8edo: Difference between revisions

{{ED intro}}

8edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1187 cents gives the tuning called [[29ed12]].

Of all prime [[harmonic]]s up to 31, pure-octave 8edo only manages to approximate 2/1 and 19/1 within 15 [[cents]], completely missing all the others.

By contrast, 29ed12 approximates 2/1, 11/1, 13/1, 17/1 and 31/1 all within 15 cents.

Of all integer harmonics up to 30, pure-octave 8edo approximates the following within 20 cents:

* 2, 4, 8, 16, 19, 27.

 

 

The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.

! rowspan="2" | Steps

! rowspan="2" | Cents

! colspan="3" | JI approximation

! 2.5/3.11/3.13/5*

! 10:11:12:13:14*

| 0

| 0

| 1/1

|1/1, 16/15

| 1

| 150

| 12/11

| 12/11, 11/10

|10/9, 25/24

| 2

| 300

| 19/16

| 6/5

|6/5, 9/8

| 3

| 13/10

| 13/10

| 13/10

|4/3, 5/4

| 4

| 600

|  

| 7/5, 10/7

|27/20, 25/18, 36/25

| 5

| 750

| 20/13

| 20/13

| 8

| 1200

| 2/1

| 2/1

<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible

8edo can be notated as a subset of 24edo, using [[Ups and downs notation|ups and downs]]. It can also be notated as a subset of 16edo, but this is a less intuitive notation.

! colspan="2" | 24edo subset notation<br />([[Enharmonic unisons in ups and downs notation|EUs:]] vvA1 and d2)

! colspan="2" | 16edo subset notation<br />(major narrower than minor)

! colspan="2" | 16edo subset notation<br />(major wider than minor)

! [[3L&nbsp;2s]] notation<br />({{nowrap|J {{=}} 1/1}})

! Audio

| [[File:0-0 unison.mp3|frameless]]

| [[File:0-150 (8-EDO).mp3|frameless]]

| [[File:0-300 (12-EDO).mp3|frameless]]

| [[File:0-450 (8-EDO).mp3|frameless]]

| [[File:0-600 (12-EDO).mp3|frameless]]

| [[File:0-750 (8-EDO).mp3|frameless]]

| [[File:0-900 (12-EDO).mp3|frameless]]

| [[File:0-1050 (8-EDO).mp3|frameless]]

| [[File:0-1200 octave.mp3|frameless]]

P1 - A1/ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7/d8d - P8d  (s = sub-, d = -oid, see [[5edo#Alternative notations|5edo]] notation)

 

 

 

 

 

 

 

 

8edo chords are very ambiguous, with many chord homonyms. Even the major and minor triads are [[Chord homonym|homonyms]]. Chord components usually default to M2, M3, P4, P5, M6, m7, M9, P11 and M13. Thus D7 has a M3, P5 and m7. 8-edo chord names using 24edo subset names are greatly simplified by using different defaults: instead of the conventional M2, M3, P4, P5, M6, m7, M9, P11 and M13, we have  ~2, ^M3, v4, ^5, M6, ~7, ~9, v11 and M13. Thus D7 becomes ^M3, ^5 and ~7.

! Chord edosteps

! Chord notes

! Full name

! Abbreviated name

! Homonyms

| 0 – 3 – 5

| D ^F♯ ^A

| D^(^5)

| D

| ^F♯m or vGm

| 0 – 2 – 5

| D F ^A

| Dm(^5)

| Dm

| ^A or vB♭

| 0 – 3 – 5 – 7

| D ^F♯ ^A ^C

| D^7(^5)

| D7

| ^F♯m♯11 or vGm♯11

| 0 – 3 – 5 – 6

| D ^F♯ ^A B

| D6(^3,^5)

| D6

| Bm7 and vG,♯9

| 0 – 2 – 5 – 7

| D F ^A ^C

| Dm,~7(^5)

| Dm7

| F6 and vB♭,♯9

| 0 – 2 – 5 – 6

| D F ^A B

| Dm6(^5)

| Dm6

| Bm7(♭5)

| 0 – 2 – 4 – 7

| D F A♭ ^C

| Ddim,~7

| Dm7(♭5)

| Fm6

[[File:8ed2-001.svg]]

{{Uniform map|edo=8}}

8et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 8 13 19 22 28 30 }}.

! [[Harmonic limit|Prime<br />limit]]

!  

!  

8edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[3L 2s|21212]]. In terms of temperaments, in the 5-limit this is best interpreted as [[father]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[mother]], but the ideal tuning for that is much closer to [[5edo]]. The 8d val supports septimal father and [[pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the {{nowrap| 3 & 5 }}. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)|diminished]] scale.

 

A [[Lumatone mapping for 8edo]] is available.