Ed5

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Ed5 means Division of the Fifth Harmonic (5/1) into n equal parts.

Division of the fifth harmonic into n equal parts

The fifth 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 fact shapes one's musical approach dramatically. Following this, 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 nth root of 5 include finding temperaments like orwell, meantone, and thuja. This approach can of course be used indiscriminately.

Below is a large list of ED5s; additionally, some equal divisions of the pentave are known by alternate names or have special interest:

Individual pages for ED5s


ED5-EDO correspondence

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, 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. 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.
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.
79ed5 34edo 79ed5 is 34edo with ~0.83 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
81ed5 35edo 81ed5 is 35edo with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.
86ed5 37edo 86ed5 is 37edo with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.
88ed5 38edo
93ed5 40edo
95ed5 41edo
100ed5 43edo
102ed5 44edo
107ed5 46edo
109ed5 47edo
114ed5 49edo
116ed5 50edo
121ed5 52edo
123ed5 53edo
128ed5 55edo
130ed5 56edo
137ed5 59edo
144ed5 62edo
146ed5 63edo
151ed5 65edo
153ed5 66edo
158ed5 68edo
160ed5 69edo
165ed5 71edo
167ed5 72edo
172ed5 74edo
174ed5 75edo
179ed5 77edo
181ed5 78edo
186ed5 80edo
188ed5 81edo
193ed5 83edo
195ed5 84edo
202ed5 87edo
209ed5 90edo
211ed5 91edo
216ed5 93edo
218ed5 94edo
223ed5 96edo
225ed5 97edo
230ed5 99edo
232ed5 100edo
237ed5 102edo
239ed5 103edo
244ed5 105edo
246ed5 106edo
251ed5 108edo
253ed5 109edo
258ed5 111edo
260ed5 112edo
267ed5 115edo
274ed5 118edo
281ed5 121edo
283ed5 122edo
288ed5 124edo
290ed5 125edo
295ed5 127edo
297ed5 128edo

See also