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m →Summary for xen-math nerds: fixed typo in new sentence added previous edit defining bounds for scales that could be considered part of the Porcutone system
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added section on porcutone diminished, changed porcutone-15 to be the SNS child of porcutone diminished, updated xen-math summary, changed interval names of porcutone octatonic to have universal intervals namesfor porcutone octatonic and diminished scales.
Line 1,269: Line 1,269:
Note that the darkest mode of the LH octatonic is the brightest mode of the RH octatonic, etc.
Note that the darkest mode of the LH octatonic is the brightest mode of the RH octatonic, etc.


Tempering out 100/99, the large step (174.05488c) represents 10/9~11/10, the medium step (146.63528c) represents 27/25~12/11, and the small step (63.14327c) represents 25/24~33/32. The following tables display the JI intervals approximated by the modes of the ptolemismic Porcutone octatonic scales, along with the scale steps in cents.
Tempering out 100/99, the large step (174.05488c) represents 10/9~11/10, the medium step (146.63528c) represents 27/25~12/11, and the small step (63.14327c) represents 25/24~33/32. The following tables display the JI intervals approximated by the modes of the ptolemismic Porcutone octatonic scales, along with the scale steps in cents. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4p%263p%261ce&limit=2.3.5.11 TE tuning].


tempering out 144/143 as well, the large step is tuned to 175.89183c TE, medium step (142.77537c TE) also represents 13/12, and the small step (66.76626c TE) also represents 27/26. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning].
Tempering out 144/143 as well, the large step is tuned to 175.89183c TE, medium step (142.77537c TE) also represents 13/12, and the small step (66.76626c TE) also represents 27/26. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning].
{| class="wikitable"
{| class="wikitable"
|+Modes of the left handed ptolemismic Porcutone octatonic
|+Modes of the left handed ptolemismic Porcutone octatonic
Line 1,376: Line 1,376:
!Porcupine[8] name
!Porcupine[8] name
!Porcutone octatonic name
!Porcutone octatonic name
!JI ratios approximated
!JI ratios approximated*
!size in cents (TE)
!size in cents (TE)
!Occurence
!Occurence
Line 1,393: Line 1,393:


major step
major step
|major-minor step
|minor step
minor-major step
minor-major step


Line 1,430: Line 1,430:


major 2-step
major 2-step
|minor 2-step
|small 2-step
major-minor 2-step
minor 2-step


minor-major 2-step
major 2-step


major 2-step
large 2-step
|9/8
|9/8
15/13 (7/6 or 8/7)
15/13 (7/6 or 8/7)
Line 1,469: Line 1,469:


major 3-step
major 3-step
|major-minor 3-step
|minor 3-step
minor-major 3-step
minor-major 3-step


Line 1,506: Line 1,506:


major 4-step
major 4-step
|minor 4-step
|small 4-step
major-minor 4-step
minor 4-step


minor-major 4-step
major 4-step


major 4-step
large 4-step
|27/20, 15/11
|27/20, 15/11
25/18, 11/8, 18/13
25/18, 11/8, 18/13
Line 1,548: Line 1,548:
major-minor 5-step
major-minor 5-step


minor-major 5-step
major 5-step
|3/2
|3/2
20/13 (14/9 or 32/16)
20/13 (14/9 or 32/16)
Line 1,582: Line 1,582:


major 6-step
major 6-step
|minor 6-step
|small 6-step
major-minor 6-step
minor 6-step


minor-major 6-step
major 6-step


major 6-step
large 6-step
|18/11, 13/8
|18/11, 13/8
5/3
5/3
Line 1,624: Line 1,624:
major-minor 7-step
major-minor 7-step


minor-major 7-step
major 7-step
|9/5, 20/11
|9/5, 20/11
50/27, 11/6, 24/13
50/27, 11/6, 24/13
Line 1,639: Line 1,639:
1
1
|}
|}
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
The following two tables detail the 3-step stacked triads of the left and right handed Porcutone octatonics:
The following two tables detail the 3-step stacked triads of the left and right handed Porcutone octatonics:
{| class="wikitable"
{| class="wikitable"
Line 1,645: Line 1,647:
!Step pattern
!Step pattern
!3-step stacked triad on root
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Porcupine[8] name
!Oneirotonic name
!Porcutone octatonic name
!Porcutone octatonic name
!JI triad approximated
!JI triad approximated*
|-
|-
|Celephaïsian dark quartal
|Celephaïsian dark quartal
Line 1,661: Line 1,663:
|MLMLLMLs
|MLMLLMLs
|G♯-C-F
|G♯-C-F
|[8] minor
|[8] augmented
|[8] augmented
|[8] minor
|[8] minor augmented
|[8] augmented minor
|30:39:52 (7:9:12, 16:21:28)
|30:39:52 (7:9:12, 16:21:28)
|-
|-
Line 1,669: Line 1,671:
|LLMLsMLM
|LLMLsMLM
|C-F-A
|C-F-A
|[8] augmented
|[8] major
|[8] major
|[8] augmented
|[8] major
|[8] major
|3:4:5
|3:4:5
Line 1,677: Line 1,679:
|MLLMLsML
|MLLMLsML
|B-E-G♯
|B-E-G♯
|[8] augmented
|[8] major
|[8] major
|[8] augmented
|[8] major
|[8] major
|3:4:5
|3:4:5
Line 1,685: Line 1,687:
|LMLsMLML
|LMLsMLML
|D-G-B
|D-G-B
|[8] augmented
|[8] major
|[8] major
|[8] augmented
|[8] major
|[8] major
|3:4:5
|3:4:5
Line 1,693: Line 1,695:
|LsMLMLLM
|LsMLMLLM
|F-A-D
|F-A-D
|[8] augmented
|[8] minor
|[8] minor
|[8] augmented
|[8] minor
|[8] minor
|12:15:20
|12:15:20
Line 1,701: Line 1,703:
|MLsMLMLL
|MLsMLMLL
|E-G♯-C
|E-G♯-C
|[8] major
|[8] minor
|[8] minor
|[8] major
|[8] major minor
|[8] minor major
|8:10:13
|8:10:13
|-
|-
Line 1,709: Line 1,711:
|sMLMLLML
|sMLMLLML
|G-B-E
|G-B-E
|[8] augmented
|[8] minor
|[8] minor
|[8] augmented
|[8] minor
|[8] minor
|12:15:20
|12:15:20
|}
|}
 
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
{| class="wikitable"
{| class="wikitable"
|+3-step stacked triads of the right handed Porcutone octatonic (G-A♭ gamut)
|+3-step stacked triads of the right handed Porcutone octatonic (G-A♭ gamut)
Line 1,720: Line 1,722:
!Step pattern
!Step pattern
!3-step stacked triad on root
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Porcupine[8] name
!Oneirotonic name
!Porcutone octatonic name
!Porcutone octatonic name
!JI triad approximated
!JI triad approximated*
|-
|-
|Ultharian bright quartal
|Ultharian bright quartal
Line 1,736: Line 1,738:
|LLMLMsLM
|LLMLMsLM
|C-F-A
|C-F-A
|[8] augmented
|[8] major
|[8] major
|[8] augmented
|[8] major
|[8] major
|3:4:5
|3:4:5
Line 1,744: Line 1,746:
|MLLMLMsL
|MLLMLMsL
|B-E-A♭
|B-E-A♭
|[8] major
|[8] augmented
|[8] augmented
|[8] major
|[8] major augmented
|[8] augmented major
|15:20:26 (12:16:21)
|15:20:26 (12:16:21)
|-
|-
Line 1,752: Line 1,754:
|LMLMsLML
|LMLMsLML
|D-G-B
|D-G-B
|[8] augmented
|[8] major
|[8] major
|[8] augmented
|[8] major
|[8] major
|3:4:5
|3:4:5
Line 1,760: Line 1,762:
|LMsLMLLM
|LMsLMLLM
|F-A-D
|F-A-D
|[8] augmented
|[8] minor
|[8] minor
|[8] augmented
|[8] minor
|[8] minor
|12:15:20
|12:15:20
Line 1,768: Line 1,770:
|MLMsLMLL
|MLMsLMLL
|E-A♭-C
|E-A♭-C
|[8] minor
|[8] major
|[8] major
|[8] minor
|[8] minor major
|[8] major minor
|55:72:90 (16:21:26)
|55:72:90 (16:21:26)
|-
|-
Line 1,776: Line 1,778:
|sLMLLMLM
|sLMLLMLM
|A♭-C-F
|A♭-C-F
|[8] augmented
|[8] minor
|[8] minor
|[8] augmented
|[8] minor
|[8] minor
|12:15:20
|12:15:20
Line 1,784: Line 1,786:
|MsLMLLML
|MsLMLLML
|G-B-E
|G-B-E
|[8] augmented
|[8] minor
|[8] minor
|[8] augmented
|[8] minor
|[8] minor
|12:15:20
|12:15:20
|}
|}
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament.


We could alternatively treat the Porcutone octatonic as a bebop scale, using 2-step stacked tetrads. Since the scale has 8 notes, there are only 2 different 2-step stacked tetrads. In 12edo these are the major add 6 and the fully diminished tetrads. The meantone C major add 6 tunes to 45:55:66:75 in Porcutone. Using the G♯, as in the left-handed Porcutone octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the right-handed Porcutone octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note).
We could alternatively treat the Porcutone octatonic as a bebop scale, using 2-step stacked tetrads. Since the scale has 8 notes, there are only 2 different 2-step stacked tetrads. In 12edo these are the major add 6 and the fully diminished tetrads. The meantone C major add 6 tunes to 45:55:66:75 in Porcutone. Using the G♯, as in the left-handed Porcutone octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the right-handed Porcutone octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note).


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 exercise for the reader to determine the complexity of the Porcutone chromatic, and compare that to the Porcutone octatonic.
 
=== Porcutone Diminished or Porcupine-Diminished ===
Modifying the right or left-handed Porcutone octatonic by switching the order of adjacent pairs of large and medium steps, i.e., by modifying steps of the scale by the L-M chroma - the difference between the large and medium steps - leads to similar Porcupine[8] detempers. Since L and M temper together under Porcupine tempering, any resulting scale tempers to Porcupine[8] just as before, but the scale it tempers to under Diminished and Father temperaments are modified.
 
Take the Ultharian dark major mode of the left-handed Porcutone octatonic, for example: LMLsMLML. Raising the sixth and eighth degrees of the scale by the L-M chroma leads to the mode LMLsLMLM. Similarly, taking the Mnarian middle major mode LMLMsLML of the right-handed Porcutone octatonic and raising the root and the third degree by the L-M chroma leads to the mode MLMLsLML. We can see that LMLsLMLM and MLMLsLML are modes of the same scale. We know that this scale tempers to Porcupine[8]; tempering M=s instead leads to LsLsLsLs, i.e., Diminished[8]; and finally tempering L=s leads to LsLLLsLs, a mod of Father[8]. Like the Porcutone chromatic and diatonic scale, this scale is an SN scale, and is therefor achiral. We may name this scale perhaps the Porcupine-Diminished scale, or we may include it in the Porcutone system as the Porcutone diminished scale. It may be more wise to refer to this scale as the Porcupine-Diminished scale to avoid confusion that might result if, while the octatonic scale and the diminished scale are different names for the same scale, the Porcutone octatonic and the Porcutone diminished scale are not.
 
Every other step of any mode of the Porcutone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Porcutone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Porcutone octatonic. Although the Porcutone Diminished is simpler in this way, the Porcutone octatonic provides more major and minor traids.
 
{| class="wikitable"
|+Modes of the just Porcutone Diminished scale (Porcupine-Diminished)
!Mode in JI
!Step pattern
!Porcupine[8]
step pattern and UDP
!Porcupine[8]
mode
!Diminished[8]
step pattern and UDP
!Porcutone diminished
mode
|-
|10/9 6/5 4/3 36/25 8/5 216/125 48/25 2/1
|LMLMLMLs
|<nowiki>LLLLLLLs 7|0</nowiki>
|Bright quartal
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Bright quartal diminished
|-
|10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1
|LMLMLsLM
|<nowiki>LLLLLsLL 5|2</nowiki>
|Bright major
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Bright major diminished
|-
|27/25 6/5 162/125 36/25 972/625 216/125 9/5 2/1
|MLMLMLsL
|<nowiki>LLLLLLsL 6|1</nowiki>
|Dark quartal
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Dark quartal diminished
|-
|10/9 6/5 4/3 25/18 125/81 5/3 50/27 2/1
|LMLsLMLM
|<nowiki>LLLsLLLL 3|4</nowiki>
|Dark major
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Dark major diminished
|-
|27/25 6/5 162/125 36/25 3/2 5/3 9/5 2/1
|MLMLsLML
|<nowiki>LLLLsLLL 4|3</nowiki>
|Middle major
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Middle major diminished
|-
|10/9 125/108 625/486 25/18 125/81 5/3 50/27 2/1
|LsLMLMLM
|<nowiki>LsLLLLLL 1|6</nowiki>
|Middle minor
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Middle minor diminished
|-
|27/25 6/5 5/4 25/18 3/2 5/3 9/5 2/1
|MLsLMLML
|<nowiki>LLsLLLLL 2|5</nowiki>
|Bright minor
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Bright minor diminished
|-
|25/24 125/108 5/4 25/18 3/2 5/3 9/5 2/1
|sLMLMLML
|<nowiki>sLLLLLLL 0|7</nowiki>
|Dark minor
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Dark minor diminished
|}
{| class="wikitable"
|+Modes of the Ptolemismic Porcutone Diminished scale (Porcupine-Diminished)
!Mode name
!Step pattern
!Mode as simplest JI pre-image
!Mode in cents
|-
|Bright quartal diminished
|LMLMLMLs
|~ 10/9 6/5 4/3 13/9 8/5 26/15 48/25 2/1
|175.892 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
|-
|Bright major diminished
|LMLMLsLM
|~ 10/9 6/5 4/3 13/9 8/5 5/3 11/6 2/1
|175.892 318.667 494.559 637.334 813.226 879.993 1055.884 1198.660
|-
|Dark quartal diminished
|MLMLMLsL
|~ 12/11 6/5 13/10 13/9 39/25 26/15 9/5 2/1
|142.775 318.667 461.443 637.334 780.120 956.002 1022.768 1198.660
|-
|Dark major diminished
|LMLsLMLM
|~ 10/9 6/5 4/3 13/9 20/13 5/3 11/6 2/1
|175.892 318.667 494.559 561.325 737.218 879.993 1055.884 1198.660
|-
|Middle major diminished
|MLMLsLML
|~ 12/11 6/5 13/10 13/9 3/2 5/3 9/5 2/1
|142.775 318.667 461.443 637.334 704.101 879.993 1022.768 1198.660
|-
|Middle minor diminished
|LsLMLMLM
|~ 10/9 15/13 50/39 11/8 20/13 5/3 11/6 2/1
|175.892 242.658 418.550 561.325 737.218 879.993 1055.884 1198.660
|-
|Bright minor diminished
|MLsLMLML
|~ 12/11 6/5 5/4 11/8 3/2 5/3 9/5 2/1
|142.775 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660
|-
|Dark minor diminished
|sLMLMLML
|~ 25/24 15/13 5/4 11/8 3/2 5/3 9/5 2/1
|66.766 242.658 385.433 561.325 704.101 879.993 1022.768 1198.660
|}
{| class="wikitable"
|+Intervals of the Porcutone diminished (Porcupine-Diminished)
!Interval class
!sizes
!Diminished[8] name
!Porcupine[8] name
!Porcutone octatonic name
!JI ratios approximated*
!size in cents (TE)
!Occurence
|-
!1-step
|s
M
 
L
|minor step
minor step
 
major step
|minor step
major step
 
major step
|minor step
minor-major step
 
major step
|25/24, 33/32, 27/26
27/25, 12/11, 13/12
 
10/9, 11/10
|66.766
 
142.775
 
175.892
|1
3
 
4
|-
!2-step
|L + s
L + M
|(perfect) 2-step
(perfect) 2-step
|minor 2-step
major 2-step
|minor 2-step
major 2-step
|15/13 (7/6 or 8/7)
6/5
|242.658
 
318.667
|2
6
|-
!3-step
|L + M + s
2L + s
L + 2M
 
2L + M
|minor 3-step
major 3-step
minor 3-step
 
major 3-step
|minor 3-step
minor 3-step
major 3-step
 
major 3-step
|minor 3-step
major-minor 3-step
minor-major 3-step
 
major 3-step
|5/4
50/39
13/10 (9/7 or 21/16)
 
4/3
|385.433
418.550
 
461.433
 
494.559
|2
1
2
 
3
|-
!4-step
|2L + M + s
2L + 2M
|(perfect) 4-step
(perfect) 4-step
|minor 4-step
major 4-step
|minor 4-step
major 4-step
|25/18, 11/8, 18/13
36/25, 16/11, 13/9
|561.325
 
637.334
|4
4
|-
!5-step
|2L + 2M + s
3L + M + s
 
2L + 3M
 
3L + 2M
|minor 5-step
major 5-step
 
minor 5-step
 
major 5-step
|minor 5-step
minor 5-step
 
major 5-step
 
major 5-step
|minor 5-step
major-minor 5-step
 
minor-major 5-step
 
major 5-step
|3/2
20/13 (14/9 or 32/16)
 
39/25
 
8/5
|704.101
737.217
 
780.120
 
813.227
|3
2
1
 
2
|-
!6-step
|3L + 2M + s
3L + 3M
|(perfect) 6-step
(perfect) 6-step
|minor 6-step
major 6-step
|minor 6-step
major 6-step
|5/3
26/15 (12/7 or 7/4)
|879.992
 
956.002
|6
2
|-
!7-step
|3L + 3M + s
4L + 2M + s
 
4L + 3M
|minor 7-step
major 7-step
 
major 7-step
|minor 7-step
minor 7-step
 
major 7-step
|minor 7-step
major-minor 7-step
 
major 7-step
|9/5, 20/11
50/27, 11/6, 24/13
 
48/25, 64/33, 52/27
|1022.768
 
1055.884
 
1131.983
|4
3
 
1
|}
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
 
{| class="wikitable"
|+3-step stacked triads of the Porcutone diminished (Porcupine-Diminished)
!Mode name
!Step pattern
!Diminished[8] name
!Porcupine[8] name
!Porcutone octatonic name
!JI triad approximated*
|-
|Bright quartal diminished
|LMLMLMLs
|[8] major
|[8] augmented
|[8] major augmented
|15:20:26 (12:16:21)
|-
|Bright major diminished
|LMLMLsLM
|[8] major
|[8] major
|[8] major
|3:4:5
|-
|Dark quartal diminished
|MLMLMLsL
|[8] minor
|[8] augmented
|[8] minor augmented
|30:39:52 (7:9:12, 16:21:28)
|-
|Dark major diminished
|LMLsLMLM
|[8] major
|[8] major
|[8] major
|3:4:5
|-
| Middle major diminished
|MLMLsLML
|[8] minor
|[8] major
|[8] minor major
|55:72:90 (16:21:26)
|-
| Middle minor diminished
|LsLMLMLM
|[8] major
|[8] minor
|[8] major minor
|39:50:65
|-
| Bright minor diminished
|MLsLMLML
|[8] minor
|[8] minor
|[8] minor
|12:15:20
|-
| Dark minor diminished
|sLMLMLML
|[8] minor
|[8] minor
|[8] minor
|12:15:20
|}
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
 
== 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. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note SNS 6/5 3/2 9/5 2/1.  
The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note SNS 6/5 3/2 9/5 2/1.  
Line 1,800: Line 2,202:
For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue).
For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue).


The Porcutone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) has 1 large step of 9/8 (L x L), 3 medium steps of 10/9 (L x s), and 3 small steps of 27/25 (s x s).
The Porcutone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) comprising 1 large step of 9/8 (''L'' x ''L''), 3 medium steps of 10/9 (''L'' x ''s''), and 3 small steps of 27/25 (''s'' x ''s'').
 
The Porcutone octatonic is an 8-note rank-3 [[Porcupine]][8] x [[Father]][8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 16/15, and 648/625; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''L'').
 
<nowiki>:</nowiki>The Porcutone diminished scale is a [[step-nested scale]] and a [[Porcupine]][8] x Diminished[8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 648/625, and 16/15; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''s'').


*  
*  
Line 1,869: Line 2,275:


== Porcutone-15 ==
== Porcutone-15 ==
Alternatively, a 15-note scale can be built from the Porcutone octatonic. The resulting scale tempers to Porcupine[15].
Alternatively, a 15-note scale can be built from the Porcutone diminished. The resulting scale tempers to Porcupine[15], as well as to Hanson[11].


Let's start with the just left-handed Porcutone octatonic: 27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1 in the Kadathian bright major mode. It contains 4 large steps of 10/9, 3 medium steps of 25/24 and 1 small step of 25/24. Putting a small step into the bottom of each medium and large step results in the 15-note scale
Starting instead with the Porcutone diminished scale: MLsLMLML, shown in the bright minor mode as porcutone bright minor diminished. Putting a small step into the bottom of each medium and large step leads to the child SNS of the porcutone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e., 


25/24 27/25 9/8 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 125/72 9/5 15/8 2/1, with step pattern msmLmLmsmLmmsmL, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625. This is right-handed Porcutone-15.
25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 48/25 2/1.


Starting instead with the just right-handed Porcutone octatonic: 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 in the Mnarian middle major mode, leads to the scale
Tempering m = s (tempering out 15625/15552, the Hanson comma) results in sLsssLsssLsssLs, which is Hanson[15];


25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 25/16 5/3 125/72 9/5 15/8 2/1, with step pattern mLmsmLmsmmLmsmL. We can see that this scale is the inverse of right-handed Porcutone-15; accordingly it is left-handed Porcutone-15.
tempering L = m (tempering out 128/125, the Augmented comma) results in LLLsLLLsLLLsLLL, which is a MODMOS of Augmented[15]


Tempering out 100/99, the ptolemismic right-handed and left-handed Porcutone-15 scales approximate
tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].


25/24 12/11 9/8 6/5 5/4 4/3 11/8 16/11 3/2 8/5 5/3 55/32 9/5 15/8 2/1 and 
tempering out s would lead to sLLsLsL, which is Dicot[7];


25/24 10/9 55/48 6/5 5/4 4/3 11/8 16/11 3/2 25/16 5/3 55/32 9/5 15/8 2/1 respectively.
tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].


If 144/143 is tempered out additionally, leading to 2.3.5.11.13 ptolemismic tuning, the scales may be more simply written as approximating
Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 48/25 2/1.


25/24 12/11 9/8 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 45/26 9/5 15/8 2/1 and 
With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. Porcutone-15 comprises
 
25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 25/16 5/3 45/26 9/5 15/8 2/1 respectively.
 
With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. The Porcutone-15 right and left-handed scales comprise


4 large steps of 109.12557c, approximating 16/15;  
4 large steps of 109.12557c, approximating 16/15;  
Line 1,899: Line 2,301:
8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.
8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.


In cents, TE 2.3.5.11.13 ptolemismic right and left-handed Porcutone-15 (in the above modes) are
In cents, TE 2.3.5.11.13 ptolemismic Porcutone-15, in the symmetric mode, is
 
66.766 142.775 209.542 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.992 956.002 1022.768 1089.534 1198.660 as smsLsLsmsLssmsL and
 
66.766 175.892 242.658 318.667 385.443 494.559 561.325 637.334 704.101 770.867 879.992 956.002 1022.768 1089.534 1198.660 as sLsmsLsmssLsmsL respectively.


From smsLsLsmsLssmsL, tempering m = s would lead to sssLsLsssLssssL, which is a MODMOS of Hanson[15];
66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs.


tempering L = m would lead to sLsLsLsLsLssLsL, which is Porcupine[15];
Accordingly Porcutone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.


tempering L = s would lead to LsLLLLLsLLLLsLL, which is a MODMOS of Augmented[15];
Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 ptolemismic tempering by the inclusion of 18/13 rather than 11/8.  
 
tempering out s would lead to sLLsLsL, which is Dicot[7];
 
tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].


Accordingly the Porcutone-15 scales would temper to two steps in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
2.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1)


== Comma pump ==
== Comma pump ==
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