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autopatrolled, emailconfirmed
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Lhearne (talk | contribs)
autopatrolled, emailconfirmed
1,636 edits
tunings throughout mostly and got rid of separate 'Tuning' section
Line 593: Line 593:
461.443, 704.101
461.443, 704.101
|}
|}
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 if we also equate 13/10 with 9/7 or 21/16 respectively).  
 
The step signature, mapping, and size for Tetracot[7] is
 
6L 1s = (10/9~11/10, 27/25~12/11) = (175.8871c, 144.0106c) in the 2.3.5.11 subgroup, or
 
6L 1s = (10/9~11/10, 27/25~12/11~13/12) = (176.0044, 142.6653) in the 2.3.5.11.13 subgroup, or
 
6L 1s = (10/9~11/10, 27/25~12/11~13/12) = (175.6125, 146.2576) as 13-limit Monkey, or
 
6L 1s = (10/9~11/10~28/25, 27/25~15/14~12/11~13/12) = (176.8600, 136.3262) as Wollemia.
 
We can see that the large step of Tetracot[7] is the medium step of the porcutone diatonic, and the small step of Tetracot[7] is the small step of the porcutone diatonic. The large step of the porcutone diatonic is the augmented second of tetracot[7].


==== Tuning options ====
==== Tuning options ====
Line 614: Line 626:
41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = (146.3663c, 58.5465c, 29.2733c)  
41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = (146.3663c, 58.5465c, 29.2733c)  


For comparison, the TE step signature and sizes for the ptolemismic porcupine chromatic is   
For comparison, the TE step signature, mapping and sizes for the (2.3.5.11.13) ptolemismic porcupine chromatic is   
 
[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264f&limit=2.3.5.11.13 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c)], 
 
the TE step signature and sizes for the 13-limit Supermagic porcutone chromatic is


7L 1m 4s = (142.77537c, 66.76626c, 33.11646c),
[http://x31eq.com/cgi-bin/rt.cgi?ets=7p%261cde%264f&limit=13 7L 1m 4s = (27/25~12/11~13/12~35/32, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39~64/63) = (145.47082c, 58.39270c, 30.85183c)],  


the TE step signature and sizes for the Supermagic porcutone chromatic is  
and the TE step signature and sizes for the 13-limit Thrasher porcutone chromatic is  


7L 1m 4s = (145.47082c, 58.39270c, 30.85183c),  
[http://x31eq.com/cgi-bin/rt.cgi?ets=7d%261cdde%264f&limit=13 7L 1m 4s = (27/25~15/14~12/11~13/12, 25/24~21/20~33/32~27/26, 250/243~28/27~55/54~121/120~40/39) = (136.27690c, 81.02531c, 40.63434c)],  


and the TE step signature and sizes for the Thrasher porcutone chromatic is  
and if optimization just to the 2.3.5.11 subgroup is desired,TE step signature and sizes for the ptolemismic porcutone chromatic is  


7L 1m 4s = (136.27690c, 81.02531c, 40.63434c).  
[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264p&limit=2.3.5.11 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c)].  


== The porcutone octatonic ==
== The porcutone octatonic ==
Line 934: Line 950:
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.
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 945: Line 961:
|LMLLMLsM
|LMLLMLsM
|~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1
|~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1
|174.055 320.690 494.745 668.800 815.435 989.490 1052.633 1199.269
|175.892 318.667 494.559 670.451 815.435 989.118 1055.884 1198.660
|-
|-
|Sarnathian bright quartal
|Sarnathian bright quartal
|MLMLLMLs
|MLMLLMLs
|~ 12/11 6/5 13/10 16/11 8/5 26/15 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
|142.775 318.667 461.443 637.334 815.435 956.002 1131.893 1198.660
|-
|-
|Dylathian middle major
|Dylathian middle major
|LLMLsMLM
|LLMLsMLM
|~ 10/9 11/9 4/3 22/15 20/13 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
|175.892 351.784 494.559 670.451 737.217 879.992 1055.884 1198.660
|-
|-
|Kadathian bright major
|Kadathian bright major
|MLLMLsML
|MLLMLsML
|~ 12/11 6/5 4/3 16/11 8/5 5/3 9/5 2/1
|~ 12/11 6/5 4/3 16/11 8/5 5/3 9/5 2/1
|146.635 320.690 494.745 641.380 815.435 878.579 1025.214 1199.269
|142.775 318.667 494.559 637.334 815.435 879.992 1022.768 1198.660
|-
|-
|Ultharian dark major
|Ultharian dark major
|LMLsMLML
|LMLsMLML
|~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1
|~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1
|174.055 320.690 494.745 557.888 704.524 878.579 1025.214 1199.269
|175.892 318.667 494.559 561.325 704.101 879.992 1022.768 1198.660
|-
|-
|Illarnekian middle minor
|Illarnekian middle minor
|LsMLMLLM
|LsMLMLLM
|~ 10/9 15/13 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
|175.892 242.658 385.433 561.325 704.101 879.992 1055.884 1198.660
|-
|-
|Hlanithian bright minor
|Hlanithian bright minor
|MLsMLMLL
|MLsMLMLL
|~ 12/11 6/5 5/4 15/11 3/2 18/11 9/5 2/1
|~ 12/11 6/5 5/4 15/11 3/2 18/11 9/5 2/1
|146.635 320.690 383.834 530.469 704.524 851.159 1025.214 1199.269
|142.775 318.667 385.433 528.209 704.101 846.876 1022.768 1198.660
|-
|-
|Mnarian dark minor
|Mnarian dark minor
|sMLMLLML
|sMLMLLML
|~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|63.143 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
|66.766 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 992: Line 1,008:
|LMLLMLMs
|LMLLMLMs
|~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1
|~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1
|174.055 320.690 494.745 668.800 815.435 989.490 1136.127 1199.269
|175.892 318.667 494.559 670.451 813.227 989.118 1131.983 1198.660
|-
|-
|Illarnekian bright major
|Illarnekian bright major
|LLMLMsLM
|LLMLMsLM
|~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1
|~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1
|174.055 348.110 494.745 668.800 815.435 878.579 1052.633 1199.269
|175.892 351.784 494.559 670.451 813.227 879.992 1055.884 1198.660
|-
|-
|Hlanithian dark quartal
|Hlanithian dark quartal
|MLLMLMsL
|MLLMLMsL
|~ 12/11 6/5 4/3 16/11 8/5 26/15 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
|142.775 318.667 494.559 637.334 813.227 956.002 1022.768 1198.660
|-
|-
|Mnarian middle major
|Mnarian middle major
|LMLMsLML
|LMLMsLML
|~ 10/9 6/5 4/3 16/11 3/2 5/3 9/5 2/1
|~ 10/9 6/5 4/3 16/11 3/2 5/3 9/5 2/1
|174.055 320.690 494.745 641.380 704.524 878.579 1025.214 1199.269
|175.892 318.667 494.559 637.334 704.101 879.992 1022.768 1198.660
|-
|-
|Celephaïsian bright minor
|Celephaïsian bright minor
|LMsLMLLM
|LMsLMLLM
|~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1
|~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1
|174.055 320.690 383.834 557.888 704.524 878.579 1052.633 1199.269
|175.892 318.667 385.433 561.325 704.101 879.992 1055.884 1198.660
|-
|-
|Sarnathian dark major
|Sarnathian dark major
|MLMsLMLL
|MLMsLMLL
|~ 12/11 6/5 13/10 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
|142.775 318.667 461.443 528.209 704.101 846.876 1022.768 1198.660
|-
|-
|Dylathian dark minor
|Dylathian dark minor
|sLMLLMLM
|sLMLLMLM
|~ 25/24 15/13 5/4 11/8 20/13 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
|66.766 242.658 385.433 561.325 737.217 879.992 1055.884 1198.660
|-
|-
|Kadathian middle minor
|Kadathian middle minor
|MsLMLLML
|MsLMLLML
|~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|146.635 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
|142.775 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 1,145: Line 1,161:


10/9, 11/10
10/9, 11/10
|
|66.766
 
142.775
 
175.892
|1
|1
3
3
Line 1,160: Line 1,180:
|9/8
|9/8
15/13 (7/6 or 8/7)
15/13 (7/6 or 8/7)
6/5
6/5


11/9, 16/13
11/9, 16/13
|
|209.542
 
242.658
 
318.667
 
351.784
|1
|1
1
1
Line 1,180: Line 1,207:


4/3
4/3
|
|385.433
 
461.433
 
494.559
|3
|3
1
1
Line 1,199: Line 1,230:


40/27, 22/15
40/27, 22/15
|
|528.209
 
561.325
 
637.334
 
670.451
|2
|2
2
2
Line 1,216: Line 1,253:


8/5
8/5
|
|704.101
 
737.217
 
813.227
|4
|4
1
1
Line 1,235: Line 1,276:


16/9
16/9
|
|846.876
 
879.992
 
956.002
 
989.118
|1
|1
5
5
Line 1,252: Line 1,299:


48/25, 64/33, 52/27
48/25, 64/33, 52/27
|
|1022.768
 
1055.884
 
1131.983
|4
|4
3
3
Line 1,299: Line 1,350:
* smsLsmm Phrygian ♭4 symmetric minor
* smsLsmm Phrygian ♭4 symmetric minor
* smmsmsL Locrian magical ♭♭7
* smmsmsL Locrian magical ♭♭7
== Tunings ==
We could tune the scale in many different ways. The TE tuning given above consists of 7 large steps of 146.6352c, 1 medium step of 63.1434c, and 4 small steps of 27.4197c.
We could instead tune to [[POTE tuning|POTE]] no-7 ptolemismic, resulting in a very similar 7L 1m 4s = [http://x31eq.com/cgi-bin/rt.cgi?limit=2_3_5_11&ets=7_1ce_4p&tuning=po (146.7247c, 63.1818c, 27.4363c)].
For reference, the 5-limit JI tuning of (27/25, 25/24, 250/243) is equal to (133.2376c, 70.6724c, 49.1661c). There are also least squares and minimax. I hope to figure those out.
We could also tune to edos. Tuning to [[15edo]], [[22edo]] or [[29edo]] collapses the scale to a [[Porcupine]][8] scale, and tuning to [[19edo]] or [[31edo]] tempers the scale to a Meantone[12] scale. We can retain three step sizes if we tune to [[27edo]] (using 27e), [[34edo]], or to [[41edo]].
27edo: 7L 1m 4s = (3, 2, 1) = (133.3333c, 88.8889c, 44.4444c)
34edo: 7L 1m 4s = (4, 2, 1) = (141.1765c, 70.5882c, 35.2941c)
41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c)


== Chords ==
== Chords ==
Retrieved from "https://en.xen.wiki/w/Pinetone"