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{{Infobox ET}}
{{Infobox ET}}
'''8[[EDF]] (8ED3/2)''' is the equal division of the fifth into 8 steps of 87.7444 [[cent]]s each, making it very nearly [[88cET]]. It is related to the [[Tetracot family|octacot temperament]]. 8EDF corresponds to 13.6761 edo (similar to every third step of [[41edo]]).
'''8 equal divisions of the fifth''' ('''8edf''', '''8ed3/2''') is the [[tuning system]] that divides the fifth into 8 steps of 87.7444 [[cent]]s each, making it very nearly [[88cET]]. It is related to the [[Octacot|octacot temperament]]. 8edf corresponds to about 13.6761edo (similar to every third step of [[41edo]]).


==Intervals==
== Intervals ==
 
8edf can be notated either using native uranian (sesquitave) notation, where the notation repeats every period (i.e. [[Just fifth|just diatonic fifth]]), or using double sesquitave (Annapolis) notation, where the notation repeats every two periods (i.e. [[9/4|major diatonic ninth]]). This interprets 8edf as [[16ed9/4]], resulting in a tuning for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. It can also be notated using tetratonic [[4edf]]-based notation.
8EDF can be notated either using native uranian (sesquitave) notation, where the notation repeats every period (i.e. [[Just fifth|just diatonic fifth]]), or using double sesquitave (Annapolis) notation, where the notation repeats every two periods (i.e. [[9/4|major diatonic ninth]]). This interprets 8edf as 16[[edIX]], resulting in a tuning for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. It can also be notated using equalized [[1L 3s (fifth-equivalent)|neptunian]] (1L 3s<3/2>) notation as a superset of [[4edf]].


The naturals result from a [[semiwolf]] generator (~[[7/6]]). For sesquitave notation, letters A-E can be used. For double sesquitave notation, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι).
The naturals result from a [[semiwolf]] generator (~[[7/6]]). For sesquitave notation, letters A-E can be used. For double sesquitave notation, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι).
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!Uranian
!Uranian
!Annapolis
!Annapolis
!Equalized [[1L 3s (fifth-equivalent)|Neptunian]] using 6\8edf
!Tetratonic notation
|-
|-
|'''0'''
|'''0'''
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|}
|}
[https://sevish.com/scaleworkshop/?name=8EDF%20Uranian&data=87.74437510817343%0A175.48875021634686%0A263.2331253245203%0A350.9775004326937%0A438.72187554086713%0A526.4662506490406%0A614.210625757214%0A701.9550008653874&freq=220&midi=57&vert=5&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white&waveform=square&ampenv=organ Scale workshop link for a keyboard/MIDI playable 8EDF (with graphical uranian scale, A=220Hz)]
[https://sevish.com/scaleworkshop/?name=8EDF%20Uranian&data=87.74437510817343%0A175.48875021634686%0A263.2331253245203%0A350.9775004326937%0A438.72187554086713%0A526.4662506490406%0A614.210625757214%0A701.9550008653874&freq=220&midi=57&vert=5&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white&waveform=square&ampenv=organ Scale workshop link for a keyboard/MIDI playable 8EDF (with graphical uranian scale, A=220Hz)]
==Scale tree==
EDF scales can be approximated in [[EDO]]s by subdividing diatonic fifths. If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.
If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
Generator range: 85.7143 cents (4\7/8 = 1\14) to 90 cents (3\5/8 = 3\40)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || ||  || 85.714||
|-
|  || || || || || ||27\47||86.170 ||
|-
| || || || ||  ||23\40||  ||86.250||
|-
|  || || || || || ||42\73||86.301||
|-
| || || || ||19\33|| || ||86.{{Overline|36}}||
|-
| || || || || || ||53\92|| 86.413||
|-
| || || || || ||34\59|| ||86.441||
|-
| || || || || || ||49\85||86.471 ||
|-
| || || ||15\26||  || || ||86.5385||
|-
| ||  || || || || ||56\97||86.598||
|-
| || || || || ||41\71|| ||86.620||
|-
| || || || || ||  ||67\116 ||86.638||
|-
| || || || ||26\45|| || ||86.{{Overline|6}}||[[Flattone]] is in this region
|-
| || ||  || || || || 63\109||86.697||
|-
| || || || || ||37\64|| ||86.719||
|-
| || || || || || ||48\83|| 86.747||
|-
| || ||11\19|| || || ||  ||86.842||
|-
| || || || || || ||51\88||86.93{{Overline|18}}||
|-
|  || || || || ||40\69|| ||86.9565||
|-
| || || || || || ||69\119||86.975||
|-
| || || || ||29\50|| || ||87.000||
|-
| ||  || || || || ||76\131||87.023||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81||  ||87.{{Overline|037}}||
|-
| || || || || || ||65\112 ||87.054||
|-
| || || ||18\31|| || || || 87.097||[[Meantone]] is in this region
|-
| || || || || || ||61\105||87.143||
|-
| || || || || ||43\74|| ||87.{{Overline|162}}||
|-
| || || || || || ||68\117||87.1795||
|-
| || || || ||25\43|| || ||87.209||
|-
| || || || || || ||57\98||87.245||
|-
| || || || || ||32\55|| ||87.{{Overline|27}} ||
|-
| || ||  || || || ||39\67||87.313||
|-
| ||7\12|| || || || || ||87.500||
|-
| ||  || || || || ||38\65||87.692||
|-
| || || || || ||31\53|| ||87.736 ||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||87.766||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||87.805||
|-
| || || || || || ||65\111||87.{{Overline|837}}||
|-
| || || || || ||41\70|| ||87.857 ||
|-
| || || || || || ||58\99||87.{{Overline|87}}||
|-
| || || ||17\29|| || || ||87.931||
|-
| || || || || || ||61\104||87.981||
|-
| || || || || ||44\75|| ||88.000||
|-
| || || || || || ||71\121||88.0165||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||88.0435||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||88.073||
|-
| || || || || ||37\63|| ||88.095||
|-
| || || || || || ||47\80||88.125||
|-
| || ||10\17|| || || || ||88.235||
|-
| || || || || || ||43\73||88.356||
|-
|  || ||  || || ||33\56|| ||88.392||
|-
| || || || || || ||56\95||88.421||The generator closest to a just [[5/3]] for EDOs less than 1600
|-
| || ||  || ||23\39|| || ||88.4615||
|-
| || || || || || ||59\100 ||88.500||
|-
| || || || || ||36\61|| ||88.525||
|-
|  || || || ||  || ||49\83||88.554||
|-
| || || ||13\22||  || || ||88.{{Overline|63}}||[[Archy]] is in this region
|-
| || || || || || ||42\71 ||88.732||
|-
| || || || || ||29\49|| ||88.7755||
|-
| ||  || || || || ||45\76||88.816||
|-
| || || || ||16\27|| ||  ||88.{{Overline|8}}||
|-
| || || ||  || || ||35\59||88.931||
|-
| || || || || ||19\32|| ||89.0625||
|-
| || || || || || ||22\37||89.{{Overline|189}}||
|-
|3\5|| || || || || || ||90.000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.


Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
== Rank 2 temperaments ==
== Rank 2 temperaments ==


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| | (augmented-like)
| | (augmented-like)
|}
|}
== Music ==
* [https://m.youtube.com/watch?v=rzcuLiE3HhE Funny Snakecharmer] - [[Sven Karma]] (Dec 2023) - uses the [[Kartvelian scales#8edf Kartvelian tetradecatonic scale|8edf Kartvelian tetradecatonic scale]], alternating blocks of [[8edf]] and [[6ed4/3]]


== Images ==
== Images ==
[[File:8EDF_cheat_sheet_circle.png|400px|thumb|left|8EDF cheat sheet (sesquitave notation) - [[Media:8EDF cheat sheet.pdf|table only]]]]
[[File:8EDF_cheat_sheet_circle.png|400px|thumb|left|8EDF cheat sheet (sesquitave notation) - [[Media:8EDF cheat sheet.pdf|table only]]]]


[[Category:Edf]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
{{Todo| review }}

Latest revision as of 06:41, 10 November 2024

← 7edf 8edf 9edf →
Prime factorization 23
Step size 87.7444 ¢ 
Octave 14\8edf (1228.42 ¢) (→ 7\4edf)
Twelfth 22\8edf (1930.38 ¢) (→ 11\4edf)
Consistency limit 3
Distinct consistency limit 3

8 equal divisions of the fifth (8edf, 8ed3/2) is the tuning system that divides the fifth into 8 steps of 87.7444 cents each, making it very nearly 88cET. It is related to the octacot temperament. 8edf corresponds to about 13.6761edo (similar to every third step of 41edo).

Intervals

8edf can be notated either using native uranian (sesquitave) notation, where the notation repeats every period (i.e. just diatonic fifth), or using double sesquitave (Annapolis) notation, where the notation repeats every two periods (i.e. major diatonic ninth). This interprets 8edf as 16ed9/4, resulting in a tuning for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. It can also be notated using tetratonic 4edf-based notation.

The naturals result from a semiwolf generator (~7/6). For sesquitave notation, letters A-E can be used. For double sesquitave notation, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι).

Scale Cents Approximate intervals Uranian Diatonic Notation
degree value 7-limit 11-limit 19-limit interval equivalent Uranian Annapolis Tetratonic notation
0 0 1 unison A Α C
1 87.7444 21/20 22/21 20/19, 19/18 min mos2nd minor second A# Α# ^C, vD
2 175.48875 10/9 21/19 maj mos2nd major second B Β D
3 263.2331 7/6 perf mos3rd subminor third C Γ ^D, vE
4 350.9775 60/49, 49/40 11/9 aug mos3rd neutral third C# Γ# E
5 438.7219 9/7 perf mos4th supermajor third D Δ ^E, vF
6 526.46625 27/20 19/14 min mos5th wolf fourth D# Δ# F
7 614.2106 10/7 27/19 maj mos5th augmented fourth E Ε ^F, vC
8 701.955 3/2 sesquitave just fifth A Ϛ/Ϝ C
9 789.6994 63/40 11/7 30/19 min mos7th minor sixth A# Ϛ#/Ϝ# ^C, vD
10 877.44375 5/3 63/38 maj mos7th major sixth B Ζ D
11 965.1881 7/4 perf mos8th subminor seventh C Η ^D, vE
12 1052.9325 90/49, 35/18 11/6 aug mos8th neutral seventh C# Η# E
13 1140.6769 27/14 perf mos9th supermajor seventh D Θ ^E, vF
14 1228.42125 81/40 57/28 min mos10th acute octave D# Θ# F
15 1316.1656 15/7 81/38 maj mos10th minor ninth E Ι ^F, vC
16 1403.91 9/4 double sesquitave major ninth A Α C

Scale workshop link for a keyboard/MIDI playable 8EDF (with graphical uranian scale, A=220Hz)

Rank 2 temperaments

MOS scales and temperament listed by generator size and period:

Periods

per octave

Generator Scale pattern Temperaments
1 1\8 1L5s, 1L6s (pathological)
1 3\8 3L2s (uranian) Semiwolf
2 3\8 2L 2s (augmented-like)

Music

Images

8EDF cheat sheet (sesquitave notation) - table only
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