27edo: Difference between revisions

{{EDO intro|27}}

Assuming pure octaves, 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. Its fifth and harmonic seventh are both sharp by 9 cents, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7 cents. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from [[superpyth|1/3-septimal-comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3-comma meantone|1/3-syntonic-comma meantone]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.

27edo, with its 400-cent major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55 cents, respectively.

Since 27 factors into primes as 3³, 27edo contains [[3edo]] and [[9edo]] as subsets.

! colspan="3" | [[Ups and downs notation]]

| {a, b, 0, 1}

| {a, b}, b < −1

| {a, b, -1}

| {a, b, 0, 0, 0, 1}

| {a, b, 0, 0, 0, -1}

| {a, b, 1}

| {a, b}, b > 1

| {a, b, 0, -1}

! rowspan="2" | [[Cent]]s !! colspan="6" | Note from C

! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation

| 0

| 711.11

| 222.22

| B{{sesquisharp2}} || F{{sesquiflat2}}

| 933.33

| 444.44

| 1155.55

| 666.66

| colspan="2" | F&#x266F;

| 177.77

| colspan="2" | C&#x266F;

| 888.88

| colspan="2" | G&#x266F;

| 400

| colspan="2" | D&#x266F;

| 1111.11

| colspan="2" | A&#x266F;

| 622.22

| colspan="2" | E&#x266F;

| 133.33

| B&#x266F;

| F&#x1D12B;

| 844.44

| F&#x1D12A;

| C&#x1D12B;

| 355.56

| C&#x1D12A;

| G&#x1D12B;

| 1066.67

| G&#x1D12A;

| D&#x1D12B;

| 577.78

| D&#x1D12A;

| A&#x1D12B;

| 88.89

| A&#x1D12A;

| E&#x1D12B;

| 800

| E&#x1D12A;

| B&#x1D12B;

| 311.11

| B&#x1D12A;

| F&#x266D;

| 1022.22

| colspan="2" | C&#x266D;

| 533.33

| colspan="2" | G&#x266D;

| 44.44

| colspan="2" | D&#x266D;

| 755.56

| colspan="2" | A&#x266D;

| 266.67

| colspan="2" | E&#x266D;

| 977.78

| colspan="2" | B&#x266D;

| 488.89

| 0

The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. With standard [[circle-of-fifths notation]], a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats.The notes from C to D are C, D&#x266D;, C{{demisharp2}}, D{{demiflat2}}, C&#x266F;, and D, with some ascending intervals appearing to be descending on the staff.

Another notational implication is that, being a Superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C–E is a 9/7 supermajor third and so the note located one major third above C must be notated as D♯ or E{{naturaldown}}. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D–F is a 7/6 subminor third and so the note located one minor third above D must be notated as either G&#x266D; or F{{naturalup}}. The composer can decide for themselves which additional accidental pair is appropriate if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B&#x266D; is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.

{{sharpness-sharp4}}

This notation is a subset of the notation for [[54edo#Sagittal notation|54-EDO]].

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

The notation of Tetracot[7]. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 step in this instance, is denoted by &#x266F; and &#x266D;.  

| aug 1sn, double-dim 2nd

| double-aug 1sn, dim 2nd

| minor 2nd

| major 2nd

| minor 7th

| major 7th

| aug 7th, double-dim 8ve

| double-aug 7th, dim 8ve

{{15-odd-limit|27}}

{{15-odd-limit|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}}

 

|}

| −3.70

* Superpyth pentatonic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6

* Superpyth hyperchromatic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1

* [[The Pinetone System#The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4

* [[The Pinetone System#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4

* [[The Pinetone System#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3

* [[The Pinetone System#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4  

* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4

* [[The Pinetone System#Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3

 

Direct sunlight (this is its original/default tuning; subset of [[Sensi]][19])

* 44.444

 

Hypersakura (this is its original/default tuning; subset of Sensi[19])