27edo: Difference between revisions

27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, 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.

However, assuming just octaves, 27edo's fifth and harmonic seventh are both sharp by nine 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 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]].

27edo, with its 400 cent major third, 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 allegedly Bohlen-Pierce 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.

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]] 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-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 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 triad, 3:5:7, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.

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 ''a'' = 2 and ''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.

=== Notation ===

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♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with some ascending intervals appearing to be descending on the staff.

 

* B♯ = F𝄫

* F𝄪 = C𝄫

* C𝄪 = G𝄫

* G𝄪 = D𝄫

* D𝄪 = A𝄫

* A𝄪 = E𝄫

 

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 technically 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♭ 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♭ is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.

! Approximate Ratios*

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

! [[Walker Brightness Notation]]

! colspan="2" | [[6L 1s]] Notation

! colspan="2" |[[Solfege|Solfeges]]

| 0.00

| 44.44

| [[28/27]], [[36/35]], [[39/38]], [[49/48]], [[50/49]], [[81/80]]

| 88.89

| [[16/15]], [[21/20]], [[25/24]], [[19/18]], [[20/19]]

| 133.33

| 177.78

| 222.22

| 266.67

| 311.11

| 355.56

| 400.00

| 444.44

| 488.89

| 533.33

| [[27/20]], [[48/35]], [[19/14]], [[26/19]]

| 577.78

| 622.22

| 666.67

| [[40/27]], [[35/24]], [[19/13]], [[28/19]]

| 711.11

| 755.56

| 800.00

| 844.44

| 888.89

| 933.33

| 977.78

| 1022.22

| 1066.67

| [[28/15]], [[13/7]], [[24/13]]

| 1111.11

| [[15/8]], [[40/21]], [[48/25]], [[19/10]], [[36/19]]

| 1155.56

| [[27/14]], [[35/18]], [[96/49]], [[49/25]], [[160/81]]

| 1200.00

| 2/1

<nowiki/>* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.

! [[Color name|Color Name]]

! Monzo Format

| {a, b, 0, 1}

| {a, b}, b &lt; -1

| {a, b, -1}

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

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

| {a, b, 1}

| {a, b}, b &gt; 1

| {a, b, 0, -1}

! JI Chord

! Notes as Edosteps

! Notes of C Chord

! Written Name

! Spoken Name

For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. See also the [[22edo]] page.

=== 15-odd-limit interval mappings ===

{{15-odd-limit|27}}

{{15-odd-limit|27.1|title=15-odd-limit intervals by 27e val mapping}}

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

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" | Tuning Error

| [{{val| 27 43 }}]

| [{{val| 27 43 63 }}]

| [{{val| 27 43 63 76 }}]

| −3.70

| [{{val| 27 43 63 76 100 }}]

| [{{val| 27 43 63 76 100 115 }}]

! MOS Scales

27edo [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 27 43 63 76 93 100 }}.)

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

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

! [[Color name|Color Name]]

| Tetracot comma

| Undecimal diasecundal comma, telepathma

| [[76/75]]

| 22.93

| Nogugu

| Large undevicesimal ninth tone

| 19th Partial chroma

<references/>

 

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

* Superpyh diatonic - Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1

* Superpyth chromatic - Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1

* 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

* 5-limit / pental double harmonic  hexatonic (Augmented[6] [[4M]]): 2 7 2 7 7 2, 7 7 2 2 7 2

* Superpyth melodic minor - Superpyth 2|4 #6 #7 or 5|1 b3: 5 1 5 5 5 5 1

* Superpyth harmonic minor - Superpyth 2|4 #7: 5 1 5 5 1 9 1

* Superpyth harmonic major - Superpyth 5|1 b6: 5 5 1 5 1 9 1

* Superpyth double harmonic major - Superpyth 5|1 b2 b6: 1 9 1 5 1 9 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])