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For comparison, the TE tuning has step signature and mapping
For comparison, the TE tuning has step signature and mapping


[http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE ptolemismic: 1L 4m 2s] = (209.77855c, 174.05488c, 146.63528c)
[http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE Ptolemismic: 1L 4m 2s] = (209.77855c, 174.05488c, 146.63528c)
 
If we also temper 144/13 (as described below), we get the step signature and mapping
 
[http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4f_2cef&limit=2_3_5_11_13 TE 2.3.11.13 Ptolemismic: 1L 4m 2s] = (209.54162c, 175.89183c, 142.77537c)


=== Intervals and chords ===
=== Intervals and chords ===
Line 569: Line 573:
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
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.
Mode 3 approximates the JI ratios: 12/11 10/9 6/5 13/10 4/3 13/9 3/2 13/8 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
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
Line 695: Line 699:
dim 4
dim 4
|diminished
|diminished
min 4
diminished min 4
|minor diminished
|minor diminished
dim min 4
dim min 4 diminished
|15:18:22
|15:18:22
10:13:15
30:39:44
|318.667, 670.451
|318.667, 670.451
461.443, 704.101
461.443, 670.451
|-
|-
|A
|A
Line 727: Line 731:
|minor (sub) dim min 6
|minor (sub) dim min 6
dim min 4 (sub) dim min 6
dim min 4 (sub) dim min 6
|40:48:63
|25:30:39
80:105:126
50:65:78
|318.667, 780.120
|318.667, 780.120


467.325, 780.120
461.443, 780.120
|-
|-
|B
|B
Line 757: Line 761:
|aug min 2 diminished
|aug min 2 diminished
major diminished
major diminished
|135:150:198
|243:275:360
27:33:40
27:33:40
|201.475, 670.451
|209.008, 670.451


351.784, 670.451
351.784, 670.451
Line 1,171: Line 1,175:
|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 13/9 8/5 26/15 48/25 2/1
|142.775 318.667 461.443 637.334 815.435 956.002 1131.893 1198.660
|142.775 318.667 461.443 637.334 815.435 956.002 1131.893 1198.660
|-
|-
Line 1,181: Line 1,185:
|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 13/9 8/5 5/3 9/5 2/1
|142.775 318.667 494.559 637.334 815.435 879.992 1022.768 1198.660
|142.775 318.667 494.559 637.334 815.435 879.992 1022.768 1198.660
|-
|-
Line 1,196: Line 1,200:
|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 13/8 9/5 2/1
|142.775 318.667 385.433 528.209 704.101 846.876 1022.768 1198.660
|142.775 318.667 385.433 528.209 704.101 846.876 1022.768 1198.660
|-
|-
Line 1,223: Line 1,227:
|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 13/9 8/5 26/15 10/9 2/1
|142.775 318.667 494.559 637.334 813.227 956.002 1022.768 1198.660
|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 13/9 3/2 5/3 9/5 2/1
|175.892 318.667 494.559 637.334 704.101 879.992 1022.768 1198.660
|175.892 318.667 494.559 637.334 704.101 879.992 1022.768 1198.660
|-
|-
Line 1,238: Line 1,242:
|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 13/8 9/5 2/1
|142.775 318.667 461.443 528.209 704.101 846.876 1022.768 1198.660
|142.775 318.667 461.443 528.209 704.101 846.876 1022.768 1198.660
|-
|-
Line 1,733: Line 1,737:
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).  
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).  


In 2.3.5.11.13 Tetracot, the left handed porcutone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 18/11 5/3 16/9 9/5 11/6 39/20 2/1, and the right handed porcutone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 16/11 3/2 20/13 18/11 5/3 27/16 9/5 11/6 39/20 2/1.
In 2.3.5.11.13 Tetracot, the left handed porcutone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right handed porcutone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.


Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left handed porcutone hyperchromatic in cents is
Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left handed porcutone hyperchromatic in cents is
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