31edo: Difference between revisions

== Theory ==

31edo's perfect fifth is flat of the just interval [[3/2]] (over five cents), as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].

! colspan="3" | [[Ups and Downs Notation]]

! colspan="3" | Extended pythagorean notation

|-

| 0

| rowspan="2" | minor

| fourthward wa

| 32/27, 16/9

|-

|-

| fifthward wa

| 9/8, 27/16

|-

| 9/7, 12/7

|}

All 31edo chords can be named using ups and downs. 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). Here are the zo, gu, ilo, yo and ru triads:

== Notation ==

{{Sharpness-sharp2}}

=== Neutral circle-of-fifths notation ===

[[Neutral circle-of-fifths notation]] is much like ups and downs notation. Between C and D (do and re) for example, we have the following notes:

The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:

* 12\31 generator generates a [[semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to [[19/13]].

* A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.

* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s|orwelloid (4L 5s)]] MOS could be treated as a 9-tone well temperament.

See [[#Rank-2 temperaments]] for a table of MOSes and their temperament interpretations.

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! colspan="2" | Tuning Error

|-

{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

| Schismina

|}

=== Rank-2 temperaments ===

{| class="wikitable center-1"

|-

! Generator

| 4\31

| 154.84

| [[Nusecond]] / [[greeley]]

| (P8, P11/11)

| 5\31

| 193.55

| [[Luna]] / [[didacus]] / [[hemithirds]] /  

[[hemiwürschmidt]] / [[tutone]]

| 6\31

| 232.26

| (P8, P5/3)

|-

| 7\31

| 270.97

| [[Orson]] / [[orwell]] / [[winston]]

| (P8, P12/7)

| 8\31

| 309.68

| [[Myna]] / [[triwell]]

| (P8, ccP5/10)

| 9\31

| 348.39

| (P8, P5/2)

|-

| 10\31

| 387.10

| [[Würschmidt]] / [[worschmidt]]

| (P8, ccP5/8)

| 11\31

| 425.81

| [[Squares]] / [[Sentinel]]

| (P8, P11/4)

| 12\31

| 464.52

| (P8, c⁵P4/14)

|-

| 13\31

| 503.23

| [[Meantone]] / [[meanpop]]

| (P8, P5)

| 14\31

| 541.94

| (P8, c⁵P4/12)

|-

| 15\31

| 580.65

| [[Tritonic]] / [[tritoni]]

| (P8, ccP4/5)

|}

=== Keyboard Instruments ===

* [https://www.huygens-fokker.org/instruments/fokkerorgan.html Fokker Organ]

* [[Pentachords of 31edo]]

* [[Tricesimoprimal Tetrachordal Tesseract]]

* [[CG-31]]

=== Videos ===

* [https://www.youtube.com/watch?v=7cv-nuSjbY4&list=PLiWv7dE90L6CsQmQySVdAiRSIIDaAymiJ&pp=iAQB Playlist of 31edo music theory videos on YouTube] by [[Zhea Erose]]

=== Software ===

* [http://31et.com/keyboard.php Virtual Piano Keyboard in 31-Tone Equal Temperament]

=== Diagrams ===