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.
8ed5 Mohajira generator without octaves
9ed5 4edo Very rough correspondence, but patent vals agree through the 7-limit.
10ed5 Every tenth note of 43edo, getting ~.44 cent flatter as it ascends the scale
11ed5 Every fourth note of 19edo, getting ~.67 cent sharper as it ascends the scale
12ed5 5edo Very rough correspondence, but patent 2s also agree
13ed5 Every fifth note of 28edo, getting ~.05 cent sharper as it ascends the scale
14ed5 6edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
15ed5 Every other note of 13edo, getting ~1.1 cents sharper as it ascends the scale
16ed5 7edo Only rough correspondence. Patent vals differ in the 7-limit.
17ed5 This Hyperpyth tuning is every third note of 22edo, getting ~.26 cent sharper as it ascends the scale
18ed5 Every fourth note of 31edo, getting ~.04 cent flatter as it ascends the scale
19ed5 8edo Very rough correspondence, but patent vals agree through the 5-limit.
20ed5 Hieronymus Tuning is every fifth note of 43edo, getting ~.22 cent flatter as it ascends the scale
21ed5 9edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 59-limit.
22ed5 Every other note of 19edo, getting ~.335 cent sharper as it ascends the scale
23ed5 10edo 23ed5 is 10edo with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.
24ed5 Every third note of 31edo, getting ~.03 cent flatter as it ascends the scale
25ed5 Every fourth note of 43edo, getting ~.18 cent flatter as it ascends the scale
26ed5 11edo Very rough correspondence, but patent vals agree through the 7-limit with the exception of 3 (which neither represents well).
27ed5 Every third note of 35edo, getting ~.03 cent sharper as it ascends the scale
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).
29ed5 Every other note of 25edo, getting ~.08 cent sharper as it ascends the scale
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).
31ed5 Every third note of 40edo, getting ~.12 cent flatter as it ascends the scale
32ed5 Every fourth note of 55edo, getting ~.2 cent flatter as it ascends the scale
33ed5 Every fifth note of 71edo, getting ~.07 cent flatter as it ascends the scale
34ed5 Every third note of 44edo, getting ~.13 cent sharper as it ascends the scale
35ed5 15edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 13-limit.
36ed5 Every other note of 31edo, getting ~.02 cent flatter as it ascends the scale
37ed5 16edo 37ed5 is 16edo with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.
38ed5 Every third note of 49edo, getting ~.145 cent flatter as it ascends the scale
39ed5 Every fifth note of 84edo, getting ~.015 cent sharper as it ascends the scale
40ed5 Every fourth note of 69edo, getting ~.09 cent sharper as it ascends the scale
41ed5 Every third note of 53edo, getting ~.03 cent flatter as it ascends the scale
42ed5 18edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.
43ed5 Every other note of 37edo, getting ~.07 cent flatter as it ascends the scale
44ed5 19edo 44ed5 is 19edo with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.
45ed5 Every fifth note of 97edo, getting ~.09 cent sharper as it ascends the scale
46ed5 20edo Same ~11 cent octave stretch as 23ed5~10edo. Patent vals match through the 23-limit, with the exception of 13 and 17.
47ed5 Every fourth note of 81edo, getting ~.02 cent sharper as it ascends the scale
48ed5 Same ~.03 cent/note progressive flattening as 24ed5~31ed8
49ed5 21edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
50ed5 Every other note of 43edo, getting ~.09 cent flatter as it ascends the scale
51ed5 22edo 51ed5 is 22edo with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.
52ed5 Same ~.05 cent/note progressive sharpening as 13ed5~28ed32
53ed5 23edo 58ed5 is 23edo with ~9.1 cent stretched octaves. Patent vals match through the 5-limit.
54ed5 Same ~.04 cent/note progressive flattening as 18ed5~31ed16
55ed5 Every third note of 71edo, getting ~.04 cent flatter as it ascends the scale
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 (which neither represents well).
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 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.
137ed5 59edo 137ed5 is 59edo with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.
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.
202ed5 87edo 202ed5 is 87edo with ~0.05 cent stretched octaves. Patent vals match through the 71-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.
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.
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.
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

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