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Added interval names to Porcutone octatonic intervals table, plus a few other little things
Line 172: Line 172:
|+Intervals of the porcutone diatonic
|+Intervals of the porcutone diatonic
!Interval class
!Interval class
!sizes
!size
!Meantone[7] names
!Meantone[7] name
!Porcupine[7] names
!Porcupine[7] name
!Porcutone diatonic names
!Porcutone diatonic names
!JI ratios approximated
!JI ratios approximated
Line 194: Line 194:
major 2nd
major 2nd
|small 2nd, minor 2nd
|small 2nd, minor 2nd
medium 2nd, major minor 2nd
medium 2nd, major-minor 2nd


large 2nd, major 2nd
large 2nd, major 2nd
Line 224: Line 224:
major 3rd
major 3rd
|small 3rd, minor 3rd
|small 3rd, minor 3rd
medium 3rd, major minor 3rd
medium 3rd, major-minor 3rd


large 3rd, major third
large 3rd, major third
Line 254: Line 254:
major 4th
major 4th
|small 4th, minor 4th
|small 4th, minor 4th
medium 4th, minor major 4th
medium 4th, minor-major 4th


large 4th, major 4th
large 4th, major 4th
Line 284: Line 284:
major 5th
major 5th
|small 5th, minor 5th
|small 5th, minor 5th
medium 5th, major minor 5th
medium 5th, major-minor 5th


large 5th, major 5th
large 5th, major 5th
Line 314: Line 314:
major 6th
major 6th
|small 6th, minor 6th
|small 6th, minor 6th
medium 6th, minor major 6th
medium 6th, minor-major 6th


large 6th, major 6th
large 6th, major 6th
Line 344: Line 344:
major 7th
major 7th
|small 7th, minor 7th
|small 7th, minor 7th
medium 7th, minor major 7th
medium 7th, minor-major 7th


large 7th, major 7th
large 7th, major 7th
Line 607: Line 607:
minor
minor
|minor
|minor
major minor
major-minor
|10:12:15
|10:12:15
18:22:27
18:22:27
Line 666: Line 666:
|major
|major
major
major
|minor major
|minor-major
major
major
|22:27:33
|22:27:33
Line 925: Line 925:
The step signature and mapping of Father[8] is 5L 3s = (10/9~25/24, 27/25~81/80),
The step signature and mapping of Father[8] is 5L 3s = (10/9~25/24, 27/25~81/80),
{| class="wikitable"
{| class="wikitable"
|+Father[8] oneirotonic mode names
|+Father[8] oneirotonic modes
!Mode number
!Mode number
!Step pattern
!Step pattern
Line 991: Line 991:
For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the [[oneirotonic]] mode names associated with Father[8]. Like in the tables of modes of the porcutone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.
For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the [[oneirotonic]] mode names associated with Father[8]. Like in the tables of modes of the porcutone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.
{| class="wikitable"
{| class="wikitable"
|+modes of the left handed just porcutone octatonic
|+Modes of the left handed just porcutone octatonic
!Mode in JI
!Mode in JI
!Step pattern
!Step pattern
Line 1,070: Line 1,070:
|}
|}
{| class="wikitable"
{| class="wikitable"
|+modes of the right handed just porcutone octatonic
|+Modes of the right handed just porcutone octatonic
!Mode in JI
!Mode in JI
!Step pattern
!Step pattern
Line 1,249: Line 1,249:


=== Intervals and triads ===
=== Intervals and triads ===
{| class="wikitable"
|+3-step stacked triads of the left handed porcutone octatonic (G♯-G gamut)
!Mode name
!Step pattern
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Porcutone octatonic name
!JI triad approximated
|-
|Celephaïsian dark quartal
|LMLLMLsM
|A-D-G
|3-step major major
|3-step major major
|3-step major major
|9:12:16
|-
|Sarnathian bright quartal
|MLMLLMLs
|G♯-C-F
|3-step minor major
|3-step major major
|3-step minor-major major
|30:39:52 (7:9:12, 16:21:28)
|-
|Dylathian middle major
|LLMLsMLM
|C-F-A
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
|Kadathian bright major
|MLLMLsML
|B-E-G♯
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
| Ultharian dark major
|LMLsMLML
|D-G-B
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
| Illarnekian middle minor
|LsMLMLLM
|F-A-D
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|-
| Hlanithian bright minor
|MLsMLMLL
|E-G♯-C
|3-step major minor
|3-step minor major
|3-step major-minor minor-major
|8:10:13
|-
| Mnarian dark minor
|sMLMLLML
|G-B-E
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|}
{| class="wikitable"
|+3-step stacked triads of the right handed porcutone octatonic (G-A♭ gamut)
!Mode name
!Step pattern
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Porcutone octatonic name
!JI triad approximated
|-
|Ultharian bright quartal
|LMLLMLMs
|A-D-G
|3-step major major
|3-step major major
|3-step major major
|9:12:16
|-
|Illarnekian bright major
|LLMLMsLM
|C-F-A
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
|Hlanithian dark quartal
|MLLMLMsL
|B-E-A♭
|3-step major minor
|3-step major major
|3-step major major
|15:20:26 (12:16:21)
|-
|Mnarian middle major
|LMLMsLML
|D-G-B
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
| Celephaïsian bright minor
|LMsLMLLM
|F-A-D
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|-
| Sarnathian dark major
|MLMsLMLL
|E-A♭-C
|3-step minor major
|3-step major minor
|3-step minor-major major-minor
|55:72:90 (16:21:26)
|-
| Dylathian dark minor
|sLMLLMLM
|A♭-C-F
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|-
| Kadathian middle minor
|MsLMLLML
|G-B-E
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|}
The following table gives all intervals of the porcutone octatonic.
The following table gives all intervals of the porcutone octatonic.
{| class="wikitable"
{| class="wikitable"
Line 1,404: Line 1,254:
!Interval class
!Interval class
!sizes
!sizes
!Oneirotonic name
!Porcupine[8] name
!Porcutone octatonic name
!JI ratios approximated
!JI ratios approximated
!size in cents (TE)
!size in cents (TE)
Line 1,413: Line 1,266:


L
L
|major step
minor step
major step
|minor step
major step
major step
|major-minor step
minor-major step
major step
|25/24, 33/32, 27/26
|25/24, 33/32, 27/26
27/25, 12/11, 13/12
27/25, 12/11, 13/12
Line 1,434: Line 1,299:


L + L
L + L
|minor 2-step
major 2-step
minor 2-step
major 2-step
|minor 2-step
minor 2-step
major 2-step
major 2-step
|minor 2-step
major-minor 2-step
minor-major 2-step
major 2-step
|9/8
|9/8
15/13 (7/6 or 8/7)
15/13 (7/6 or 8/7)
Line 1,459: Line 1,342:


2L + M
2L + M
|major 3-step
minor 3-step
major 3-step
|minor 3-step
major 3-step
major 3-step
|major-minor 3-step
minor-major 3-step
major 3-step
|5/4
|5/4
13/10 (9/7 or 21/16)
13/10 (9/7 or 21/16)
Line 1,480: Line 1,375:


3L + M
3L + M
|minor 4-step
major 4-step
minor 4-step
major 4-step
|minor 4-step
minor 4-step
major 4-step
major 4-step
|minor 4-step
major-minor 4-step
minor-major 4-step
major 4-step
|27/20, 15/11
|27/20, 15/11
25/18, 11/8, 18/13
25/18, 11/8, 18/13
Line 1,505: Line 1,418:


3L + 2M
3L + 2M
|minor 5-step
major 5-step
minor 5-step
|minor 5-step
minor 5-step
major 5-step
|minor 5-step
major-minor 5-step
minor-major 5-step
|3/2
|3/2
20/13 (14/9 or 32/16)
20/13 (14/9 or 32/16)
Line 1,526: Line 1,451:


4L + 2M
4L + 2M
|minor 6-step
major 6-step
minor 6-step
major 6-step
|minor 6-step
minor 6-step
major 6-step
major 6-step
|minor 6-step
major-minor 6-step
minor-major 6-step
major 6-step
|18/11, 13/8
|18/11, 13/8
5/3
5/3
Line 1,551: Line 1,494:


4L + 3M
4L + 3M
|minor 7-step
major 7-step
minor 7-step
|minor 7-step
minor 7-step
major 7-step
|minor 7-step
major-minor 7-step
minor-major 7-step
|9/5, 20/11
|9/5, 20/11
50/27, 11/6, 24/13
50/27, 11/6, 24/13
Line 1,565: Line 1,520:
1
1
|}
|}
The following two tables detail the 3-step stacked triads of the left and right handed porcutone octatonics:
{| class="wikitable"
|+3-step stacked triads of the left handed porcutone octatonic (G♯-G gamut)
!Mode name
!Step pattern
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Porcutone octatonic name
!JI triad approximated
|-
|Celephaïsian dark quartal
|LMLLMLsM
|A-D-G
|3-step major major
|3-step major major
|3-step major major
|9:12:16
|-
|Sarnathian bright quartal
|MLMLLMLs
|G♯-C-F
|3-step minor major
|3-step major major
|3-step minor-major major
|30:39:52 (7:9:12, 16:21:28)
|-
|Dylathian middle major
|LLMLsMLM
|C-F-A
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
|Kadathian bright major
|MLLMLsML
|B-E-G♯
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
| Ultharian dark major
|LMLsMLML
|D-G-B
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
| Illarnekian middle minor
|LsMLMLLM
|F-A-D
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|-
| Hlanithian bright minor
|MLsMLMLL
|E-G♯-C
|3-step major minor
|3-step minor major
|3-step major-minor minor-major
|8:10:13
|-
| Mnarian dark minor
|sMLMLLML
|G-B-E
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|}
{| class="wikitable"
|+3-step stacked triads of the right handed porcutone octatonic (G-A♭ gamut)
!Mode name
!Step pattern
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Porcutone octatonic name
!JI triad approximated
|-
|Ultharian bright quartal
|LMLLMLMs
|A-D-G
|3-step major major
|3-step major major
|3-step major major
|9:12:16
|-
|Illarnekian bright major
|LLMLMsLM
|C-F-A
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
|Hlanithian dark quartal
|MLLMLMsL
|B-E-A♭
|3-step major minor
|3-step major major
|3-step major major
|15:20:26 (12:16:21)
|-
|Mnarian middle major
|LMLMsLML
|D-G-B
|3-step major major
|3-step major minor
|3-step major major-minor
|3:4:5
|-
| Celephaïsian bright minor
|LMsLMLLM
|F-A-D
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|-
| Sarnathian dark major
|MLMsLMLL
|E-A♭-C
|3-step minor major
|3-step major minor
|3-step minor-major major-minor
|55:72:90 (16:21:26)
|-
| Dylathian dark minor
|sLMLLMLM
|A♭-C-F
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|-
| Kadathian middle minor
|MsLMLLML
|G-B-E
|3-step major major
|3-step minor major
|3-step major-minor major
|12:15:20
|}
Unlike the Porcutone diatonic, and chromatic scales, the porcutone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the porcutone diatonic. The Porcutone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the porcutone octatonic is more complex than the porcutone pentatonic as well. It is left as an excercise for the reader to determine the complexity of the porcutone chromatic, and compare that to the porcutone octatonic.
Unlike the Porcutone diatonic, and chromatic scales, the porcutone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the porcutone diatonic. The Porcutone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the porcutone octatonic is more complex than the porcutone pentatonic as well. It is left as an excercise for the reader to determine the complexity of the porcutone chromatic, and compare that to the porcutone octatonic.
== Summary for xen-math nerds ==
== Summary for xen-math nerds ==
The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.  
The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.  
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