Pinetone: Difference between revisions

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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 Pinetone 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 Pinetone 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.

In 2.3.5.11.13 Tetracot, the left-handed Pinetone 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 Pinetone 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 Pinetone hyperchromatic in cents is

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 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 Pinetone 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 Pinetone 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.
+In 2.3.5.11.13 Tetracot, the left-handed Pinetone 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 Pinetone 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 Pinetone hyperchromatic in cents is