33edo: Difference between revisions
Cmloegcmluin (talk | contribs) add Notation section, and Sagittal notation |
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{{EDO intro|33}} | {{EDO intro|33}} | ||
== Theory == | == Theory == | ||
Because the [[chromatic semitone]] in 33edo is 1 step, 33edo can be notated using only naturals, sharps, and flats. However, many key signatures will require double- and triple-sharps and flats, making notation unwieldy in distant keys. | |||
=== Harmonics === | |||
33edo is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N_subgroups|3*33 subgroup]] 2.27.15.21.11.13. On this subgroup it tunes things to the same tuning as [[99edo]], and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc. In particular, the [[Subgroup temperaments#Terrain|terrain]] 2.7/5.9/5 subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for [[Mint_temperaments#Slurpee|slurpee temperament]] in the 5, 7, 11 and 13 limits. | 33edo is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N_subgroups|3*33 subgroup]] 2.27.15.21.11.13. On this subgroup it tunes things to the same tuning as [[99edo]], and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc. In particular, the [[Subgroup temperaments#Terrain|terrain]] 2.7/5.9/5 subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for [[Mint_temperaments#Slurpee|slurpee temperament]] in the 5, 7, 11 and 13 limits. | ||
So while it might not be the most harmonically accurate temperament, it's structurally quite interesting, and it approximates the full 19-limit consort in it's way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th. | |||
{{Harmonics in equal|33}} | |||
=== Structural properties === | |||
While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of [[11edo]], it approximates the 7th and 11th harmonics via [[orgone]] temperament (see [[26edo]]). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having a [[3L 7s|3L 7s]] with {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c mapping (which has val {{val| 33 52 76 }}) tempers out [[81/80]] and can be used to represent [[1/2-comma meantone]], a [[Meantone family#Flattertone|"flattertone"]] tuning where the whole tone is 10/9 in size. Indeed, the perfect fifth is tuned about 11 cents flat, and two stacked fifths fall only 0.6 cents flat of 10/9. Leaving the scale be would result in the standard diatonic scale ([[5L 2s]]) having minor seconds of four steps and whole tones of five steps. This also results in common practice minor and major chords becoming more supraminor and submajor in character, making everything sound almost neutral in quality. | While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of [[11edo]], it approximates the 7th and 11th harmonics via [[orgone]] temperament (see [[26edo]]). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having a [[3L 7s|3L 7s]] with {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c mapping (which has val {{val| 33 52 76 }}) tempers out [[81/80]] and can be used to represent [[1/2-comma meantone]], a [[Meantone family#Flattertone|"flattertone"]] tuning where the whole tone is 10/9 in size. Indeed, the perfect fifth is tuned about 11 cents flat, and two stacked fifths fall only 0.6 cents flat of 10/9. Leaving the scale be would result in the standard diatonic scale ([[5L 2s]]) having minor seconds of four steps and whole tones of five steps. This also results in common practice minor and major chords becoming more supraminor and submajor in character, making everything sound almost neutral in quality. | ||
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33edo contains an accurate approximation of the Bohlen-Pierce scale with 4\33 near 1\[[13edt]]. | 33edo contains an accurate approximation of the Bohlen-Pierce scale with 4\33 near 1\[[13edt]]. | ||
Other notable 33edo scales are [[diasem]] with L:m:s = 5:3:1 and [[5L 4s]] with L:s = 5:2. This step ratio for 5L 4s is great for its semitone size of 72.7¢. | Other notable 33edo scales are [[diasem]] with L:m:s = 5:3:1 and [[5L 4s]] with L:s = 5:2. This step ratio for 5L 4s is great for its semitone size of 72.7¢. | ||
33 is also the number of years in the Iranian calendar's leap cycle, where leap year is inserted once every 4 or 5 years. This corresponds to the [[1L 7s]] with the step ratio of 5:4. | 33 is also the number of years in the Iranian calendar's leap cycle, where leap year is inserted once every 4 or 5 years. This corresponds to the [[1L 7s]] with the step ratio of 5:4. | ||
== Intervals == | == Intervals == | ||