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

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[[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
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