12edo: Difference between revisions

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{{EDO intro|12}} It is the predominating tuning system in the world today.

12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3¹²/2¹⁹, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

{{Harmonics in equal}}

|+ Intervals of 12edo

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

! [[3-limit]]

! [[5-limit]]

! [[7-limit]]

! Other

| [[25/24]] (+29.328)<br>[[16/15]] (-11.731)

| [[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (-19.443)

| [[18/17]] (+1.045)<br>[[17/16]] (-4.955)

| [[9/8]] (-3.910)

| [[10/9]] (+17.596)

| [[19/17]] (+7.442)<br>[[55/49]] (+0.020)<br>[[64/57]] (-0.532)<br>[[17/15]] (-16.687)

| [[32/27]] (+5.865)

| [[6/5]] (-15.641)

| [[7/6]] (+33.129)<br>[[25/21]] (-1.847)

| [[19/16]] (+2.487)<br>[[44/37]] (+0.026)

| [[81/64]] (-7.820)

| [[5/4]] (+13.686)

| [[63/50]] (-0.108)<br>[[9/7]] (-35.084)

| [[34/27]] (+0.910)<br>[[24/19]] (-4.442)

| [[4/3]] (+1.955)

| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)

| [[24/17]] (+3.000)<br>[[99/70]] (-0.088)<br>[[17/12]] (-3.000)

| [[3/2]] (-1.955)

| [[128/81]] (+7.820)

| [[8/5]] (-13.686)

| [[14/9]] (+35.084)<br>[[100/63]] (+0.108)

| [[19/12]] (+4.442)<br>[[27/17]] (-0.910)

| [[27/16]] (-5.865)

| [[5/3]] (+15.641)

| [[42/25]] (+1.847)<br>[[12/7]] (-33.129)

| [[37/22]] (-0.026)<br>[[32/19]] (-2.487)

| [[16/9]] (+3.910)

| [[9/5]] (-17.596)

| [[30/17]] (+16.687)<br>[[57/32]] (+0.532)<br>[[98/55]] (-0.020)<br>[[34/19]] (-7.442)

| [[15/8]] (+11.731)<br>[[48/25]] (-29.328)

| [[28/15]] (+19.443)<br>[[40/21]] (-15.533)<br>[[27/14]] (-37.039)

| [[32/17]] (+4.955)<br>[[17/9]] (-1.045)

12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

|+ Notation of 12edo

! [[5L 2s|Diatonic]] interval names

* [[ups and downs notation]] is identical to standard notation;

* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

|+ Solfege of 12edo

== JI approximation ==

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.

 

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)

| 17.5

| [[7/6]], [[12/7]]

| 33.1

|-

| [[11/10]], [[20/11]]

| 34.996

| 35.0

{{15-odd-limit}}

 

 

 

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

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

! colspan="2" | Tuning Error

| {{monzo| -19 12 }}

| [{{val| 12 19 }}]

| +0.617

| 0.617

| 0.617

| [{{val| 12 19 28 }}]

| -1.56

| [{{val| 12 19 28 34 }}]

| -3.95

| [{{val| 12 19 28 34 49 }}]

| -2.92

| [{{val| 12 19 28 34 49 51 }}]

| -2.53

 

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

{{Uniform map|13|11.5|12.5}}  

12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

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

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

! [[Color name|Color Name]]

| Lalawa

| Quadgu

| Greater diesis, diminished comma

| Trigu

| Lesser diesis, augmented comma

| Gu

| Syntonic comma, Didymus comma, meantone comma

| Sagugu

| Sasa-trigu

| Layo

| Sepbisa-quadtrigu

| Rugu

| Septimal quartertone

| Biruyo

| Ru

| Triru-aquinyo

| Gariboh

| Zotrigu

| Rurutriyo

| Labizogugu

| Ruyoyo

| Zozoquingu

| Hemimean

| Saruyo

| Hemifamity

| Sasaru

| Latriru-asepyo

| [[Meter]]

| Trizogugu

| 1uu2

| Loruru

| Luyoyo

| Lorugugu

| Saluzo

| Luzozogu

| Bilorugu

| Thozogu

| Superleap

| Thulu

| Sogu

| Sugu

| {{monzo| -5 -2 2 }}

| Soso

<references/>

! Periods <br> per 8ve

! Generator

| [[Ripple]] / [[passion]]

| [[Meantone]] / [[dominant]]

| 1\12

| [[Srutal]] / [[pajara]] / [[injera]]

| 1\12

| [[Augmented]] / [[lithium]]

| 1\12

| [[Diminished]]

| 1\12

{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].

* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)

* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)

* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===

Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132

* Melodic major – 2212122

* [[:purdal:12-EDD]]