Pinetone: Difference between revisions
Added section: Porcutone hyperchromatic scales |
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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 F♭ and C♭, or sDssDsDsmdmsDssDsDs with E♯ and B♯, or sDsDssDsmdmsDssDsDs with E♯ and C♭. D is the largest step, at 111.924c, followed by d at 76.009c, then by m (66.766c) and s (30.852c). | sDsDssDsmdmsDsDssDs with F♭ and C♭, or sDssDsDsmdmsDssDsDs with E♯ and B♯, 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). | |||
D is the largest step, at 111.924c, followed by d at 76.009c, then by m (66.766c) and s (30.852c). | |||
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-F♭ 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: | |||
40/39 12/11 10/9 121/108 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: | |||
33.116 142.775 175.892 206.744 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 1022.768 1055.884 1165.543 1198.660 | |||
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: | |||
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 | |||
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: | |||
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 | |||
== Comma pump == | == Comma pump == | ||