Ed6

Equal divisions of the 6th harmonic (ed6) are tunings obtained by dividing the 6th harmonic in a certain number of equal steps.

The 6th harmonic, sextuple, or hexatave, is particularly wide as far as equivalences go, and there are only about 3.9 instances of the 6th harmonic in the human hearing range. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the 10th, and to a lesser extent, the 12th share this property).

However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between octave and twelfth based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, 44ed6 gives us an excellent compromise between 17edo and 27edt, and 49ed6 achieves the same with respect to 19edo and 30edt. In fact, ed6's optimize for the 1:2:3:6 chord, with equal magnitudes and opposite signs of error on 2 and 3.

As generator chains for temperaments

Taking the n-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately. The ed6's serve as generator chains for the temperaments:

• 4ed6 – squares generator (with octaves)
• 5ed6 – tritonic generator (with octaves)
• 6ed6 – harry generator (with octaves)
• 7ed6 – sensi generator (with octaves)
• 8ed6 – würschmidt generator (with octaves)
• 10ed6 – myna generator (with octaves)
• 14ed6 – hemisensi generator (with octaves)
• 16ed6 – hemiwürschmidt generator (with octaves)
• 17ed6 – minortone generator (with octaves)
• 19ed6 – porcupine generator (with octaves)
• 21ed6 – progression generator (with octaves)
• 24ed6 – twothirdtonic generator (with octaves)
• 26ed6 – septidiasemi generator (with octaves)
• 35ed6 – octacot generator (with octaves)
• 40ed6 – valentine generator (with octaves)

Individual pages for ed6's

0…99

0	1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99

200 and beyond

• 256, 269, 287, 323, 512

Ed6–edo correspondence

Following ed6's (up to 200) contain good correspondences to edo tunings.

Ed6	Edo	Comments
13ed6	5edo	13ed6 is 5edo with ~6.9 cent compressed octaves. Equivalently, 5edo is 13ed6 with hexataves stretched by ~18 cents. Patent vals match through the 13-limit.
18ed6	7edo	18ed6 is 7edo with ~6.3 cent stretched octaves. Patent vals match through the 13-limit.
21ed6	8edo	21ed6 is 8edo with ~18 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
23ed6	9edo	23ed6 is 9edo with ~14 cent stretched octaves. Patent vals match through the 13-limit.
26ed6	10edo	Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals match through the 17-limit.
31ed6	12edo	31ed6 is 12edo with ~0.76 cent stretched octaves. Patent vals match through the 31-limit, with the exception of 11.
36ed6	14edo	Same ~6.3 cent octave stretch as 18ed6~7edo. Patent vals match through the 43-limit, with the exception of 5.
39ed6	15edo	Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals match through the 13-limit.
44ed6	17edo	44ed6 is 17edo with ~1.5 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 5.
49ed6	19edo	49ed6 is 19edo with ~2.8 cent stretched octaves. Patent vals match through the 13-limit.
52ed6	20edo	Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals differ in the 5-limit.
54ed6	21edo	Same ~6.3 cent octave stretch as 18ed6~7edo. Patent vals match through the 7-limit.
57ed6	22edo	57ed6 is 22edo with ~2.8 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 13.
62ed6	24edo	Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 19-limit.
67ed6	26edo	67ed6 is 26edo with ~3.7 cent stretched octaves. Patent vals match through the 19-limit.
70ed6	27edo	70ed6 is 27edo with ~3.5 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 11.
75ed6	29edo	75ed6 is 29edo with ~0.58 cent compressed octaves. Patent vals match through the 137-limit.
80ed6	31edo	80ed6 is 31edo with ~2.0 cent stretched octaves. Patent vals match through the 13-limit.
83ed6	32edo	83ed6 is 32edo with ~4.1 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 5.
85ed6	33edo	85ed6 is 33edo with ~4.3 cent stretched octaves. Patent vals differ in the 5 and 7-limit.
88ed6	34edo	Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 17-limit, with the exception of 7.
93ed6	36edo	Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 31-limit, with the exception of 11.
98ed6	38edo	Same ~2.8 cent octave stretch as 49ed6~19edo. Patent vals match through the 11-limit, with the exception of 7.
101ed6	39edo	101ed6 is 39edo with ~2.2 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 7.
106ed6	41edo	106ed6 is 41edo with ~0.19 cent compressed octaves. Patent vals match through the 41-limit.
111ed6	43edo	111ed6 is 43edo with ~1.7 cent stretched octaves. Patent vals match through the 17-limit.
114ed6	44edo	Same ~2.8 cent octave compression as 57ed6~22edo. Patent vals match through the 31-limit, with the exception of 11.
116ed6	45edo	116ed6 is 45edo with ~3.3 cent stretched octaves. Patent vals match through the 7-limit.
119ed6	46edo	119ed6 is 46edo with ~0.92 cent compressed octaves. Patent vals match through the 17-limit.
124ed6	48edo	Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 43-limit.
129ed6	50edo	129ed6 is 50edo with ~2.3 cent stretched octaves. Patent vals match through the 23-limit.
132ed6	51edo	Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 13-limit, with the exceptions of 5 and 11.
137ed6	53edo	137ed6 is 53edo with ~0.026 cent stretched octaves. Patent vals match through the 61-limit.
142ed6	55edo	142ed6 is 55edo with ~1.46 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 13.
145ed6	56edo	145ed6 is 56edo with ~2.0 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 13.
150ed6	58edo	Same ~0.58 cent octave compression as 75ed6~29edo. Patent vals match through the 113-limit.
155ed6	60edo	Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 31-limit, with the exception of 11.
158ed6	61edo	158ed6 is 61edo with ~2.4 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 7.
160ed6	62edo	Same ~2.0 cent octave stretch as 80ed6~31edo. Patent vals match through the 31-limit.
163ed6	63edo	163ed6 is 63edo with ~1.1 cent compressed octaves. Patent vals match through the 61-limit.
168ed6	65edo	168ed6 is 65edo with ~0.16 cent stretched octaves. Patent vals match through the 157-limit, with the exception of 13.
173ed6	67edo	173ed6 is 67edo with ~1.3 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 5.
176ed6	68edo	Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 23-limit, with the exception of 11.
181ed6	70edo	181ed6 is 70edo with ~0.35 cent compressed octaves. Patent vals match through the 67-limit.
186ed6	72edo	Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 19-limit.
189ed6	73edo	189ed6 is 73edo with ~1.9 cent compressed octaves. Patent vals match through the 11-limit.
191ed6	74edo	191ed6 is 74edo with ~1.8 cent stretched octaves. Patent vals match through the 11-limit, with the exception of 7.
194ed6	75edo	194ed6 is 75edo with ~0.79 cent compressed octaves. Patent vals match through the 23-limit, with the exception of 11.
199ed6	77edo	199ed6 is 77edo with ~0.25 cent stretched octaves. Patent vals match through the 37-limit.

See also

• Relative errors of small ED6s
• Ed12