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→The porcutone pentatonic and the porcutone chromatic: added intervals table, added to Porcupine[8] and Father[8] tables
Line 595: Line 595:
Note the major minor third approximating 11/9 and the minor major third approximating 27/22 are very similar in size in this tuning (351.784 and 352.317 respectively). If we equate these intervals, we additionally temper out 243/242, leading to an extension of Tetracot (Wollemia or Monkey is we also equate 13/10 with 9/7 or 21/16 respectively). Tetracot[7] comprises 6 of our porcutone diatonic medium step (tuned to 176.0044c) and one remaining step of our porcutone diatonic small step (tuned to 142.6653c).
Note the major minor third approximating 11/9 and the minor major third approximating 27/22 are very similar in size in this tuning (351.784 and 352.317 respectively). If we equate these intervals, we additionally temper out 243/242, leading to an extension of Tetracot (Wollemia or Monkey is we also equate 13/10 with 9/7 or 21/16 respectively). Tetracot[7] comprises 6 of our porcutone diatonic medium step (tuned to 176.0044c) and one remaining step of our porcutone diatonic small step (tuned to 142.6653c).


==== Tuning options ====
As with the porcutone diatonic, tuning the porcutone chromatic to 19edo collapses it to the Meantone[12] (Flattone[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step signatures, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:
As with the porcutone diatonic, tuning the porcutone chromatic to 19edo collapses it to the Meantone[12] (Flattone[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step signatures, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:


Line 624: Line 625:


7L 1m 4s = (136.27690c, 81.02531c, 40.63434c).  
7L 1m 4s = (136.27690c, 81.02531c, 40.63434c).  
I like to keep the small step large enough to sound like a melodic interval, so I either tune to TE ptolemismic, or to 41edo if an equal tuning is desired.


== The porcutone octatonic ==
== The porcutone octatonic ==
Line 632: Line 631:
On my [[Lumatone]] I chose to colour the G♯/A♭ pink, and the rest of the chromatic notes blue, so the porcutone octatonic is on the white and pink keys, while there's a porcutone diatonic on the white keys and a porcutone pentatonic on the blue and pink keys.  
On my [[Lumatone]] I chose to colour the G♯/A♭ pink, and the rest of the chromatic notes blue, so the porcutone octatonic is on the white and pink keys, while there's a porcutone diatonic on the white keys and a porcutone pentatonic on the blue and pink keys.  


If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' ''[[UDP]]'' is show in the table. The UDP show the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor.
If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' ''[[UDP]]'' is show in the table. The UDP show the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our porcutone chromatic and octatonic scales.
 
The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80)
{| class="wikitable"
{| class="wikitable"
|+Porcupine[8] functional mode names
|+Porcupine[8] modes (G♯-G gamut)
!Mode number
!Mode number
!Step pattern
!Step pattern
!UDP
!UDP
!Mode name
!Mode name
!3-step stacked triad on root (with G#)
!(with A♭ = H)
!JI triad approximated
|-
|-
|4
|4
Line 644: Line 648:
|<nowiki>7|0</nowiki>
|<nowiki>7|0</nowiki>
|Bright quartal
|Bright quartal
|G#-C-F
|A-D-G
|9:12:16
|-
|-
|3
|3
Line 649: Line 656:
|<nowiki>6|1</nowiki>
|<nowiki>6|1</nowiki>
|Dark quartal
|Dark quartal
|A-D-G
|B-E-A♭ = B-E-H
|9:12:16
|-
|-
|2
|2
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|<nowiki>5|2</nowiki>
|<nowiki>5|2</nowiki>
|Bright major
|Bright major
|B-E-G#
|C-F-A
|3:4:5
|-
|-
|1
|1
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|<nowiki>4|3</nowiki>
|<nowiki>4|3</nowiki>
|Middle major
|Middle major
|C-F-A
|D-G-B
|3:4:5
|-
|-
| -1
| -1
|LLLsLLLL
|LLLsLLLL
|<nowiki>3|4</nowiki>
|<nowiki>3|4</nowiki>
|Dark major
|Dark major
|D-G-B
|E-A♭-C = E-H-C
|3:4:5
|-
|-
| -2
| -2
|LLsLLLLL
|LLsLLLLL
|<nowiki>2|5</nowiki>
|<nowiki>2|5</nowiki>
|Bright minor
|Bright minor
|E-G#-C
|F-A-D
|12:15:20
|-
|-
| -3
| -3
|LsLLLLLL
|LsLLLLLL
|<nowiki>1|6</nowiki>
|<nowiki>1|6</nowiki>
|Middle minor
|Middle minor
|F-A-D
|G-B-E
|12:15:20
|-
|-
| -4
| -4
|sLLLLLLL
|sLLLLLLL
|<nowiki>0|7</nowiki>
|<nowiki>0|7</nowiki>
|Dark minor
|Dark minor
|G-B-E
|A♭-C-F = H-C-F
|12:15:20
|}
|}


All of these triads are pretty consonant, shoutout to Porcupine[8]!
We get Father[8], instead, if we temper out the difference (16/15) between the large step and the small step. Recall that the porcupine pentatonic reduces to Father[5], a subset of Father[8]. Father scales are generated by an interval representing both 5/4 and 4/3 (the 3-step interval of 8-note scales). The modes of Father[8] have names in use already, as an [[oneirotonic]]. These are shown in the table below with the mode number, step patter, and UDP.


We get Father[8], instead, if we temper out the difference (16/15) between the large step and the small step. Recall that the porcupine pentatonic reduces to Father[5], a subset of Father[8]. Father scales are generated by an interval representing both 5/4 and 4/3 (the 3-step interval of 8-note scales). The modes of Father[8] have names in use already, as an [[oneirotonic]]. These are shown in the table below with the mode number, step patter, and UDP.
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 mode names
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!UDP
!UDP
!Mode name
!Mode name
!JI triads approximated by
3-step stacked triad on root
|-
|-
|4
|4
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|<nowiki>7|0</nowiki>
|<nowiki>7|0</nowiki>
|Dylathian (də-LA(H)TH-iən)
|Dylathian (də-LA(H)TH-iən)
|3:4:5, 9:12:16
|-
|-
|3
|3
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|<nowiki>6|1</nowiki>
|<nowiki>6|1</nowiki>
|Illarnekian (ill-ar-NEK-iən)
|Illarnekian (ill-ar-NEK-iən)
|3:4:5, 9:12:16
|-
|-
|2
|2
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|<nowiki>5|2</nowiki>
|<nowiki>5|2</nowiki>
|Celephaïsian (kel-ə-FAY-zhən)
|Celephaïsian (kel-ə-FAY-zhən)
|3:4:5, 9:12:16
|-
|-
|1
|1
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|<nowiki>4|3</nowiki>
|<nowiki>4|3</nowiki>
|Ultharian (ul-THA(I)R-iən)
|Ultharian (ul-THA(I)R-iən)
|3:4:5, 9:12:16
|-
|-
| -1
| -1
|LsLsLLsL
|LsLsLLsL
|<nowiki>3|4</nowiki>
|<nowiki>3|4</nowiki>
|Mnarian (mə-NA(I)R-iən)
|Mnarian (mə-NA(I)R-iən)
|3:4:5, 9:12:16
|-
|-
| -2
| -2
|sLLsLLsL
|sLLsLLsL
|<nowiki>2|5</nowiki>
|<nowiki>2|5</nowiki>
|Kadathian (kə-DA(H)TH-iən)
|Kadathian (kə-DA(H)TH-iən)
|3:4:5, 9:12:16
|-
|-
| -3
| -3
|sLLsLsLL
|sLLsLsLL
|<nowiki>1|6</nowiki>
|<nowiki>1|6</nowiki>
|Hlanithian (lə-NITH-iən)
|Hlanithian (lə-NITH-iən)
|160:200:243
|-
|-
| -4
| -4
|sLsLLsLL
|sLsLLsLL
|<nowiki>0|7</nowiki>
|<nowiki>0|7</nowiki>
|Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|200:243:324
|}
|}
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.
Line 892: Line 933:


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.
tempering out 144/143 as well, the medium step also represents 13/12, and the small step also represents 27/26.
{| class="wikitable"
{| class="wikitable"
|+Modes of the left handed ptolemismic porcutone octatonic
|+Modes of the left handed ptolemismic porcutone octatonic
Line 906: Line 949:
|Sarnathian bright quartal
|Sarnathian bright quartal
|MLMLLMLs
|MLMLLMLs
|~ 12/11 6/5 72/55 16/11 8/5 96/55 48/25 2/1
|~ 12/11 6/5 13/10 16/11 8/5 26/15 48/25 2/1
|146.635 383.834 467.325 641.380 815.435 962.070 1136.127 1199.269
|146.635 383.834 467.325 641.380 815.435 962.070 1136.127 1199.269
|-
|-
|Dylathian middle major
|Dylathian middle major
|LLMLsMLM
|LLMLsMLM
|~ 10/9 11/9 4/3 22/15 55/36 5/3 11/6 2/1
|~ 10/9 11/9 4/3 22/15 20/13 5/3 11/6 2/1
|174.055 348.110 494.745 668.800 731.943 878.579 1052.633 1199.269
|174.055 348.110 494.745 668.800 731.943 878.579 1052.633 1199.269
|-
|-
Line 926: Line 969:
|Illarnekian middle minor
|Illarnekian middle minor
|LsMLMLLM
|LsMLMLLM
|~ 10/9 55/48 5/4 11/8 3/2 5/3 11/6 2/1
|~ 10/9 15/13 5/4 11/8 3/2 5/3 11/6 2/1
|174.055 237.198 383.834 557.888 704.524 878.579 1052.633 1199.269
|174.055 237.198 383.834 557.888 704.524 878.579 1052.633 1199.269
|-
|-
Line 958: Line 1,001:
|Hlanithian dark quartal
|Hlanithian dark quartal
|MLLMLMsL
|MLLMLMsL
|~ 12/11 6/5 4/3 16/11 8/5 96/55 10/9 2/1
|~ 12/11 6/5 4/3 16/11 8/5 26/15 10/9 2/1
|146.635 320.690 494.745 641.380 815.435 962.070 1025.214 1199.269
|146.635 320.690 494.745 641.380 815.435 962.070 1025.214 1199.269
|-
|-
Line 973: Line 1,016:
|Sarnathian dark major
|Sarnathian dark major
|MLMsLMLL
|MLMsLMLL
|~ 12/11 6/5 72/55 15/11 3/2 18/11 9/5 2/1
|~ 12/11 6/5 13/10 15/11 3/2 18/11 9/5 2/1
|146.635 320.690 467.325 530.469 704.524 851.159 1025.214 1199.269
|146.635 320.690 467.325 530.469 704.524 851.159 1025.214 1199.269
|-
|-
|Dylathian dark minor
|Dylathian dark minor
|sLMLLMLM
|sLMLLMLM
|~ 25/24 55/48 5/4 11/8 55/36 5/3 11/6 2/1
|~ 25/24 15/13 5/4 11/8 20/13 5/3 11/6 2/1
|63.143 237.198 383.834 557.888 731.943 878.579 1052.633 1199.269
|63.143 237.198 383.834 557.888 731.943 878.579 1052.633 1199.269
|-
|-
Line 986: Line 1,029:
|146.635 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
|146.635 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
|}
|}
{| class="wikitable"
|+3-step stacked triads of the right handed porcutone octatonic (G♯-G gamut)
!Mode name
!Step pattern
!3-step stacked triad on root
!JI triad approximated
|-
|Celephaïsian dark quartal
|LMLLMLsM
|A-D-G
|9:12:16
|-
|Sarnathian bright quartal
|MLMLLMLs
|G#-C-F
|30:39:52 (7:9:12 Thrasher, 16:21:28 Supermagic)
|-
|Dylathian middle major
|LLMLsMLM
|B-E-G#
|3:4:5
|-
|Kadathian bright major
|MLLMLsML
|C-F-A
|3:4:5
|-
| Ultharian dark major
|LMLsMLML
|D-G-B
|3:4:5
|-
| Illarnekian middle minor
|LsMLMLLM
|F-A-D
|12:15:20
|-
| Hlanithian bright minor
|MLsMLMLL
|E-G#-C
|8:10:13
|-
| Mnarian dark minor
|sMLMLLML
|G-B-E
|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
!JI triad approximated
|-
|Ultharian bright quartal
|LMLLMLsM
|A-D-G
|9:12:16
|-
|Illarnekian bright major
|MLMLLMLs
|A♭-C-F
|9:12:16
|-
|Hlanithian dark quartal
|LLMLsMLM
|C-F-A
|3:4:5
|-
|Mnarian middle major
|MLLMLsML
|B-E-A♭
|27:36:44
|-
| Celephaïsian bright minor
|LMLsMLML
|D-G-B
|3:4:5
|-
| Sarnathian dark major
|LsMLMLLM
|F-A-D
|12:15:20
|-
| Dylathian dark minor
|MLsMLMLL
|E-A♭-C
|99:121:162
|-
| Kadathian middle minor
|sMLMLLML
|G-B-E
|12:15:20
|}
To me the left handed porcutone octatonic is superior because of the 3-step stacked triads it offers.
The following table gives all intervals of the porcutone octatonic.
{| class="wikitable"
|+Intervals of the porcutone octatonic
!Interval class
!sizes
!JI ratios approximated
!size in cents (TE)
!Occurence
|-
!1-step
|s
M
L
|25/24, 33/32, 27/26
27/25, 12/11, 13/12
10/9, 11/10
|
|1
3
4
|-
!2-step
|M + s
L + s
L + M
L + L
|9/8
15/13 (7/6 or 8/7)
6/5
11/9, 16/13
|
|1
1
5
1
|-
!3-step
|L + M + s
L + 2M
2L + M
|5/4
13/10 (9/7 or 21/16)
4/3
|
|3
1
4
|-
!4-step
|L + 2M + s
2L + M + s
2L + 2M
3L + M
|27/20, 15/11
25/18, 11/8, 18/13
36/25, 16/11, 13/9
40/27, 22/15
|
|2
2
2
2
|-
!5-step
|2L + 2M + s
3L + M + s
3L + 2M
|3/2
20/13 (14/9 or 32/16)
8/5
|
|4
1
3
|-
!6-step
|2L + 3M + s
3L + 2M + s
3L + 3M
4L + 2M
|18/11, 13/8
5/3
26/15 (12/7 or 7/4)
16/9
|
|1
5
1
1
|-
!7-step
|3L + 3M + s
4L + 2M + s
4L + 3M
|9/5, 20/11
50/27, 11/6, 24/13
48/25, 64/33, 52/27
|
|4
3
1
|}
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 ==
Retrieved from "https://en.xen.wiki/w/Pinetone"