Comments
1\3 171.429 1 1 1.000 Equalised
6\17 180.000 6 5 1.200
5\14 181.818 5 4 1.250
14\39 182.609 14 11 1.273
9\25 183.051 9 7 1.286
4\11 184.615 4 3 1.333
11\30 185.915 11 8 1.375
7\19 186.667 7 5 1.400
10\27 187.500 10 7 1.429
13\35 187.952 13 9 1.444
16\43 188.253 16 11 1.4545
3\8 189.474 3 2 1.500 Mahuric-Meantone starts here
14\37 190.909 14 9 1.556
11\29 191.304 11 7 1.571
8\21 192.000 8 5 1.600
5\13 193.548 5 3 1.667
12\31 194.595 12 7 1.714
7\18 195.348 7 4 1.750
9\23 196.364 9 5 1.800
11\28 197.015 11 6 1.833
13\33 197.468 13 7 1.857
15\38 197.802 15 8 1.875
17\43 198.058 17 9 1.889
19\48 198.261 19 10 1.900
21\53 198.425 21 11 1.909
23\58 198.561 23 12 1.917
25\63 198.675 25 13 1.923
27\68 198.773 27 14 1.929
29\73 198.857 29 15 1.933
31\78 198.930 31 16 1.9375
33\83 198.995 33 17 1.941
35\88 199.052 35 18 1.944
2\5 200.000 2 1 2.000 Mahuric-Meantone ends, Mahuric-Pythagorean begins
17\42 201.980 17 8 2.125
15\37 202.247 15 7 2.143
13\32 202.597 13 6 2.167
11\27 203.077 11 5 2.200
9\22 203.774 9 4 2.250
7\17 204.878 7 3 2.333
12\29 205.714 12 5 2.400
5\12 206.897 5 2 2.500 Mahuric-Neogothic heartland is from here…
18\43 207.693 18 7 2.571
13\31 208.000 13 5 2.600
8\19 208.696 8 3 2.667 …to here
11\26 209.524 11 4 2.750
14\33 210.000 14 5 2.800
3\7 211.755 3 1 3.000 Mahuric-Pythagorean ends, Mahuric-Superpyth begins
22\51 212.903 22 7 3.143
19\44 213.084 19 6 3.167
16\37 213.333 16 5 3.200
13\30 213.699 13 4 3.250
10\23 214.286 10 3 3.333
7\16 215.385 7 2 3.500
11\25 216.393 11 3 3.667
15\34 216.867 15 4 3.750
19\43 217.143 19 5 3.800
4\9 218.182 4 1 4.000
13\29 219.718 13 3 4.333
9\20 220.408 9 2 4.500
14\31 221.053 14 3 4.667
5\11 222.222 5 1 5.000 Mahuric-Superpyth ends
11\24 223.728 11 2 5.500
17\37 224.176 17 3 5.667
6\13 225.000 6 1 6.000
1\2 240.000 1 0 → inf Paucitonic

See also

2L 1s (4/3-equivalent) - idealized tuning

4L 2s (7/4-equivalent) - Mixolydian and Dorian hexatonic Archytas temperament

4L 2s (39/22-equivalent) - Mixolydian and Dorian hexatonic Neogothic temperament

4L 2s (Komornik–Loreti constant-equivalent) - Mixolydian and Dorian hexatonic Komornik–Loreti temperament

4L 2s (9/5-equivalent) - Mixolydian and Dorian hexatonic Meantone temperament

6L 3s (7/3-equivalent) - Mahuric-Archytas temperament

6L 3s (26/11-equivalent) - Mahuric-Neogothic temperament

6L 3s (12/5-equivalent) - Mahuric-Meantone temperament

8L 4s (28/9-equivalent) - Bijou Archytas temperament

8L 4s (22/7-equivalent) and 8L 4s ([math]π[/math]-equivalent) - Bijou Neogothic temperament

8L 4s (16/5-equivalent) - Bijou Meantone temperament

10L 5s (112/27-equivalent) - Hyperionic Archytas temperament

10L 5s (88/21-equivalent) - Hyperionic Neogothic temperament

10L 5s (32/15-equivalent) - Hyperionic Meantone temperament

10L 5s (30/7-equivalent) - Hyperionic septimal Meantone temperament

12L 6s (16/3-equivalent) - Warped Pythagorean Subsextal temperament

12L 6s (343/64-equivalent) - 1/2 comma Archytas Subsextal temperament]

12L 6s (11/2-equivalent) - Low undecimal Subsextal temperament

12L 6s (448/81-equivalent) - 1/6 comma Archytas Subsextal temperament

12L 6s (4096/729-equivalent) - Pythagorean Subsextal temperament

12L 6s (28/5-equivalent) - Low septimal (meantone) Subsextal temperament

12L 6s (256/45-equivalent) - 1/6 comma meantone Subsextal temperament

12L 6s (40/7-equivalent) - High septimal Subsextal temperament

12L 6s (64/11-equivalent) - High undecimal Subsextal temperament

12L 6s (729/125-equivalent) - 1/2 comma meantone Subsextal temperament

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