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Porcupine: Difference between revisions

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m →Interval chain: cleanup; bold to odd harmonics; sort intervals by complexity
→Interval chain: adopt CWE tuning
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|-
|-
| 0
| 0
| 0.00
| 0.0
| '''1/1'''
| '''1/1'''
| P1
| P1
| 0
| 0
| 1200.00
| 1200.0
| '''2/1'''
| '''2/1'''
| P8
| P8
|-
|-
| 1
| 1
| 162.75
| 162.8
| 10/9, 11/10, 12/11
| 10/9, 11/10, 12/11
| vM2 = ^^m2
| vM2 = ^^m2
| -1
| -1
| 1037.25
| 1037.2
| 9/5, 11/6, 20/11
| 9/5, 11/6, 20/11
| ^m7 = vvM7
| ^m7 = vvM7
|-
|-
| 2
| 2
| 325.50
| 325.6
| 6/5, 11/9
| 6/5, 11/9
| ^m3 = vvM3
| ^m3 = vvM3
| -2
| -2
| 874.50
| 874.4
| 5/3, 18/11
| 5/3, 18/11
| vM6 = ^^m6
| vM6 = ^^m6
|-
|-
| 3
| 3
| 488.25
| 488.4
| 4/3
| 4/3
| P4
| P4
| -3
| -3
| 711.75
| 711.6
| '''3/2'''
| '''3/2'''
| P5
| P5
|-
|-
| 4
| 4
| 651.00
| 651.3
| 16/11, 22/15
| 16/11, 22/15
| v5 = ^^d5
| v5 = ^^d5
| -4
| -4
| 549.00
| 548.7
| '''11/8''', 15/11
| '''11/8''', 15/11
| ^4 = vvA4
| ^4 = vvA4
|-
|-
| 5
| 5
| 813.75
| 814.1
| 8/5
| 8/5
| ^m6 = vvM6
| ^m6 = vvM6
| -5
| -5
| 386.25
| 385.9
| '''5/4'''
| '''5/4'''
| vM3 = ^^m3
| vM3 = ^^m3
|-
|-
| 6
| 6
| 976.50
| 976.9
| '''7/4''', 16/9
| '''7/4''', 16/9
| m7
| m7
| -6
| -6
| 223.50
| 223.1
| 8/7, '''9/8'''
| 8/7, '''9/8'''
| M2
| M2
|-
|-
| 7
| 7
| 1139.25
| 1139.7
| 48/25, 160/81
| 48/25, 160/81
| v8 = ^^d8
| v8 = ^^d8
| -7
| -7
| 60.75
| 60.3
| 25/24, 81/80
| 25/24, 81/80
| ^1 = vvA1
| ^1 = vvA1
|-
|-
| 8
| 8
| 102.00
| 102.5
| 16/15, 21/20
| 16/15, 21/20
| ^m2 = vvM2
| ^m2 = vvM2
| -8
| -8
| 1098.00
| 1097.5
| 15/8, 40/21
| 15/8, 40/21
| vM7 = ^^m7
| vM7 = ^^m7
|-
|-
| 9
| 9
| 264.75
| 265.3
| 7/6
| 7/6
| m3
| m3
| -9
| -9
| 935.25
| 934.7
| 12/7
| 12/7
| M6
| M6
|-
|-
| 10
| 10
| 427.50
| 428.2
| 14/11
| 14/11
| v4 = ^^d4
| v4 = ^^d4
| -10
| -10
| 772.50
| 771.8
| 11/7
| 11/7
| ^5 = vvA5
| ^5 = vvA5
|-
|-
| 11
| 11
| 590.25
| 591.0
| 7/5
| 7/5
| ^d5 = vv5
| ^d5 = vv5
| -11
| -11
| 609.75
| 609.0
| 10/7
| 10/7
| vA4 = ^^4
| vA4 = ^^4
|-
|-
| 12
| 12
| 753.00
| 753.8
| 14/9
| 14/9
| m6
| m6
| -12
| -12
| 447.00
| 446.2
| 9/7
| 9/7
| M3
| M3
|}
|}


The specific tuning shown is the full 11-limit [[POTE tuning]], but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents ([[15edo]]) and as large as 165.5 cents ([[29edo]]). (However, the 29edo patent val does not support 11-limit porcupine proper, since it does not temper out [[64/63]].)
The specific tuning shown is the full 11-limit [[CWE tuning]], but of course there is a range of acceptable porcupine tunings that includes generators as small as 160 cents ([[15edo]]) and as large as 165.5 cents ([[29edo]]). However, the 29edo patent val does not support 11-limit porcupine proper, since it does not temper out [[64/63]].


[[12/11]], [[11/10]], and [[10/9]] are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.
[[12/11]], [[11/10]], and [[10/9]] are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.
Line 165: Line 165:
The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". This means that the [[27/20]] "acute fourth" of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone).
The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". This means that the [[27/20]] "acute fourth" of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone).


The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both [[25/24]] and [[81/80]].
The characteristic small interval of porcupine, which is 60.3 cents in this tuning but can range from < 50 to 80 cents in general, represents both [[25/24]] and [[81/80]].


== Chords ==
== Chords ==
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