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 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).

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]])

* [[12ed5]]

* [[13ed5]]

* [[15ed5]]

* [[17ed5]]

* [[20ed5]]

* [[22ed5]]

* [[24ed5]]

* [[25ed5]]

* [[28ed5]]

* [[33ed5]]

* [[67ed5]]

 

 

! | 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.

|[[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|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]]

|

|[[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]]

|[[36ed5]]

|

|Every other note of [[31edo]], getting ~.02 cent flatter as it ascends the scale

| | [[37ed5]]

| | [[16edo]]

|[[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

|-

| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.

|-

| | 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.

|Every other note of [[49edo]], getting ~.1 cent flatter as it ascends the scale

| | 58ed5 is 25edo with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.

|[[59ed5]]

|Same ~7.4 cent octave stretch as 30ed5~13edo. Patent vals match through the 29-limit.

|Every other note of [[55edo]], getting ~.1 cent flatter as it ascends the scale

|-

|[[77ed5]]

|[[33edo]]

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence continues to break down. Patent vals match through the 7-limit.

|[[78ed5]]

|

|Every other note of [[67edo]], getting ~.1 cent flatter as it ascends the scale

| | [[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).

|[[80ed5]]

|

|Every other note of [[69edo]], getting ~.05 cent sharper as it ascends the scale

| | [[81ed5]]

| | [[35edo]]

| | 81ed5 is 35edo with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.

|[[82ed5]]

|

|Every third note of [[106edo]], getting ~.02 cent sharper as it ascends the scale

|[[83ed5]]

|

|Every fourth note of [[143edo]], getting ~.04 cent sharper as it ascends the scale

|[[84ed5]]

|[[36edo]]

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence is momentarily fixed again. Patent vals match through the 11-limit, with the exception of 7.

|[[85ed5]]

|

|Every other note of [[73edo]], getting ~.1 cent flatter as it ascends the scale

| | [[86ed5]]

| | [[37edo]]

| | 86ed5 is 37edo with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.

|[[87ed5]]

|

|Every other note of [[75edo]], getting ~.03 cent sharper as it ascends the scale

| | [[88ed5]]

| | [[38edo]]

| | Same 3.2 cent octave stretch as 44ed5~19edo. Patent vals differ in the 7-limit.

|[[89ed5]]

|

|Every third note of [[115edo]], getting ~.0025 cent sharper as it ascends the scale

|[[90ed5]]

|

|Same ~7.4 cent octave stretch as 30ed5~13edo, but really matches the tuning with generator 4\155 of an octave.

|[[91ed5]]

|[[39edo]]

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence begins to break down again. Patent vals match through the 7-limit.

|[[92ed5]]

|

|Every fifth note of [[198edo]], getting ~.02 cent flatter as it ascends the scale

| | [[93ed5]]

| | [[40edo]]

| | 93ed5 is 40edo with ~1.6 cent compressed octaves. Patent vals match through the 7-limit.

|[[94ed5]]

|

|Every other note of [[81edo]], getting ~.01 cent sharper as it ascends the scale

| | [[95ed5]]

| | [[41edo]]

| | 95ed5 is 41edo with ~2.5 cent stretched octaves. Patent vals match through the 11-limit.

|[[96ed5]]

|

|Every third note of [[124edo]], getting ~.01 cent flatter as it ascends the scale

|[[97ed5]]

|

|Every fourth note of [[167edo]], getting ~.02 cent flatter as it ascends the scale

|[[98ed5]]

|[[42edo]]

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence momentarily stops breaking down further.

|[[99ed5]]

|

|Every third note of [[128edo]], getting ~.02 cent sharper as it ascends the scale

| | [[100ed5]]

| | [[43edo]]

| | 100ed5 is 43edo with ~1.9 cent compressed octaves. Patent vals match through the 37-limit.

|[[101ed5]]

|

|Every other note of [[87edo]], getting ~.001 cent sharper as it ascends the scale

| | [[102ed5]]

| | [[44edo]]

| | Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 23-limit, with the exception of 7.

|[[103ed5]]

|

|Every other note of [[89edo]], getting ~.009 cent sharper as it ascends the scale

|[[104ed5]]

|[[45edo]]

|104ed5 is 45edo with ~5.6 cent stretched octaves. Patent vals match through the 7-limit.

|[[105ed5]]

|

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence begins to break down again. Patent vals match through the 5-limit, with the exception of 3.

|[[106ed5]]

|

|Every third note of [[137edo]], getting ~.009 cent sharper as it ascends the scale

| | [[107ed5]]

| | [[46edo]]

| | 107ed5 is 46edo with ~2.1 cent compressed octaves. Patent vals match through the 11-limit.

|[[108ed5]]

|

|Every other note of [[93edo]], getting ~.007 cent flatter as it ascends the scale

| | [[109ed5]]

| | [[47edo]]

| | 109ed5 is 47edo with ~1.4 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 11.

|[[110ed5]]

|

|Every other note of [[95edo]], getting ~.67 cent sharper as it ascends the scale

|[[111ed5]]

|[[48edo]]

|Same ~4.9 cent octave stretch as 37ed5~16edo. Patent vals match through the 5-limit.

|[[112ed5]]

|

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence is momentarily fixed again. Patent vals match through the 7-limit, with the exception of 5.

|[[113ed5]]

|

|Every third note of [[146edo]], getting ~.0001 cent sharper as it ascends the scale

| | [[114ed5]]

| | [[49edo]]

| | 114ed5 is 49edo with ~2.4 cent compressed octaves. Patent vals match through the 11-limit.

|[[115ed5]]

|

|Every other note of [[99edo]], getting ~.001 cent flatter as it ascends the scale

| | [[116ed5]]

| | [[50edo]]

| | Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals agree through the 43-limit.

|[[117ed5]]

|

|Every other note of [[101edo]], getting ~.05 cent sharper as it ascends the scale

|[[118ed5]]

|[[51edo]]

|118ed5 is 51edo with ~4.25 cent stretched octaves. Patent vals match through the 17-limit with the exception of 13.

|[[119ed5]]

|

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence has finally broken down more or less altogether. Patent vals match through the 3-limit.

|[[120ed5]]

|

|Same 7.4 cent octave stretch as 30ed5~13edo, but really matches the tuning with generator 3\155 of an octave.

| | [[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).

|[[122ed5]]

|

|Every other note of [[105edo]], getting ~.002 cent flatter as it ascends the scale

| | [[123ed5]]

| | [[53edo]]

| | 123ed5 is 53edo with ~0.61 cent stretched octaves. Patent vals match through the 29-limit.

|[[124ed5]]

|

|Every other note of [[107edo]], getting ~.04 cent sharper as it ascends the scale

|[[125ed5]]

|[[54edo]]

|125ed5 is 54edo with ~3.7 cent stretched octaves, but the correspondence is rough. Patent vals match through the 5-limit, with the exception of 3 (which neither represents well).

|[[126ed5]]

|

|Same rough correspondence as 119ed5~51edo

|[[127ed5]]

|

|Every third note of [[164edo]], getting ~.01 cent flatter as it ascends the scale

| | [[128ed5]]

| | [[55edo]]

| | 128ed5 is 55edo with ~2.8 cent compressed octaves, but the correspondence is rough. Patent vals differ in the 7-limit.

|[[129ed5]]

|

|Every other note of [[111edo]], getting ~.02 cent flatter as it ascends the scale

| | [[130ed5]]

| | [[56edo]]

| | Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 103-limit.

|[[131ed5]]

|

|Every other note of [[113edo]], getting ~.03 cent sharper as it ascends the scale

|[[132ed5]]

|[[57edo]]

|Same 3.2 cent octave stretch as 44ed5~19edo. Patent vals differ in the 11-limit.

|[[133ed5]]

|

|This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence has finally broken down more or less altogether. Patent vals match through the 2-limit.

|[[134ed5]]

|

|Every third note of [[173edo]], getting ~.02 cent flatter as it ascends the scale

|[[135ed5]]

|[[58edo]]

|135ed5 is 58edo with ~2.9 cent compressed octaves. Patent vals match through the 13-limit.

|[[136ed5]]

|

|Every other note of 117edo, getting ~.025 cent flatter as it ascends the scale

| | [[137ed5]]

| | [[59edo]]

| | 137ed5 is 59edo with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.

|[[138ed5]]

|

|Every other note of [[119edo]], getting ~.02 cent sharper as it ascends the scale

|[[139ed5]]

|[[60edo]]

|139ed5 is 60edo with ~2.7 cent stretched octaves. Pantent vals match through the 31-limit with the exception of 11.

|[[140ed5]]

|

|Same rough correspondence as 133ed5~57edo

|[[141ed5]]

|

|Every third note of [[182edo]], getting ~.02 cent flatter as it ascends the scale

|[[142ed5]]

|[[61edo]]

|142ed5 is 61edo with ~3.1 cent compressd octaves. Patent vals match through the 13-limit, with the exception of 7 and 11.

|[[143ed5]]

|

|Every other note of [[123edo]], getting ~.03 cent flatter as it ascends the scale

| | [[144ed5]]

| | [[62edo]]

| | Same 0.34 cent octave compression as 72ed5~31edo. Patent vals match through the 19-limit, with the exception of 11.

|[[145ed5]]

|

|Every other note of [[125edo]], getting ~.02 cent sharper as it ascends the scale

| | [[146ed5]]

| | [[63edo]]

| | 146ed5 is 63edo with ~2.3 cent stretched octaves. Patent vals match through the 13-limit.

|[[147ed5]]

|

|Same rough correspondence as 133ed5~57edo

|[[148ed5]]

|

|Same ~4.9 cent octave stretch as 37ed5~16edo, but really matches the tuning with generator 4\255 of an octave.

|[[149ed5]]

|[[64edo]]

|149ed5 is 64edo with ~3.2 cent compressed octaves. Patent vals match through the 7-limit, with the exception of 3

|[[150ed5]]

|

|Every other note of [[129edo]], getting ~.02 cent sharper as it ascends the scale

| | [[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).

|[[152ed5]]

|

|Every other note of [[131edo]], getting ~.001 cent sharper as it ascends the scale

| | [[153ed5]]

| | [[66edo]]

| | Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 13-limit, with the exception of 3.

|[[154ed5]]

|

|Same rough correspondence as 133ed5~57edo

|[[155ed5]]

|

|Every fourth note of [[267edo]], getting ~.001 cent flatter as it ascends the scale

|[[156ed5]]

|[[67edo]]

|156ed5 is 67edo with ~3.3 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 7

|[[157ed5]]

|

|Every other note of [[135edo]], getting ~.03 cent flatter as it ascends the scale

| | [[158ed5]]

| | [[68edo]]

| | Same 0.83 cent octave compression as 79ed5~34edo. Patent vals agree through the 23-limit.

|[[159ed5]]

|

|Every other note of [[137edo]], getting ~.006 cent sharper as it ascends the scale

| | [[160ed5]]

| | [[69edo]]

| | 160ed5 is 69edo with ~1.6 cent stretched octaves. Patent vals differ in the 7-limit.

|[[161ed5]]

|

|Same 11 cent octave stretch as 23ed5~10edo, but really matches the tuning with generator 3\208 of an octave.

|[[162ed5]]

|

|Every third note of [[209edo]], getting ~.025 cent flatter as it ascends the scale

|[[163ed5]]

|[[70edo]]

|163ed5 is 70edo with ~3.4 cent compressed octaves. Patent vals match through the 7-limit.

|[[164ed5]]

|

|Every other note of [[141edo]], getting ~.03 cent flatter as it ascends the scale

| | [[165ed5]]

| | [[71edo]]

| | 165ed5 is 71edo with ~1.0 cent compressed octaves. Patent vals match through the 37-limit.

|[[166ed5]]

|

|Every other note of [[143edo]], getting ~.02 cent sharper as it ascends the scale

| | [[167ed5]]

| | [[72edo]]

| | 167ed5 is 72edo with ~1.3 cent stretched octaves. Patent vals agree through the 19-limit.

|[[168ed5]]

|

|Same rough correspondence as 133ed5~57edo

|[[169ed5]]

|

|Every third note of [[218edo]], getting ~.03 cent flatter as it ascends the scale

|[[170ed5]]

|[[73edo]]

|170ed5 is 73edo with ~3.5 cent compressed octaves, but the correspondence is rough. Patent vals match through the 7-limit.

|[[171ed5]]

|

|Every other note of [[147edo]], getting ~.03 cent flatter as it ascends the scale