Pinetone: Difference between revisions
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Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively. | Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively. | ||
Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8. | Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8. | ||
2.3.5.13 325/324 may be | In this tuning the medium and small steps are within 1c of the same size. Tempering them together results in Cata temperament, an extension of Hanson. 2.3.5.13 325/324 may alternatively be tuned to 46edo, with (L, m, s) = (4, 2, 3), or as Cata[15] in 53edo, 72edo, or 87edo as (5, 3, 3), (7, 4, 4), or (8, 5, 5). | ||
Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name. | Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name. | ||