36edo: Difference between revisions
→Harmonics: Change to odd harmonics to include those referenced in 2.3.25.7.55.13.17 subgroup already in text; also consistent with lesser multiples of 9edo |
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That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33{{c}}, one can arrive at a 24-tone subset of 36edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut. | That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33{{c}}, one can arrive at a 24-tone subset of 36edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut. | ||
=== Odd | === Odd harmonics === | ||
In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} [[49/48]] × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17 (since the 25th harmonic is more accurate than the 5th harmonic, and the 55th harmonic is more accurate than the 5th and 11th harmonics), and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]]. | In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} [[49/48]] × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17 (since the 25th harmonic is more accurate than the 5th harmonic, and the 55th harmonic is more accurate than the 5th and 11th harmonics), and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]]. | ||