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

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
No edit summary
Line 7: Line 7:
[[Basic]] diatonic is in [[5ed4/3]], which is a very good fourth-based equal tuning similar to [[12edo]].
[[Basic]] diatonic is in [[5ed4/3]], which is a very good fourth-based equal tuning similar to [[12edo]].
==Notation==
==Notation==
There are 4 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, it may be more convenient to notate diatonic scales as repeating at the double, triple,  quadruple or quintuple fourth (minor seventh, tenth, thirteenth or sixteenth), however it does make navigating the [[Generator|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] or a minor sixteenth which is the Phrygian mode of Hyperionic. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth and 15 in quintuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal or hex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 with flats written F molle) may be used.
There are 4 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, it may be more convenient to notate diatonic scales as repeating at the double, triple,  quadruple or quintuple fourth (minor seventh, tenth, thirteenth or sixteenth), however it does make navigating the [[Generator|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 notation 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.
{| class="wikitable"
{| class="wikitable"
|+Cents<ref name=":05">Fractions repeating more than 4 digits written as continued fractions</ref>
|+Cents<ref name=":05">Fractions repeating more than 4 digits written as continued fractions</ref>
Line 324: Line 324:
|}
|}
{| class="wikitable"
{| class="wikitable"
! colspan="3" |Notation
! colspan="4" |Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 336: Line 336:
!Bijou
!Bijou
!Hyperionic
!Hyperionic
!Subsextal
!~11ed4/3
!~11ed4/3
!~8ed4/3
!~8ed4/3
Line 345: Line 346:
|-
|-
|G#
|G#
|0#, D#
|0#, E#
|1#
|1#
|0#
|1\11
|1\11
46; 6.5
46; 6.5
Line 366: Line 368:
|1b, 1d
|1b, 1d
|2f
|2f
|1f
|3\11
|3\11
138; 3.25
138; 3.25
Line 382: Line 385:
|'''1'''
|'''1'''
|'''2'''
|'''2'''
|'''1'''
|'''4\11'''
|'''4\11'''
'''184; 1.625'''
'''184; 1.625'''
Line 400: Line 404:
|1#
|1#
|2#
|2#
|1#
|5\11
|5\11
230; 1.3
230; 1.3
Line 418: Line 423:
|'''2b, 2d'''
|'''2b, 2d'''
|'''3f'''
|'''3f'''
|2f
|'''7\11'''
|'''7\11'''
'''323; 13'''
'''323; 13'''
Line 434: Line 440:
|2
|2
|3
|3
|'''2'''
|8\11
|8\11
369; 4.{{Overline|3}}
369; 4.{{Overline|3}}
Line 452: Line 459:
|2#
|2#
|3#
|3#
|2#
|9\11
|9\11
415; 2.6
415; 2.6
Line 470: Line 478:
|3b, 3d
|3b, 3d
|4f
|4f
|'''3f'''
|10\11
|10\11
461; 1, 1.1{{Overline|6}}
461; 1, 1.1{{Overline|6}}
Line 486: Line 495:
!3
!3
!4
!4
!3
!'''11\11'''
!'''11\11'''
'''507; 1.{{Overline|4}}'''
'''507; 1.{{Overline|4}}'''
Line 504: Line 514:
|3#
|3#
|4#
|4#
|3#
|12\11
|12\11
553; 1.{{Overline|18}}
553; 1.{{Overline|18}}
Line 522: Line 533:
|4b, 4d
|4b, 4d
|5f
|5f
|4f
|14\11
|14\11
646; 6.5
646; 6.5
Line 538: Line 550:
|'''4'''
|'''4'''
|'''5'''
|'''5'''
|'''4'''
|'''15\11'''
|'''15\11'''
'''692; 3.25'''
'''692; 3.25'''
Line 556: Line 569:
|4#
|4#
|5#
|5#
|4#
|16\11
|16\11
738; 2.1{{Overline|6}}
738; 2.1{{Overline|6}}
Line 574: Line 588:
|'''5b, 5d'''
|'''5b, 5d'''
|'''6f'''
|'''6f'''
|5f
|'''18\11'''
|'''18\11'''
'''830; 1.3'''
'''830; 1.3'''
Line 590: Line 605:
|5
|5
|6
|6
|'''5'''
|19\11
|19\11
876; 1.08{{Overline|3}}
876; 1.08{{Overline|3}}
Line 608: Line 624:
|5#
|5#
|6#
|6#
|5#
|20\11
|20\11
923: 13
923: 13
Line 626: Line 643:
|6b, 6d
|6b, 6d
|7f
|7f
|'''6f'''
|21\11
|21\11
969; 4.{{Overline|3}}
969; 4.{{Overline|3}}
Line 642: Line 660:
!6
!6
!7
!7
!6
!22\11
!22\11
1015; 2.6
1015; 2.6
Line 660: Line 679:
|6#
|6#
|7#
|7#
|6#
|23\11
|23\11
1061; 1, 1.1{{Overline|6}}
1061; 1, 1.1{{Overline|6}}
Line 678: Line 698:
|7b, 7d
|7b, 7d
|8f
|8f
|7f
|25\11
|25\11
1153; 1.{{Overline|18}}
1153; 1.{{Overline|18}}
Line 694: Line 715:
|'''7'''
|'''7'''
|'''8'''
|'''8'''
|7
|'''26\11'''
|'''26\11'''
'''1200'''
'''1200'''
Line 712: Line 734:
|7#
|7#
|8#
|8#
|7#
|27\11
|27\11
1246; 6,5
1246; 6,5
Line 728: Line 751:
|-
|-
|'''Ff'''
|'''Ff'''
|'''8b, Fd'''
|'''8b, Gd'''
|'''9f'''
|'''9f'''
|8f
|'''29\11'''
|'''29\11'''
'''1338; 3.25'''
'''1338; 3.25'''
Line 744: Line 768:
|-
|-
|F
|F
|8, F
|8, G
|9
|9
|'''8'''
|30\11
|30\11
1384; 1.625
1384; 1.625
Line 762: Line 787:
|-
|-
|F#
|F#
|8#, F#
|8#, G#
|9#
|9#
|8#
|31\11
|31\11
1430; 1.3
1430; 1.3
Line 780: Line 806:
|-
|-
|Gf
|Gf
|9b, Gd
|9b, Ad
|Af
|Af
|9f
|32\11
|32\11
1476; 1.08{{Overline|3}}
1476; 1.08{{Overline|3}}
Line 796: Line 823:
|-
|-
!G
!G
!'''9, G'''
!'''9, A'''
!A
!A
!9
!33\11
!33\11
1523; 13
1523; 13
Line 814: Line 842:
|-
|-
|G#
|G#
|9#, G#
|9#, A#
|A#
|A#
|9#
|34\11
|34\11
1569; 4.{{Overline|3}}
1569; 4.{{Overline|3}}
Line 832: Line 861:
|-
|-
|Jf, Af
|Jf, Af
|Xb, Ad
|Xb, Bd
|Bf
|Bf
|Xb
|36\11
|36\11
1661; 1, 1.1{{Overline|6}}
1661; 1, 1.1{{Overline|6}}
Line 848: Line 878:
|-
|-
|'''J, A'''
|'''J, A'''
|'''X, A'''
|'''X, B'''
|'''B'''
|'''B'''
|'''X'''
|'''37\11'''
|'''37\11'''
'''1707; 1.{{Overline|4}}'''
'''1707; 1.{{Overline|4}}'''
Line 867: Line 898:
|-
|-
|J#, A#
|J#, A#
|X#, A#
|X#, B#
|B#
|B#
|X#
|38\11
|38\11
1753; 1.{{Overline|18}}
1753; 1.{{Overline|18}}
Line 885: Line 917:
|-
|-
|'''Af, Bf'''
|'''Af, Bf'''
|'''Eb, Bd'''
|'''Eb, Dd'''
|'''Cf'''
|'''Cf'''
|'''ɛf'''
|'''40\11'''
|'''40\11'''
'''1846; 6.5'''
'''1846; 6.5'''
Line 902: Line 935:
|-
|-
|A, B
|A, B
|E, B
|E, D
|C
|C
|ɛ
|41\11
|41\11
1892; 3.25
1892; 3.25
Line 920: Line 954:
|-
|-
|A#, B#
|A#, B#
|E#, B#
|E#, D#
|C#
|C#
|ɛ#
|42\11
|42\11
1938; 2.1{{Overline|6}}
1938; 2.1{{Overline|6}}
Line 938: Line 973:
|-
|-
|Bb, Cf
|Bb, Cf
|0b, Dd
|0b, Ed
|Df
|Df
|43\15
|Af
|43\11
1984; 1.625
1984; 1.625
|50\13
|50\13
Line 954: Line 990:
|-
|-
!B, C
!B, C
!0, D
!0, E
!D
!D
!A
!44\11
!44\11
2030; 1.3
2030; 1.3
Line 973: Line 1,010:
|-
|-
|B#, C#
|B#, C#
|0#, D#
|0#, E#
|D#
|D#
|A#
|45\11
|45\11
2076; 1.08{{Overline|3}}
2076; 1.08{{Overline|3}}
Line 993: Line 1,031:
|1b, 1d
|1b, 1d
|Ef
|Ef
|Bf
|47\11
|47\11
2169; 4.{{Overline|3}}
2169; 4.{{Overline|3}}
Line 1,009: Line 1,048:
|'''1'''
|'''1'''
|'''E'''
|'''E'''
|'''B'''
|'''48\11'''
|'''48\11'''
'''2215; 2.6'''
'''2215; 2.6'''
Line 1,027: Line 1,067:
|1#
|1#
|E#
|E#
|B#
|49\11
|49\11
2261; 1, 1.1{{Overline|6}}
2261; 1, 1.1{{Overline|6}}
Line 1,045: Line 1,086:
|'''2b, 2d'''
|'''2b, 2d'''
|'''Ff'''
|'''Ff'''
|'''Cf'''
|'''51\11'''
|'''51\11'''
'''2353; 1.{{Overline|18}}'''
'''2353; 1.{{Overline|18}}'''
Line 1,061: Line 1,103:
|2
|2
|F
|F
|C
|52\11
|52\11
2400
2400
Line 1,079: Line 1,122:
|2#
|2#
|F#
|F#
|C#
|53\11
|53\11
2446; 6.5
2446; 6.5
Line 1,097: Line 1,141:
|3b, 3d
|3b, 3d
|1f
|1f
|Df
|54\11
|54\11
2492; 3.25
2492; 3.25
Line 1,113: Line 1,158:
!3
!3
!1
!1
!D
!55\11
!55\11
2538; 2.1{{Overline|6}}
2538; 2.1{{Overline|6}}
Line 1,127: Line 1,173:
!45\9
!45\9
2454.{{Overline|54}}
2454.{{Overline|54}}
|-
|D#, S#
|3#
|1#
|D#
|56\11
2584; 1.625
|41\8
2589; 2.1̄
|
| rowspan="2" |26\5
2600
|63\12
2606; 1, 8.{{Overline|6}}
|37\7
2611; 1, 3.25
|48\9
2618.{{Overline|18}}
|-
|Ef
|4b, 4d
|2f
|Ef
|58\11
2676; 1.08{{Overline|3}}
|42\8
2652; 1.58{{Overline|3}}
|
|62\12
2565; 1.9{{Overline|3}}
|36\7
2541; 5.{{Overline|6}}
|46\9
2509.{{Overline|09}}
|-
|'''E'''
|'''4'''
|'''2'''
|'''E'''
|'''59\11'''
'''2723; 13'''
|'''43\8'''
'''2715; 1.2{{Overline|6}}'''
|
|'''27\5'''
'''2700'''
|'''65\12'''
'''2689; 1, 1.9'''
|'''38\7'''
'''2682; 2.8{{Overline|3}}'''
|'''49\9'''
'''2672.{{Overline|72}}'''
|-
|E#
|4#
|2#
|E#
|60\11
2769; 4.{{Overline|3}}
|44\8
2778; 1.05̄
|
| rowspan="2" |'''28\5'''
'''2800'''
|68\12
2813; 1, 3.8{{Overline|3}}
|40\7
2823; 1.{{Overline|8}}
|52\9
2836.{{Overline|36}}
|-
|'''Ff'''
|'''5b, 5d'''
|'''3f'''
|'''Ff'''
|'''62\11'''
'''2861; 1, 1.1{{Overline|6}}'''
|'''45\8'''
'''2842; 9.5'''
|
|'''67\12'''
'''2772; 29'''
|'''39\7'''
'''2752; 1.0625'''
|'''50\9'''
'''2727.{{Overline|27}}'''
|-
|F
|5
|3
|F
|63\11
2907; 1.{{Overline|4}}
|46\8
2905; 3.8
|
|29\5
2900
|70\12
2896; 1.8125
|41\7
2894; 8.5
|53\9
2890.{{Overline|90}}
|-
|F#
|5#
|3#
|F#
|64\11
2953; 1.{{Overline|18}}
| rowspan="2" |47\8
2968; 2.375
|
|30\5
3000
|73\12
3020; 1.45
|43\7
3035; 3,4
|55\9
3000
|-
|Gf
|6b, 6d
|4f
|0f
|65\11
3000
|
|29\5
2900
|69\29
2855; 5.8
|40\7
2823; 1.{{Overline|8}}
|52\9
2836.{{Overline|36}}
|-
!G
!6
!4
!0
!66\11
3046; 6.5
!48\8
30'''31; 1.{{Overline|72}}'''
!
!30\5
3000
!72\12
29'''79; 3.{{Overline|2}}'''
!42\7
2964; 1, 3.25
!54\9
2945.{{Overline|45}}
|}
|}
==AIntervals==
==Intervals==
{| class="wikitable"
{| class="wikitable"
!Generators
!Generators

Revision as of 05:31, 4 July 2023

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 4/3 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 4 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, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple or quintuple fourth (minor seventh, tenth, thirteenth or sixteenth), 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 notation 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[1]
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Fourth Seventh ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
Do#, Sol# Sol# 1\11

46; 6.5

1\8

63; 6.3

2\13

77; 2, 2.6

1\5

100

3\12

124; 7.25

2\7

141; 5.6

3\9

163.63

Reb, Lab Lab 3\11

138; 3.25

2\8

126; 3.16

3\13

116; 7.75

2\12

82; 1.318

1\7

70; 1.7

1\9

54.54

Re, La La 4\11

184; 1.625

3\8

189; 2.1

5\13

193; 1, 1, 4.6

2\5

200

5\12

206; 1, 8.6

3\7

211; 1, 3.25

4\9

218.18

Re#, La# La# 5\11

230; 1.3

4\8

252; 1.583

7\13

270; 1.03

3\5

300

8\12

331; 29

5\7

352; 1.0625

7\9

381.81

Mib, Sib Sib 7\11

323; 13

5\8

315; 1.26

8\13

309; 1, 2.1

7\12

289; 1, 1.9

4\7

282; 2.83

5\9

272.72

Mi, Si Si 8\11

369; 4.3

6\8

378; 1.05

10\13

387; 10.3

4\5

400

10\12

413; 1, 3.83

6\7

423; 1.8

8\9

436.36

Mi#, Si# Si# 9\11

415; 2.6

7\8

442; 9.5

12\13

464; 1.9375

5\5

500

13\12

537; 14.5

8\7

564; 1.416

11\9

600

Dob, Solb Dob 10\11

461; 1, 1.16

11\13

425; 1.24

4\5

400

9\12

372; 2.416

5\7

352; 1.0625

6\9

327.27

Do, Sol Do 11\11

507; 1.4

8\8

505; 3.8

13\13

503; 4, 2.3

5\5

500

12\12

496; 1.8125

7\7

494; 8.5

9\9

490.90

Do#, Sol# Do# 12\11

553; 1.18

9\8

568; 2.375

15\13

580; 1.55

6\5

600

15\12

620; 1.45

9\7

635; 3.4

12\9

654.54

Reb, Lab Reb 14\11

646; 6.5

10\8

631; 1.72

16\13

619; 2.81

14\12

579; 3.2

8\7

564; 1.416

10\9

545.45

Re, La Re 15\11

692; 3.25

11\8

694; 1, 2.8

18\13

696; 1.2916

7\5

700

17\12

703; 2, 2.16

10\7

705; 1.13

13\9

709.09

Re#, La# Re# 16\11

738; 2.16

12\8

757; 1, 8.5

20\13

774; 5.16

8\5

800

20\12

827; 1, 1.416

12\7

847; 17

16\9

872.72

Mib, Sib Mib 18\11

830; 1.3

13\8

821; 19

21\13

812; 1, 9.3

19\12

786; 4.83

11\7

776; 2.125

14\9

763.63

Mi, Si Mi 19\11

876; 1.083

14\8

884; 4.75

23\13

890; 3.1

9\5

900

22\12

910; 2.9

13\7

917; 1.54

17\9

927.27

Mi#, Si# Mi# 20\11

923: 13

15\8

947; 2, 1.4

25\13

967; 1, 2.875

10\5

1000

25\12

1034; 2, 14

15\7

1058; 1, 4.6

20\9

1090.90

Dob, Solb Solb 21\11

969; 4.3

24\13

929; 31

9\5

900

21\12

868; 1, 28

11\7

776; 2.125

15\9

818.18

Do, Sol Sol 22\11

1015; 2.6

16\8

1010; 1.9

26\13

1006; 2, 4.6

10\5

1000

24\12

993; 9.6

14\7

988; 4.25

18\9

981.81

Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Mahur Bijou Hyperionic Subsextal ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
G# 0#, E# 1# 0# 1\11

46; 6.5

1\8

63; 6.3

2\13

77; 2, 2.6

1\5

100

3\12

124; 7.25

2\7

141; 5.6

3\9

163.63

Jf, Af 1b, 1d 2f 1f 3\11

138; 3.25

2\8

126; 3.16

3\13

116; 7.75

2\12

82; 1.318

1\7

70; 1.7

1\9

54.54

J, A 1 2 1 4\11

184; 1.625

3\8

189; 2.1

5\13

193; 1, 1, 4.6

2\5

200

5\12

206; 1, 8.6

3\7

211; 1, 3.25

4\9

218.18

J#, A# 1# 2# 1# 5\11

230; 1.3

4\8

252; 1.583

7\13

270; 1.03

3\5

300

8\12

331; 29

5\7

352; 1.0625

7\9

381.81

Af, Bf 2b, 2d 3f 2f 7\11

323; 13

5\8

315; 1.26

8\13

309; 1, 2.1

7\12

289; 1, 1.9

4\7

282; 2.83

5\9

272.72

A, B 2 3 2 8\11

369; 4.3

6\8

378; 1.05

10\13

387; 10.3

4\5

400

10\12

413; 1, 3.83

6\7

423; 1.8

8\9

436.36

A#, B# 2# 3# 2# 9\11

415; 2.6

7\8

442; 9.5

12\13

464; 1.9375

5\5

500

13\12

537; 14.5

8\7

564; 1.416

11\9

600

Bb, Cf 3b, 3d 4f 3f 10\11

461; 1, 1.16

11\13

425; 1.24

4\5

400

9\12

372; 2.416

5\7

352; 1.0625

6\9

327.27

B, C 3 4 3 11\11

507; 1.4

8\8

505; 3.8

13\13

503; 4, 2.3

5\5

500

12\12

496; 1.8125

7\7

494; 8.5

9\9

490.90

B#, C# 3# 4# 3# 12\11

553; 1.18

9\8

568; 2.375

15\13

580; 1.55

6\5

600

15\12

620; 1.45

9\7

635; 3.4

12\9

654.54

Cf, Qf 4b, 4d 5f 4f 14\11

646; 6.5

10\8

631; 1.72

16\13

619; 2.81

14\12

579; 3.2

8\7

564; 1.416

10\9

545.45

C, Q 4 5 4 15\11

692; 3.25

11\8

694; 1, 2.8

18\13

696; 1.2916

7\5

700

17\12

703; 2, 2.16

10\7

705; 1.13

13\9

709.09

C#, Q# 4# 5# 4# 16\11

738; 2.16

12\8

757; 1, 8.5

20\13

774; 5.16

8\5

800

20\12

827; 1, 1.416

12\7

847; 17

16\9

872.72

Qf, Df 5b, 5d 6f 5f 18\11

830; 1.3

13\8

821; 19

21\13

812; 1, 9.3

19\12

786; 4.83

11\7

776; 2.125

14\9

763.63

Q, D 5 6 5 19\11

876; 1.083

14\8

884; 4.75

23\13

890; 3.1

9\5

900

22\12

910; 2.9

13\7

917; 1.54

17\9

927.27

Q#, D# 5# 6# 5# 20\11

923: 13

15\8

947; 2, 1.4

25\13

967; 1, 2.875

10\5

1000

25\12

1034; 2, 14

15\7

1058; 1, 4.6

20\9

1090.90

Df, Sf 6b, 6d 7f 6f 21\11

969; 4.3

24\13

929; 31

9\5

900

21\12

868; 1, 28

11\7

776; 2.125

15\9

818.18

D, S 6 7 6 22\11

1015; 2.6

16\8

1010; 1.9

26\13

1006; 2, 4.6

10\5

1000

24\12

993; 9.6

14\7

988; 4.25

18\9

981.81

D#, S# 6# 7# 6# 23\11

1061; 1, 1.16

17\8

1073; 1, 2.16

28\13

1083; 1.148

11\5

1100

27\12

1117; 4, 7

16\7

1129; 2, 2.3

24\9

1309.09

Ef 7b, 7d 8f 7f 25\11

1153; 1.18

18\8

1136; 1.1875

29\13

1122; 1.72

26\12

1075; 1.16

15\7

1058; 1, 4.6

19\9

1036.36

E 7 8 7 26\11

1200

19\8

1200

31\13

1200

12\5

1200

29\12

1200

17\7

1200

22\9

1200

E# 7# 8# 7# 27\11

1246; 6,5

20\8

1263; 6.3

33\13

1277; 2, 2.6

13\5

1300

32\12

1324; 7.25

19\7

1341; 5.6

25\9

1363.63

Ff 8b, Gd 9f 8f 29\11

1338; 3.25

21\8

1326; 3.16̄

34\13

1316; 7.75

31\12

1282; 1.318

18\7

1270; 1.7

23\9

1254.54

F 8, G 9 8 30\11

1384; 1.625

22\8

1389; 2.1̄

36\13

1393; 1, 1, 4.6

14\5

1400

34\12

1406; 1, 8.6

20\7

1411; 1, 3.25

26\9

1418.18

F# 8#, G# 9# 8# 31\11

1430; 1.3

23\8

1452; 1.583

38\13

1470; 1.03

15\5

1500

37\12

1531; 29

22\7

1552; 1.0625

29\9

1581.81

Gf 9b, Ad Af 9f 32\11

1476; 1.083

37\13

1432: 3.875

14\5

1400

33\12

1365; 1.93

19\7

1341; 5.3

24\9

1309.09

G 9, A A 9 33\11

1523; 13

24\8

1515; 1.26

39\13

1509; 1, 2.1

15\5

1500

36\12

1489; 1, 1.9

21\7

1482; 2.83

27\9

1472.72

G# 9#, A# A# 9# 34\11

1569; 4.3

25\8

1578; 1.05̄

41\13

1587; 10.3

16\5

1600

39\12

1613; 1, 3.83

23\7

1623; 1.8

30\9

1636.36

Jf, Af Xb, Bd Bf Xb 36\11

1661; 1, 1.16

26\8

1642; 9.5

42\13

1625; 1.24

38\12

1572; 29

22\7

1552; 1.0625

28\9

1527.27

J, A X, B B X 37\11

1707; 1.4

27\8

1705; 3.8

44\13

1703; 4, 2.3̄

17\5

1700

41\12

1696; 1.8125

24\7

1694; 8.5

31\9

1690.90

J#, A# X#, B# B# X# 38\11

1753; 1.18

28\8

1768; 2.375

46\13

1780; 1.55

18\5

1800

44\12

1820; 1.45

26\7

1835; 3,4

34\9

1854.54

Af, Bf Eb, Dd Cf ɛf 40\11

1846; 6.5

29\8

1831; 1.72

47\13

1819; 2.81

43\12

1779; 3.2

25\7

1764; 1, 3.25

32\9

1745.45

A, B E, D C ɛ 41\11

1892; 3.25

30\8

1894; 1, 2.8

49\13

1896; 1.2916

19\5

1900

46\12

1903; 2, 2.16

27\7

1905; 1, 7.5

35\9

1909.09

A#, B# E#, D# C# ɛ# 42\11

1938; 2.16

31\8

1957; 1, 8.5

51\13

1974; 5.16

20\5

2000

49\12

2027; 1, 1.416

29\7

2047; 17

38\9

2072.72

Bb, Cf 0b, Ed Df Af 43\11

1984; 1.625

50\13

1935; 2.06

19\5

1900

45\12

1862; 14.5

26\7

1835; 3,4

33\9

1800

B, C 0, E D A 44\11

2030; 1.3

32\8

2021; 19

52\13

2012; 1, 9.3

20\5

2000

48\12

1986; 4.83

28\7

1976; 2.125

36\9

1963.63

B#, C# 0#, E# D# A# 45\11

2076; 1.083

33\8

2084; 4.75

54\13

2090; 3.1

21\5

2100

51\12

2110; 2.9

30\7

2117; 1.54

39\9

2127.27

Cf, Qf 1b, 1d Ef Bf 47\11

2169; 4.3

34\8

2147; 2, 1.4

55\13

2129; 31

50\12

2068; 1, 28

29\7

2047; 17

37\9

2018.18

C, Q 1 E B 48\11

2215; 2.6

35\8

2210; 1.9

57\13

2206; 2, 4.6

22\5

2200

53\12

2193; 9.6

31\7

2188; 4.25

40\9

2181.81

C#, Q# 1# E# B# 49\11

2261; 1, 1.16

36\8

2273; 1, 2.16

59\13

2083; 1.148

23\5

2300

56\12

2327; 4, 7

33\7

2329; 2, 2.3

43\9

2345.45

Qf, Df 2b, 2d Ff Cf 51\11

2353; 1.18

37\8

2336; 1.1875

61\13

2322; 1.72

55\12

2275; 1.16

32\7

2258; 1, 4.6

41\9

2236.36

Q, D 2 F C 52\11

2400

38\8

2400

62\13

2400

24\5

2400

58\12

2400

34\7

2400

44\9

2400

Q#, D# 2# F# C# 53\11

2446; 6.5

39\8

2463; 6.3

64\13

2477; 2, 2.6

25\5

2500

61\12

2524; 7.25

36\7

2541; 5.6

47/9

2563.63

Df, Sf 3b, 3d 1f Df 54\11

2492; 3.25

63\13

2438; 1.136

24\5

2400

57\12

2358; 1.61̄

33\7

2329; 2, 2.3

42\9

2390.90

D, S 3 1 D 55\11

2538; 2.16

40\8

2526; 3.16

65\13

2516; 7.75

25\5

2500

60\12

2482; 1.318

35\7

2470; 1.7

45\9

2454.54

D#, S# 3# 1# D# 56\11

2584; 1.625

41\8

2589; 2.1̄

26\5

2600

63\12

2606; 1, 8.6

37\7

2611; 1, 3.25

48\9

2618.18

Ef 4b, 4d 2f Ef 58\11

2676; 1.083

42\8

2652; 1.583

62\12

2565; 1.93

36\7

2541; 5.6

46\9

2509.09

E 4 2 E 59\11

2723; 13

43\8

2715; 1.26

27\5

2700

65\12

2689; 1, 1.9

38\7

2682; 2.83

49\9

2672.72

E# 4# 2# E# 60\11

2769; 4.3

44\8

2778; 1.05̄

28\5

2800

68\12

2813; 1, 3.83

40\7

2823; 1.8

52\9

2836.36

Ff 5b, 5d 3f Ff 62\11

2861; 1, 1.16

45\8

2842; 9.5

67\12

2772; 29

39\7

2752; 1.0625

50\9

2727.27

F 5 3 F 63\11

2907; 1.4

46\8

2905; 3.8

29\5

2900

70\12

2896; 1.8125

41\7

2894; 8.5

53\9

2890.90

F# 5# 3# F# 64\11

2953; 1.18

47\8

2968; 2.375

30\5

3000

73\12

3020; 1.45

43\7

3035; 3,4

55\9

3000

Gf 6b, 6d 4f 0f 65\11

3000

29\5

2900

69\29

2855; 5.8

40\7

2823; 1.8

52\9

2836.36

G 6 4 0 66\11

3046; 6.5

48\8

3031; 1.72

30\5

3000

72\12

2979; 3.2

42\7

2964; 1, 3.25

54\9

2945.45

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 Do, Sol perfect unison 0 Do, Sol perfect fourth
1 Mib, Sib diminished third -1 Re, La perfect second
2 Reb, Lab diminished second -2 Mi, Si perfect third
The chromatic 5-note MOS also has the following intervals (from some root):
3 Dob, Solb diminished fourth -3 Do#, Sol# augmented unison (chroma)
4 Mibb, Sibb doubly diminished third -4 Re#, La# augmented second

Genchain

The generator chain for this scale is as follows:

Mibb

Sibb

Dob

Solb

Reb

Lab

Mib

Sib

Do

Sol

Re

La

Mi

Si

Do#

Sol#

Re#

La#

Mi#

Si#

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 LsLL 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[1] L s L/s Comments
1\3 171; 2.3 1 1 1.000 Equalised
6\17 180 6 5 1.200
11\31 180; 1.216 11 9 1.222
5\14 181.81 5 4 1.250
14\39 182; 1, 1.5 14 11 1.273
9\25 183; 19.6 9 7 1.286
4\11 184; 1.625 4 3 1.333
15\41 185; 1.763 15 11 1.364
11\30 185, 1, 10.83 11 8 1.375
7\19 186.6 7 5 1.400
10\27 187.5 10 7 1.429
13\35 187; 1, 19.75 13 9 1.444
16\43 188; 4.25 16 11 1.4545
3\8 189; 2.1 3 2 1.500 Mahuric-Meantone starts here
14\37 190.90 14 9 1.556
11\29 191; 3, 2.3 11 7 1.571
8\21 192 8 5 1.600
5\13 193; 1, 1, 4.6 5 3 1.667
12\31 194.594 12 7 1.714
7\18 195; 2.86 7 4 1.750
9\23 196.36 9 5 1.800
11\28 197; 67 11 6 1.833
13\33 197; 2.135 13 7 1.857
15\38 197; 1, 2, 1, 1.54 15 8 1.875
17\43 198; 17.16 17 9 1.889
19\48 198: 3, 1, 28 19 10 1.900
21\53 198; 2.3518 21 11 1.909
23\58 198; 1, 3, 1.7 23 12 1.917
25\63 198; 1, 2, 12.25 25 13 1.923
27\68 198; 1, 3.405 27 14 1.929
29\73 198; 1, 1.16 29 15 1.933
31\78 198; 1, 12, 2.8 31 16 1.9375
33\83 198; 1.005 33 17 1.941
35\88 199; 19.18 35 18 1.944
2\5 200 2 1 2.000 Mahuric-Meantone ends, Mahuric-Pythagorean begins
17\42 201.9801 17 8 2.125
15\37 202; 4.045 15 7 2.143
13\32 202; 1, 1, 2.06 13 6 2.167
11\27 203; 13 11 5 2.200
9\22 203; 1, 3.416 9 4 2.250
7\17 204; 1. 7.2 7 3 2.333
12\29 205; 1.4 12 5 2.400
5\12 206; 1, 8.6 5 2 2.500 Mahuric-Neogothic heartland is from here…
18\43 207; 1.4 18 7 2.571
13\31 208 13 5 2.600
8\19 208; 1.4375 8 3 2.667 …to here
11\26 209; 1.90 11 4 2.750
14\33 210 14 5 2.800
3\7 211; 1, 3.25 3 1 3.000 Mahuric-Pythagorean ends, Mahuric-Superpyth begins
22\51 212; 1, 9.3 22 7 3.143
19\44 213; 11.8 19 6 3.167
16\37 213.3̄ 16 5 3.200
13\30 213; 1, 2.318 13 4 3.250
10\23 214; 3.5 10 3 3.333
7\16 215; 2.6 7 2 3.500
11\25 216; 2.5416 11 3 3.667
15\34 216; 1.1527 15 4 3.750
19\43 217; 7 19 5 3.800
4\9 218.18 4 1 4.000
13\29 219; 1, 2.55 13 3 4.333
9\20 220; 2.45 9 2 4.500
14\31 221; 19 14 3 4.667
5\11 222.2 5 1 5.000 Mahuric-Superpyth ends
11\24 223; 1, 2.6875 11 2 5.500
17\37 224; 5.72 17 3 5.667
6\13 225 6 1 6.000
1\3 240 1 0 → inf Paucitonic

See also

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

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

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

4L 2s (9/5-equivalent) - Mixolydian 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) - 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 (30/7-equivalent) - Hyperionic Meantone temperament

  1. ↑ 1.0 1.1 Fractions repeating more than 4 digits written as continued fractions
Retrieved from "https://en.xen.wiki/index.php?title=User:Moremajorthanmajor/2L_1s_(perfect_fourth-equivalent)&oldid=118336"