8edo: Difference between revisions

 

{{EDO intro|8}}

Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out.

=== Odd harmonics ===

{{Harmonics in equal|8|intervals=odd}}

=== Octave shrinking ===

8edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1188 cents gives the tuning called [[1ed148.5c]].

Pure-octaves 8edo approximates only 6 of the first 27 integer harmonics within 20 cents. 1ed148.5c approximates 10 of them.

In this way, 1ed148.5c (compressed-octaves 8edo) is able to provide a wider and stronger palette of [[consonance]]s compared to pure-octaves 8edo.

There are [[zeta peak index]] tunings nearby 8edo as well, however they damage the octave by over 20 cents, rendering them unrecognisable as stretched or compressed 8edo and more like entirely new scales in their own right.

8edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], ….

|0

|0

|1/1

|1/1

|1

|150

|12/11

|11/10

|2

|300

|19/16

|6/5

|3

|450

|13/10

|13/10

|4

|600

|17/12

|7/5, 10/7

|5

|750

|20/13

|20/13

|6

|900

|32/19

|5/3

|7

|1050

|11/6

|20/11

|8

|1200

|2/1

|2/1

(^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

([[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 2s]] notation (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]]

=== Chord names ===

[[Ups and Downs Notation #Chords and Chord Progressions|Ups and downs]] can name any 8edo chord. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).  

! | 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|alt=alt : Your browser has no SVG support.]]

 

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

{{Uniform map|13|7.5|8.5}}

8edo [[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_family|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 [[Father_family#Mother|Mother]], but the ideal tuning for that is much closer to [[5edo]]. The d val supports septimal father and [[Father_family#Pater|Pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3&5. In terms of multi-period temperaments, it makes for a near perfect [[Jubilismic_clan#Walid|Walid]] or a much less accurate [[Dimipent_family#Diminished|Diminished]] scale.