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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | {{ED intro}} | ||
== Theory == | == Theory == | ||
=== Structural properties === | === 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]] with {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c ({{val| 33 52 76 93 }}) and 33cd ({{val| 33 52 76 92 }}) mappings temper 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{{c}} flat, and two stacked fifths fall only 0.6{{c}} 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]] with {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c ({{val| 33 52 76 93 }}) and 33cd ({{val| 33 52 76 92 }}) mappings temper 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{{c}} flat, and two stacked fifths fall only 0.6{{c}} 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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Instead of the flat 19-step fifth you may use the 20-step sharp fifth, over 25{{c}} sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a {{nowrap|6\33 {{=}} 2\[[11edo|11]]}} interval of 218{{c}}. Together, these add up to {{nowrap|6\33 + 5\33 {{=}} 11\33 {{=}} 1\3}}, or 400{{c}}, the same major third as 12edo. We also have both a 327{{c}} minor third ({{nowrap|9\33 {{=}} 6\22 {{=}} 3\11}}), the same as that of [[22edo]], and a flatter 8\33 third of 291{{c}}, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7{{c}} (if you use the patent val it is an extremely inaccurate 6/5). Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the [[cuthbert triad]]. The 8\33 generator, with MOS of size 5, 9, and 13, gives plenty of scope for these, as well as the 11th, 13th, and 19th harmonics (taking the generator as a 19/16) which are relatively well in tune. | Instead of the flat 19-step fifth you may use the 20-step sharp fifth, over 25{{c}} sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a {{nowrap|6\33 {{=}} 2\[[11edo|11]]}} interval of 218{{c}}. Together, these add up to {{nowrap|6\33 + 5\33 {{=}} 11\33 {{=}} 1\3}}, or 400{{c}}, the same major third as 12edo. We also have both a 327{{c}} minor third ({{nowrap|9\33 {{=}} 6\22 {{=}} 3\11}}), the same as that of [[22edo]], and a flatter 8\33 third of 291{{c}}, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7{{c}} (if you use the patent val it is an extremely inaccurate 6/5). Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the [[cuthbert triad]]. The 8\33 generator, with MOS of size 5, 9, and 13, gives plenty of scope for these, as well as the 11th, 13th, and 19th harmonics (taking the generator as a 19/16) which are relatively well in tune. | ||
33edo contains an accurate approximation of the | 33edo contains an accurate approximation of the [[Bohlen–Pierce]] scale with 4\33 near [[13edt|1\13edt]]. | ||
Other notable 33edo scales are [[diasem]] with {{nowrap|L:m:s {{=}} 5:3:1}} and [[5L 4s]] with {{nowrap|L:s {{=}} 5:2}}. This step ratio for 5L 4s is great for its semitone size of 72.7{{c}}. | Other notable 33edo scales are [[diasem]] with {{nowrap|L:m:s {{=}} 5:3:1}} and [[5L 4s]] with {{nowrap|L:s {{=}} 5:2}}. This step ratio for 5L 4s is great for its semitone size of 72.7{{c}}. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|33}} | |||
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 [[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 [[slurpee]] temperament in the 5-, 7-, 11-, and 13-limits. | |||
While it might not be the most harmonically accurate temperament, it is structurally quite interesting, and it approximates the full 19-limit consort in its own 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. | |||
=== Miscellany === | |||
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. | ||
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| 690.909 | | 690.909 | ||
| [[3/2]] | | [[3/2]] | ||
| 701. | | 701.955 | ||
| −11.046 | | −11.046 | ||
| Perfect 5th | | Perfect 5th | ||
| Line 224: | Line 227: | ||
| 763.636 | | 763.636 | ||
| [[14/9]] | | [[14/9]] | ||
| 764. | | 764.916 | ||
| −1.280 | | −1.280 | ||
| Double-aug 5th | | Double-aug 5th | ||
| Line 242: | Line 245: | ||
| 836.364 | | 836.364 | ||
| [[13/8]] | | [[13/8]] | ||
| 840. | | 840.528 | ||
| −4.164 | | −4.164 | ||
| Minor 6th | | Minor 6th | ||
| Line 260: | Line 263: | ||
| 909.091 | | 909.091 | ||
| [[22/13]] | | [[22/13]] | ||
| 910. | | 910.790 | ||
| −1.699 | | −1.699 | ||
| Augmented 6th | | Augmented 6th | ||
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== Approximation to JI == | == Approximation to JI == | ||
{{Q-odd-limit intervals}} | {{Q-odd-limit intervals}} | ||
{{Q-odd-limit intervals|32. | {{Q-odd-limit intervals|32.87|apx=val|header=none|tag=none|title=15-odd-limit intervals by 33cd val mapping}} | ||
== Nearby equal temperaments == | == Nearby equal temperaments == | ||
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| 2.3.5 | | 2.3.5 | ||
| 81/80, 1171875/1048576 | | 81/80, 1171875/1048576 | ||
| {{mapping| 33 52 76 }} ( | | {{mapping| 33 52 76 }} (33c) | ||
| +5.59 | | +5.59 | ||
| 4.13 | | 4.13 | ||
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Brightest mode is listed except where noted. | Brightest mode is listed except where noted. | ||
* Deeptone[7], 5 5 5 4 5 5 4 (diatonic) | * Deeptone[7], 5 5 5 4 5 5 4 (diatonic) | ||
** Fun 5-tone subset of Deeptone[7] 9 5 5 4 10 | |||
* Deeptone[12], 4 4 1 4 1 4 4 1 4 1 4 1 (chromatic) | * Deeptone[12], 4 4 1 4 1 4 4 1 4 1 4 1 (chromatic) | ||
* Deeptone[19], 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 (enharmonic) | * Deeptone[19], 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 (enharmonic) | ||
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; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/watch?v=swyP6tB78k0 ''groove 33edo''] (2023) | * [https://www.youtube.com/watch?v=swyP6tB78k0 ''groove 33edo''] (2023) | ||
* [https://www.youtube.com/watch?v=GypR6x_Ih1I ''33edo jam''] (2025) | |||
* [https://www.youtube.com/shorts/mkaaAJEyGFU ''33edo riff''] (2025) | |||
; [[Peter Kosmorsky]] | ; [[Peter Kosmorsky]] | ||