33edo

The 33 equal division divides the octave into 33 equal parts of 36.3636 cents each. It is not especially good at representing all rational intervals in the 7-limit, but it does very well on the 7-limit 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 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 relatively uncommon, 33edo is actually quite an interesting system. As a multiple of 11edo, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see 26edo). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, leading to a near perfect 10/9. The <33 52 76| or 33c val tempers out 81/80 and so leads to a very flat meantone tuning where the major tone is approximately 10/9 in size. Leaving the scale be would result in a flattone 5L+2s of L=5 s=4

Instead of the flat 19\33 fifth you may use the sharp fifth of 20\33, over 25 cents 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 6\33 = 2\11 11edo interval of 218 cents. Now 6\33 + 5\33 = 11\33 = 1\3 of an octave, or 400 cents, the same major third as 12edo. Also, we have both a 327 minor third from 9\33 = 3\11, the same as the 22edo minor third, and a flatter 8\33 third of 291 cents, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7 cents (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 11, 13 and 19 harmonics (taking the generator as a 19/16) which are relatively well in tune.

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.

Step #	ET	Just		Difference (ET minus Just)	Ups and Downs Notation
	Cents	Interval	Cents
0	0¢	1/1	0	0	unison	1	D
1	36.364	48/47	36.448	-0.084742	Up unison		D#
2	72.727	24/23	73.6806	0.95338	Downminor 2nd		D#^
3	109.091	16/15	111.7312	-2.64037	Minor 2nd		Ebv
4	145.455	12/11	150.6371	-5.182513	Mid 2nd		Eb
5	181.818	10/9	182.4037	-0.58553	Major 2nd		E
6	218.182	17/15	216.6866	1.49521	Upmajor 2nd		E#
7	254.545	15/13	247.7410	6.804401	2nd/3rd		E#^/Fbv
8	290.909	13/11	289.2097	1.699371	Subminor 3rd		Fb
9	327.273	6/5	315.6412	11.63144	Minor 3rd		F
10	363.636	16/13	359.4723	4.164025	Neutral 3rd		F#
11	400.000	5/4	386.3137	13.686286	Major 3rd		F#^
12	436.364	9/7	435.0841	1.2795411	Supermajor 3rd		Gbv
13	472.727	21/16	470.7809	1.9463653	Diminished 4th		Gb
14	509.091	4/3	498.0449	11.0459099	Perfect 4th		G
15	545.455	11/8	551.3179	-5.863396	Augmented 4th		G#
16	581.818	7/5	582.5129	-0.694010	Low Tritone		G#^
17	618.182	10/7	617.4878	0.694010	High Tritone		Abv
18	654.545	16/11	648.6821	5.863396	Diminished 5th		Ab
19	690.909	3/2	701.9550	-11.0459099	Perfect 5th		A
20	727.273	32/21	729.2191	-1.9463653	Augmented 5th		A#
21	763.636	14/9	764.9159	-1.2795411	Subminor 6th		A#^
22	800.000	8/5	813.6862	-13.686286	Minor 6th		Bbv
23	836.364	13/8	840.5276	-4.164025	Mid 6th		Bb
24	872.727	5/3	884.3587	-11.63144	Major 6th		B
25	909.091	22/13	910.7903	-1.699371	Supermajor 6th		B#
26	945.455	12/7	933.1291	12.325451	6th/7th		B#^/Cbv
27	981.818	7/4	968.8259	12.9922753	Subminor 7th		Cb
28	1018.182	9/5	1017.596	0.58553	Minor 7th		C
29	1054.545	11/6	1049.363	5.182513	Mid 7th		C#
30	1090.909	15/8	1088.268	2.64037	Major 7th		C#^
31	1127.273	23/12	1126.319	-0.95338	Supermajor 7th		Dbv
32	1163.636	47/24	1163.551	0.084742	Down 8ve		Db
33	1200	2/1	1200	0	8ve	8	D

Nearby Equal Temperaments:

Music:

Deluge Peter Kosmorsky

5 5 1 mode of 33 equal (with video) play by Chris Vaisvil

Bach Contrapunctus 4 Claudi Meneghin version