Pinetone: Difference between revisions

Line 1,725:

Or, from the porcutone chromatic with flats (mode 3), we add another porcutone diatonic scale, mode 0 starting on D♯, leading to the right-handed porcutone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.

If 81/80 were additionally tempered out, these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the porcutone chromatic), the medium step of the porcutone chromatic, approximating 25/24 and 33/32, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.

If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the porcutone chromatic), the medium step of the porcutone chromatic, approximating 25/24 and 33/32, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.

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 Tetracot[20].

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.

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.

Tuned to TE 2.3.5.11.13 Tetracot, 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

33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917.

and the right handed porcutone hyperchromatic in cents is

33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.

== Comma pump ==

We can't use our circle of fifths (Meantone comma pump) or our Porcupine comma pumps here, as both 81/80 and 250/243 are observed. In the ptolemismic tuning we temper out 100/99 which we can can pump with chord progressions such as

@@ Line 1,725: / Line 1,725: @@
 Or, from the porcutone chromatic with flats (mode 3), we add another porcutone diatonic scale, mode 0 starting on D♯, leading to the right-handed porcutone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.
-If 81/80 were additionally tempered out, these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the porcutone chromatic), the medium step of the porcutone chromatic, approximating 25/24 and 33/32, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
+If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the porcutone chromatic), the medium step of the porcutone chromatic, approximating 25/24 and 33/32, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
-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 Tetracot[20].
+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.
 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.
-Tuned to TE 2.3.5.11.13 Tetracot, 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
+.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917.
 and the right handed porcutone hyperchromatic in cents is
+.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.
 == Comma pump ==
 We can't use our circle of fifths (Meantone comma pump) or our Porcupine comma pumps here, as both 81/80 and 250/243 are observed. In the ptolemismic tuning we temper out 100/99 which we can can pump with chord progressions such as