ED5

The equal division of the 5th harmonic (ed5) is a tuning obtained by dividing the 5th harmonic in a certain number of equal steps.

Theory

The 5th harmonic is particularly wide as far as equivalences go. There are (at absolute most) ~4.8 pentaves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with pentave equivalence, this range restriction is a crucial consideration. Pentave equivalence itself may have a basis in Western music seeing as minor chords have an octave of 5 in their root (i.e. 10:12:15).

One way to treat 5/1 as an equivalence is by eliminating the primes 2 and 3. The most fundamental chord in this paradigm is 5:7:11. This chord can be approximated in a 5.7.11 (or "no-twos-or-threes 11-limit") subgroup regular temperament by eliminating the comma 859375/823543, equating a stack of 7 7/5 generators with 11/5. Other equivalences that could be used for such "no-two-or-threes" music include equal divisions of 11/5 and equal divisions of 11/7.

The quintessential example of a pentave based tuning is hyperpyth (see 17ed5). However, perhaps the more common reason to use these scales is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus (20ed5) which itself is a zeta peak tuning (not "no-fives", full on zeta). Other reasons for taking the n-th root of 5 include finding temperaments like orwell, meantone, and thuja. This approach can of course be used indiscriminately.

Some equal divisions of the pentave are known by alternate names or have special interest:

• 3ed5 orwell generator
• 4ed5 meantone generator
• 5ed5 thuja generator
• 6ed5 uncle generator
• 8ed5 mohajira generator
• Hyperpyth tuning (e.g. 17ed5)
• 20ed5 Hieronymus Tuning
• 25ed5 (Stockhausen, McLaren)

Individual pages for ed5'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

Ed5-edo correspondence

Following ed5's (up to 339) contain good correspondences to edo tunings.

Ed5	Edo	Comments
7ed5	3edo	7ed5 is 3edo with ~5.9 cent compressed octaves. Equivalently, 3edo is 7ed5 with pentaves stretched by ~13.7 cents. Patent vals match through the 67-limit.
9ed5	4edo	Very rough correspondence (~38 cent octave stretch), but patent vals agree through the 7-limit.
14ed5	6edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
16ed5	7edo	Only rough correspondence (~19 cent octave stretch). Patent vals differ in the 7-limit.
21ed5	9edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 59-limit.
23ed5	10edo	23ed5 is 10edo with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.
28ed5	12edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 19-limit, with the exception of 13 (as expected).
30ed5	13edo	30ed5 is 13edo with ~7.4 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 3 (which neither represents well).
35ed5	15edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 13-limit.
37ed5	16edo	37ed5 is 16edo with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.
42ed5	18edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.
44ed5	19edo	44ed5 is 19edo with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.
49ed5	21edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
51ed5	22edo	51ed5 is 22edo with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.
56ed5	24edo	This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence begins to break down. Patent vals match through the 13-limit, with the exception of 7.
58ed5	25edo	58ed5 is 25edo with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.
63ed5	27edo	This is a rough correspondence, but the (7n)ed5 ~ (3n)edo sequence is momentarily fixed. Patent vals match through the 13-limit.
65ed5	28edo	65ed5 is 28edo with ~0.26 cent stretched octaves. Patent vals match through the 43-limit.
72ed5	31edo	72ed5 is 31edo with ~0.34 cent compressed octaves. Patent vals match through the 31-limit.
74ed5	32edo	Same 4.9 cent octave stretch as 37ed5~16edo. Patent vals match through the 13-limit with the exception of 7 and 11.
79ed5	34edo	79ed5 is 34edo with ~0.83 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7 (which neither represents well).
81ed5	35edo	81ed5 is 35edo with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.
84ed5	36edo	Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 11-limit, with the exception of 7.
86ed5	37edo	86ed5 is 37edo with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.
88ed5	38edo	Same 3.2 cent octave stretch as 44ed5~19edo. Patent vals differ in the 7-limit.
93ed5	40edo	93ed5 is 40edo with ~1.6 cent compressed octaves. Patent vals match through the 7-limit.
95ed5	41edo	95ed5 is 41edo with ~2.5 cent stretched octaves. Patent vals match through the 11-limit.
100ed5	43edo	100ed5 is 43edo with ~1.9 cent compressed octaves. Patent vals match through the 37-limit.
102ed5	44edo	Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 23-limit, with the exception of 7.
107ed5	46edo	107ed5 is 46edo with ~2.1 cent compressed octaves. Patent vals match through the 11-limit.
109ed5	47edo	109ed5 is 47edo with ~1.4 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 11.
114ed5	49edo	114ed5 is 49edo with ~2.4 cent compressed octaves. Patent vals match through the 11-limit.
116ed5	50edo	Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals agree through the 43-limit.
121ed5	52edo	121ed5 is 52edo with ~2.6 cent compressed octaves, but the correspondence is rough. Patent vals match through the 11-limit, with the exception of 3 (which neither represents well).
123ed5	53edo	123ed5 is 53edo with ~0.61 cent stretched octaves. Patent vals match through the 29-limit.
128ed5	55edo	128ed5 is 55edo with ~2.8 cent compressed octaves, but the correspondence is rough. Patent vals differ in the 7-limit.
130ed5	56edo	Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 103-limit.
135ed5	58edo	135ed5 is 58edo with ~2.9 cent compressed octaves. Patent vals match through the 13-limit.
137ed5	59edo	137ed5 is 59edo with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.
139ed5	60edo	139ed5 is 60edo with ~2.7 cent stretched octaves. Pantent vals match through the 31-limit with the exception of 11.
144ed5	62edo	Same 0.34 cent octave compression as 72ed5~31edo. Patent vals match through the 19-limit, with the exception of 11.
146ed5	63edo	146ed5 is 63edo with ~2.3 cent stretched octaves. Patent vals match through the 13-limit.
151ed5	65edo	151ed5 is 65edo with ~0.59 cent compressed octaves. Patent vals match through the 53-limit, with the exception of 7 (as expected).
153ed5	66edo	Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 13-limit, with the exception of 3.
158ed5	68edo	Same 0.83 cent octave compression as 79ed5~34edo. Patent vals agree through the 23-limit.
160ed5	69edo	160ed5 is 69edo with ~1.6 cent stretched octaves. Patent vals differ in the 7-limit.
165ed5	71edo	165ed5 is 71edo with ~1.0 cent compressed octaves. Patent vals match through the 37-limit.
167ed5	72edo	167ed5 is 72edo with ~1.3 cent stretched octaves. Patent vals agree through the 19-limit.
172ed5	74edo	Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 13-limit.
174ed5	75edo	Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals differ in the 7-limit.
179ed5	77edo	179ed5 is 77edo with ~1.4 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11.
181ed5	78edo	181ed5 is 78edo with ~0.73 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 13.
186ed5	80edo	Same 1.6 cent octave compression as 93ed5~40edo. Patent vals agree through the 29-limit.
188ed5	81edo	188ed5 is 81edo with ~0.49 cent stretched octaves. Patent vals match through the 41-limit.
193ed5	83edo	193ed5 is 83edo with ~1.7 cent compressed octaves. Patent vals match through the 7-limit.
195ed5	84edo	Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 31-limit.
200ed5	86edo	Same 1.9 cent octeave compression as 100ed5~43edo. Patent vals match alternate primes through the 17-limit.
202ed5	87edo	202ed5 is 87edo with ~0.05 cent stretched octaves. Patent vals match through the 71-limit.
204ed5	88edo	Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 7-limit.
209ed5	90edo	209ed5 is 90edo with ~0.15 cent compressed octaves. Patent vals match through the 53-limit.
211ed5	91edo	211ed5 is 91edo with ~1.7 cent stretched octaves. Patent vals match through the 7-limit.
216ed5	93edo	Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 31-limit.
218ed5	94edo	Same 1.4 cent octave stretch as 109ed5~47edo. Patent vals match through the 23-limit, with the exception of 13.
223ed5	96edo	223ed5 is 96edo with ~0.51 cent compressed octaves. Patent vals match through the 13-limit.
225ed5	97edo	225ed5 is 97edo with ~1.2 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 11.
230ed5	99edo	230ed5 is 99edo with ~0.67 cent compressed octaves. Patent vals match through the 7-limit.
232ed5	100edo	Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 13-limit.
237ed5	102edo	Same 0.83 cent octave compression as 79ed5~34edo. Patent vals differ in the 7-limit.
239ed5	103edo	239ed5 is 103edo with ~0.80 cent stretched octaves. Patent vals match through the 17-limit.
244ed5	105edo	244ed5 is 105edo with ~0.97 cent compressed octaves. Patent vals disagree in the 5-limit and higher prime limit.
246ed5	106edo	Same 0.61 cent octave stretch as 123ed5~53edo. Patent vals match through the 43-limit, with the exception of 7.
251ed5	108edo	251ed5 is 108edo with ~1.1 cent compressed octaves. Patent vals match through the 13-limit.
253ed5	109edo	253ed5 is 109edo with ~0.43 cent stretched octaves. Patent vals match through the 13-limit.
258ed5	111edo	Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 19-limit.
260ed5	112edo	Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 47-limit, with the exception of 3.
265ed5	114edo	265ed5 is 114edo with ~2.2 cent compressed octaves. Patent vals match through the 43-limit, with the exception of 11, 19 and 37.
267ed5	115edo	267ed5 is 115edo with ~0.10 cent stretched octaves. Patent vals match through the 17-limit.
274ed5	118edo	Same 0.05 cent octave compression as 137ed5~59edo. Patent vals agree through the 103-limit.
276ed5	119edo	276ed5 is 119edo with ~1.35 cent stretched octaves. Patent vals match through the 23-limit with the exception of 13 and 19.
281ed5	121edo	281ed5 is 121edo with ~0.20 cent compressed octaves. Patent vals match through the 29-limit.
283ed5	122edo	283ed5 is 122edo with ~1.2 cent stretched octaves. Patent vals match through the 13-limit.
288ed5	124edo	Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 23-limit.
290ed5	125edo	Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 37-limit, with the exception of 13.
295ed5	127edo	295ed5 is 127edo with ~0.47 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 11.
297ed5	128edo	297ed5 is 128edo with ~0.84 cent stretched octaves. Patent vals match through the 7-limit.
302ed5	130edo	Same 0.59 cent octave compression as 151ed5~65edo. Patent vals match through the 41-limit, with the exception of 11, 19 and 23.
304ed5	131edo	304ed5 is 131edo with ~0.68 cent stretched octaves. Patent vals match through the 41-limit, with the exception of 13 and 23.
309ed5	133edo	309ed5 is 133edo with ~0.73 cent compressed octaves. Patent vals match through the 71-limit, with the exception of 7 and 59.
311ed5	134edo	311ed5 is 134edo with ~0.53 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 11, 19 and 53.
316ed5	136edo	Same 0.83 cent octave compression as 79ed5~34edo. Patent vals match through the 41-limit, with the exception of 13 and 23.
318ed5	137edo	318ed5 is 137edo with ~0.39 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 7.
323ed5	139edo	323ed5 is 139edo with ~0.94 cent compressed compressed octaves. Patent vals match alternating primes through the 11-limit.
325ed5	140edo	Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 73-limit, with the exception of 31 and 67.
330ed5	142edo	Same 1.0 cent octave compression as 165ed5~71edo. Patent vals match through the 13-limit, with the exception of 11.
332ed5	143edo	32ed5 is 143edo with ~0.13 cent stretched octaves. Patent vals match through the 29-limit, with the exception of 7, 17 and 19.
337ed5	145edo	337ed5 is 145edo with ~1.1 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
339ed5	146edo	339ed5 is practically identical to 146edo, with a slight octave stretch (~0.0053 cents). Patent vals match through the 827-limit.

See also

• Pentave Reduced Harmonics
• Pentave Reduced Subharmonics
• Relative errors of small ED5s

External links

• Nonoctave.com: tuning: equal division of the fifth harmonic