Pinetone: Difference between revisions
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We might want to add in another two notes for E♯ or F♭ and B♯ or C♭. We'll call L-s 'D' (for large diminished) and L-m 'd' (for small diminished), resulting in 19-note scales with 4 step sizes: | We might want to add in another two notes for E♯ or F♭ and B♯ or C♭. We'll call L-s 'D' (for large diminished) and L-m 'd' (for small diminished), resulting in 19-note scales with 4 step sizes: | ||
sDsDssDsmdmsDsDssDs with | sDsDssDsmdmsDsDssDs with E♯ and B♯, or sDssDsDsmdmsDssDsDs with F♭ and C♭, or sDsDssDsmdmsDssDsDs with E♯ and C♭. | ||
We note that the the interval D-F♭ approximates 121/108, and is very close to 9/8. Accordingly we temper out the difference between 121/108 and 9/8, 243/242, leading to Tetracot temperament, where 2*s = m (and D = d + s). | We note that the the interval D-F♭ approximates 121/108, and is very close to 9/8. Accordingly we temper out the difference between 121/108 and 9/8, 243/242, leading to Tetracot temperament, where 2*s = m (and D = d + s). | ||
| Line 1,733: | Line 1,733: | ||
We will call these scales the left handed, right handed, and symmetric porcutone hyperchromatic scales respectively. | We will call these scales the left handed, right handed, and symmetric porcutone hyperchromatic scales respectively. | ||
Given that D- | Given that D-E♯ is 206.744c, very close to 9/8, but maps to 121/108, we can't really avoid tempering out 243/242, tuning to Tetracot | ||
The right handed porcutone hyperchromatic approximates the JI ratios: | The right handed porcutone hyperchromatic approximates the JI ratios: | ||
40/39 12/11 10/9 | 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 9/5 11/6 39/20 2/1, with a TE tuning of: | ||
33.116 142.775 175.892 | 33.116 142.775 175.892 287.815 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 991.916 1022.768 1055.884 1165.543 1198.660 | ||
The left handed porcutone hyperchromatic approximates the JI ratios: | The left handed porcutone hyperchromatic approximates the JI ratios: | ||
40/39 12/11 10/9 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 9/5 11/6 39/20 2/1, with a TE tuning of: | 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 27/16 9/5 11/6 39/20 2/1, with a TE tuning of: | ||
33.116 142.775 175.892 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 1022.768 1055.884 1165.543 1198.660 | 33.116 142.775 175.892 206.744 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 910.845 1022.768 1055.884 1165.543 1198.660 | ||
The symmetric porcutone hyperchromatic approximates the JI ratios: | The symmetric porcutone hyperchromatic approximates the JI ratios: | ||
40/39 12/11 10/9 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 9/5 11/6 39/20 2/1, with a TE tuning of: | 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 27/16 9/5 11/6 39/20 2/1, with a TE tuning of: | ||
33.116 142.775 175.892 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 1022.768 1055.884 1165.543 1198.660 | 33.116 142.775 175.892 287.815 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 910.815 1022.768 1055.884 1165.543 1198.660 | ||