36edo: Difference between revisions

{{EDO intro|36}}

== Theory ==

Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.

That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

36edo also offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36.

The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is {{val| 36 65 116 }}, which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.  

Another 5-limit alternative val is {{val| 36 57 83 }} (36c-edo), which is similar to the patent val but has 5/4 mapped to the 367-cent submajor third rather than the major third. This mapping supports very sharp [[porcupine]] temperament using 5\36 as a generator.  

Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.

=== Divisors ===

36edo is the 7th [[highly composite EDO]], with subset edos {{EDOs|1, 2, 3, 4, 6, 9, 12, 18}}.

{| class="wikitable center-all right-2"

! Degree

! [[cent|cents]]

! Approximate <br> ratios of 2.3.7

! Additional ratios <br> of 2.3.7.13.17

! colspan="3" | [[Ups and Downs Notation|ups and downs <br> notation]]

| 0.00

| 33.33

| 64/63, [[49/48]]

| style="text-align:right;" |66.67

| 100

| 133.33

| 166.67

| 200

|  

| 233.33

| 266.67

| 300

|  

| 333.33

| 366.67

| 400

|  

| 433.33

| 466.67

| 500.00

| 533.33

| 566.67

| 600

| 633.33

| 666.67

| 700

| 733.33

| 766.67

| 800

|  

| 833.33

| 866.67

| 900

|  

| 933.33

| 966.67

| 1000

|  

| 1033.33

| 1066.67

| 1100

| 1133.33

| 1166.67

| 1200.00

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].

One way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, '''A''' is 33.333 cents above '''<span style="background-color: #6ee8e8; color: #071ac7;">A</span>''' and 33.333 cents below '''<span style="background-color: #eda2a2; color: #ff0000;">A</span>'''. Or the colors could be written out (red A, blue C#, etc.) or abbreviated as rA, bC#, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo).

Another way to notate the microtonal notes which doesn't rely on colors is to use [[Alternative symbols for ups and downs notation#Sharp-3|ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz-Ellis notation]]:

In some cases, accidentals may be best denoted using double arrows:

{{Sharpness-sharp3-extended|36}}

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. Unlike [[24edo]], which has genuinely foreign intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366.667 cents and the flat 6/5 is 300 cents. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.

36edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.

The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still not overly unpleasant). In contrast, most people consider 24edo's 50 cent step to sound much more discordant when used as a subminor second.

People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo resulting in other pitches and intervals sounding out of tune. This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes.

In particle physics, [https://en.wikipedia.org/wiki/Baryon baryons], which are the main building blocks of atomic nuclei, are always comprised of three quarks. One could draw an analogy between baryons and semitones (the main building block of 12edo); the baryon is comprised of three quarks and the semitone of three sixth-tones. The number of quarks in a [https://en.wikipedia.org/wiki/Color_charge colorless] particle is always a multiple of three; similarly, the width of "colorless" intervals (the 12-edo intervals, which are neither red nor blue), expressed in terms of sixth-tones, is always a multiple of three. Because of this amusing coincidence, Mason Green proposes referring to the 33.333-cent sixth-tone interval as a "quark".

This is a partial list of the [[commas]] that 36edo [[tempers out]] with its patent [[val]], {{val| 36 57 84 101 125 133 }}.

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

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

| Greater diesis, diminished comma

| Lesser diesis, augmented comma

| Syntonic comma, Didymus comma, meantone comma

| Large Tetracot diesis

{| class="wikitable center-all"

! Periods<br> per 8ve

! Generator<br>(Reduced)

! Cents<br>(Reduced)

! Associated Ratio<br>(Reduced)

! Temperament

! MoS Scales

| rowspan="5" | 1

| 166.667

| [[Squirrel]], [[Porcupine]](36c)  

| [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]]

| 233.333

| [[Slendric]], [[Mothra]], [[Guiron]]

| [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]]

| 11\36 (25/36)

| 366.66 (833.33)

| &phi;

| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[10L 3s]], [[13L 10s]]

| 433.333

| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[11L 14s]]

| 566.667

| [[Liese]], [[Pycnic]]

| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]] ... [[2L 15s]], [[17L 2s]]

| rowspan="2" | 2

| 166.667

| [[Echidna]], [[Hedgehog]]

| [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]]

| 233.333

| [[Baladic]], [[Echidnic]]

| [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]]

| 166.66

| [[6L 3s]], [[6L 9s]], [[15L 6s]]

| rowspan="2" | 4

| 66.66

| [[4L 4s]], [[4L 8s]], [[4L 12s]] [[16L 4s]]

| [[4L 4s]], [[8L 4s]], [[8L 12s]], [[8L 20s]]  

| 33.333

| 1\36

| 33.333

| 50/49, 49/48

| [[Niner]], [[Tritrizo]]

| 1\36

| 33.333

| 50/49, 49/48

| [[Catler]], [[Compton]]

| 33.333

| 50/49, 49/48

 

* [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]]): '''9 6 2 4 7 2 6'''

 

* Palace (subset of Hedgehog[14]): '''5 5 5 6 5 5 5'''

 

== Tuning by Ear ==

36edo can be played on the [[Lumatone]] (see [[Lumatone mapping for 36edo]]) and using three instruments tuned to 12edo with different root notes.