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The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269
The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269
Note the more complex intervals: 55/54, 55/36, 72/55, and 108/55. If we temper out an additional comma, we can equate these with simpler intervals, adding prime 13: Tempering out 144/143, these four interval approximate 40/39, 20/13, 13/10, and 39/20 respectively. Tempering out 144/143 also means that the small step of the porcutone diatonic, equivalently the large step of the porcutone chromatic approximates 13/12, which, when all three are justly tuned, lies between the other intervals approximated by the step - 27/25, and 12/11.
This leads to a step signature, mapping, and TE tuning of 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).
Mode -3 approximates the JI ratios: 40/39 10/9 6/5 11/9 4/3 11/8 3/2 20/13 5/3 9/5 11/6 2/1, with step pattern sLLsLmLsLLsL.
The TE tuning in cents is: 33.116 175.892 318.667 351.784 494.559 561.325 704.101 737.217 879.992 1022.768 1055.884 1198.660 as D♯ E F F♯ G G♯ A A♯ B C C♯ D
Mode 3 approximates the JI ratios: 12/11 10/9 6/5 13/10 4/3 16/11 3/2 18/11 5/3 9/5 39/20 2/1, with step pattern LsLLsLmLsLLs.
The TE tuning in cents is: 142.775 175.892 318.667 461.443 494.559 637.334 704.101 846.876 879.992 1022.768 1165.543 1198.660 as E♭ E F G♭ G A♭ A B♭ B C D♭ D
I find this tuning to be melodically superior, given the small step is 6 cents large, now a sixth tone rather than an eight tone.
If a full 13-limit tuning is desired, there are two options. The interval approximating 13/10 may either be tempered to approximate 21/16, leading to Supermagic, or 7/6, leading to Thrasher. The Supermagic tuning decreases the size of the small step, and the Starling tuning increases it. The Supermagic tuning reduces to Flattone (where 7/4 is found at a diminished 7th) and Porcupine (where 7/4 is found at a minor seventh), and the Starling tuning reduces to Meanenneadecal and Opossum (both where 7/4 is found at an augmented 6th). If we temper to 13/10 to equate to both 9/7 and 21/16, we get Keema, an extension of Hanson temperament. Keema[7] comprises 4 large steps of 247.695c, and 3 small steps of 69.682c.


The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.
The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.
Line 409: Line 425:
|10:12:15
|10:12:15
18:22:27
18:22:27
|320.690, 704.524
|318.667, 704.101


348.110, 704.524
351.784, 704.101
|-
|-
|D♯
|D♯
Line 424: Line 440:
|10:12:15
|10:12:15
10:13:15
10:13:15
|320.690, 704.524
|318.667, 704.101


467.325, 704.524
461.443, 704.101
|-
|-
|E
|E
Line 439: Line 455:
|10:12:15
|10:12:15
4:5:6
4:5:6
|320.690, 704.524
|318.667, 704.101


383.834, 704.524
385.433, 704.101
|-
|-
|F
|F
Line 454: Line 470:
|(14:16:21 or 6:7:9)
|(14:16:21 or 6:7:9)
4:5:6
4:5:6
|237.198, 704.524
|242.658, 704.101


383.834, 704.524
385.433, 704.101
|-
|-
|F♯
|F♯
Line 469: Line 485:
|22:27:33
|22:27:33
4:5:6
4:5:6
|356.414, 704.524
|352.317, 704.101


383.834, 704.524
385.433, 704.101
|-
|-
|G
|G
Line 484: Line 500:
|(14:16:21 or 6:7:9)
|(14:16:21 or 6:7:9)
4:5:6
4:5:6
|237.198, 704.524
|242.658, 704.101


383.834, 704.524
385.433, 704.101
|-
|-
|G♯
|G♯
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|15:18:22
|15:18:22
10:13:15
10:13:15
|320.690, 668.800
|318.667, 670.451
467.325, 704.524
461.443, 704.101
|-
|-
|A
|A
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|15:18:22
|15:18:22
27:33:40
27:33:40
|320.690, 668.800
|318.667, 670.451


348.110, 668.800
351.784, 670.451
|-
|-
|A♯
|A♯
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|40:48:63
|40:48:63
80:105:126
80:105:126
|320.690, 788.015
|318.667, 780.120


467.325, 788.015
467.325, 780.120
|-
|-
|B
|B
Line 543: Line 559:
|15:18:22
|15:18:22
27:33:40
27:33:40
|320.690, 668.800
|318.667, 670.451


348.110, 668.800
351.784, 670.451
|-
|-
|C
|C
Line 558: Line 574:
|135:150:198
|135:150:198
27:33:40
27:33:40
|201.475, 668.800
|201.475, 670.451


348.110, 668.800
351.784, 670.451
|-
|-
|C♯
|C♯
Line 573: Line 589:
|10:12:15
|10:12:15
10:13:15
10:13:15
|320.690, 704.524
|318.667, 704.101


467.325, 704.524
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).
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:


27edo: 7L 1m 4s = (3, 2, 1) = (133.3333c, 88.8889c, 44.4444c) (Aug maj 2 is 7/6)
27edo: 7L 1m 4s = (3, 2, 1) = (133.3333c, 88.8889c, 44.4444c) (dim min 4 is 9/7 - Starling)


34edo: 7L 1m 4s = (4, 2, 1) = (141.1765c, 70.5882c, 35.2941c) (Aug maj 2 is 7/6 and 8/7)
34edo: 7L 1m 4s = (4, 2, 1) = (141.1765c, 70.5882c, 35.2941c) (dim min 4 is 9/7 and 21/16 - Supermagic or Starling)


41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) (Aug maj 2 is 8/7)
41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) (dim min 4 is 21/16 - Supermagic)


104edo: 7L 1m 4s = (13, 5, 2) = (150c, 57.6923c, 23.0769c) (Aug maj 2 is 8/7)
All three of these edos also temper out 243/242, so the major minor and minor major thirds collapse to a single interval - the neutral third, and the diatonic scale can be considered a MODMOS of Tetracot[7] i.e. porcutone msmLmsm = Tetracot LsLALsL.


And allowing octave stretch, the tuning may be optimized via TE tuning to:
And allowing octave stretch, the tuning may be optimized via TE tuning to:
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For comparison, the TE step signature and sizes for the ptolemismic porcupine chromatic is   
For comparison, the TE step signature and sizes for the ptolemismic porcupine chromatic is   


7L 1m 4s = (146.6352c, 63.1434c, 27.4197c).
7L 1m 4s = (142.77537c, 66.76626c, 33.11646c), 
 
the TE step signature and sizes for the Supermagic porcutone chromatic is
 
7L 1m 4s = (145.47082c, 58.39270c, 30.85183c),


and the TE step signature and sizes for the supermagic porcupine chromatic is  
and the TE step signature and sizes for the Thrasher porcutone chromatic is  


7L 1m 4s = (149.5159c, 58.8799c, 23.6254c)  
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.  
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.  
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