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]])

* [[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]]

| | [[42ed5]]

| | [[18edo]]

|-

|-

| | 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.

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

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

|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.

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

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

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

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

|-

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

|207ed5 is 89edo with ~2 cent compressed octaves. Patent vals match through the 23, with the exception of 13 and19.

|[[214ed5]]

|[[92edo]]

|Same 2.1 cent octave compression as 107ed5~46edo, but the correspondence is rough. Patent vals match through the 5-limit.

|[[215ed5]]

|

|Every other note of [[185edo]], getting ~.13 cent flatter as it ascends the scale

| | [[216ed5]]

| | [[93edo]]

| | Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 31-limit.

|[[217ed5]]

|

|Every other note of [[187edo]], getting ~.006 cent sharper as it ascends the scale

| | [[218ed5]]

| | [[94edo]]

| | Same 1.4 cent octave stretch as 109ed5~47edo. Patent vals match through the 23-limit, with the exception of 13.

|[[219ed5]]

|

|Every third note of [[286edo]], getting ~.135 cent sharper as it ascends the scale

|[[220ed5]]

|

|Same 3.2 cent octave stretch as 44ed5~19edo, but really matches the tuning with generator 3\379 of an octave..

|[[221ed5]]

|[[95edo]]

|221ed5 is 95edo with ~2.3 cent compressed octaves. Patent vals match through the 29-limit with the exception of 13 and 23

|[[222edo]]

|

|Every other note of [[191edo]], getting ~.01 cent flatter as it ascends the scale

| | [[223ed5]]

| | [[96edo]]

| | 223ed5 is 96edo with ~0.51 cent compressed octaves. Patent vals match through the 13-limit.

|[[224ed5]]

|

|Every other note of [[193edo]], getting ~.004 cent sharper as it ascends the scale

| | [[225ed5]]

| | [[97edo]]

| | 225ed5 is 97edo with ~1.2 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 11.

|[[226edo]]

|

|Every third note of [[292edo]], getting ~.000055 cent sharper as it ascends the scale

|[[227ed5]]

|

|Every fourth note of [[391edo]], getting ~.002 cent flatter as it ascends the scale

|[[228ed5]]

|[[98edo]]

|Same 2.4 octave compression as 114ed5~49edo, but the correspondence is rough. Patent vals match through the 5-limit with the exception of 3.

|[[229ed5]]

|

|Every other note of [[197edo]], getting ~.015 cent flatter as it ascends the scale

| | [[230ed5]]

| | [[99edo]]

| | 230ed5 is 99edo with ~0.67 cent compressed octaves. Patent vals match through the 7-limit.

|[[231ed5]]

|

|Every other note of [[199edo]], getting ~.002 cent sharper as it ascends the scale