ED5: Difference between revisions

'''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:

*[[3ed5]] [[orwell]] generator (with octaves)

*[[4ed5]] [[meantone]] generator (with octaves)

*[[5ed5]] [[2L_7s|thuja]] generator (with octaves)

*[[Hyperpyth|Hyperpyth tuning]] (e.g. [[17ed5]])

*[[3ed5]]

*[[4ed5]]

*[[5ed5]]

*[[6ed5]]

*[[7ed5]]

*[[8ed5]]

*[[9ed5]]

*[[10ed5]]

*[[11ed5]]

*[[14ed5]]

*[[15ed5]]

*[[16ed5]]

*[[17ed5]]

*[[20ed5]]

*[[25ed5]]

=ED5-EDO correspondence=

{| class="wikitable"

! | ED5

! | EDO

! | Comments

| | [[7ed5]]

| | [[3edo]]

| | 7ed5 is 3edo with ~5.9 cent compressed octaves. <br>Equivalently, 3edo is 7ed5 with pentaves stretched by ~13.7 cents. <br>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).

| | [[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 (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.