19edo: Difference between revisions

In music, '''19 equal temperament''', called 19-TET, 19-[[EDO|EDO]], or 19-ET, is the scale derived by dividing the [[Octave|octave]] into 19 [[Equal|equal]]ly large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 [[cent|cents]]. It is the 8th [[prime_numbers|prime]] edo, following [[17edo|17edo]] and coming before [[23edo|23edo]].

Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur_Dieu_ta_pitié|Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts.  Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://sonic-arts.org/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[Meantone_family|meantone]] temperament. It is also a suitable for [[Regular_Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths.'' For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.

Another option would be to use a stretched octave; the [[The_Riemann_Zeta_Function_and_Tuning|zeta function]]-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For instance, if we're using [[93ed30|93ed30]] (a variant of 19edo in which 30:1 is just), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.

Because 19 EDO allows for more blended, consonant harmonies than 12 EDO does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. William Lynch suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12 EDO blend much better in 19 EDO.

The narrow whole tones and wide diatonic semitones of 19edo give the diatonic scale a somewhat duller quality, but has the opposite effect on the pentatonic scale, which becomes much more expressive owing to the larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

|-

! | Degree

! | Solfege

! | Ratios*

|-

| style="text-align:center;" | 0

| style="text-align:center;" | do

| style="text-align:center;" | P1

| style="text-align:center;" | 1/1

|-

| style="text-align:center;" | 1

| style="text-align:center;" | di

| style="text-align:center;" | A1, m2

| style="text-align:center;" | 25/24, 28/27, 26/25, 27/26

|-

| style="text-align:center;" | 2

| style="text-align:center;" | ra

| style="text-align:center;" | M2, m3

| style="text-align:center;" | 15/14, 16/15, 13/12, 14/13

|-

| style="text-align:center;" | 3

| style="text-align:center;" | re

| style="text-align:center;" | M3

| style="text-align:center;" | 9/8, 10/9

|-

| style="text-align:center;" | 4

| style="text-align:center;" | ri/ma

| style="text-align:center;" | m4, a3

| style="text-align:center;" | 7/6, 8/7, 15/13

|-

| style="text-align:center;" | 5

| style="text-align:center;" | me

| style="text-align:center;" | M4, m5

| style="text-align:center;" | 6/5

|-

| style="text-align:center;" | 6

| style="text-align:center;" | mi

| style="text-align:center;" | M5

| style="text-align:center;" | 5/4, 16/13, 26/21

|-

| style="text-align:center;" | 7

| style="text-align:center;" | mo

| style="text-align:center;" | A5, d6

| style="text-align:center;" | 32/25, 9/7, 13/10

|-

| style="text-align:center;" | 8

| style="text-align:center;" | fa

| style="text-align:center;" | P6

| style="text-align:center;" | 4/3

|-

| style="text-align:center;" | 9

| style="text-align:center;" | fi

| style="text-align:center;" | A6, m7

| style="text-align:center;" | 25/18, 7/5, 18/13

|-

| style="text-align:center;" | 10

| style="text-align:center;" | se

| style="text-align:center;" | M7, d8

| style="text-align:center;" | 36/25, 10/7, 13/9

|-

| style="text-align:center;" | 11

| style="text-align:center;" | sol

| style="text-align:center;" | P8

| style="text-align:center;" | 3/2

|-

| style="text-align:center;" | 12

| style="text-align:center;" | lo

| style="text-align:center;" | A8, m9

| style="text-align:center;" | 25/16, 14/9, <span style="line-height: 1.5;">20/13</span>

|-

| style="text-align:center;" | 13

| style="text-align:center;" | le

| style="text-align:center;" | M9, m10

| style="text-align:center;" | 8/5, <span style="line-height: 1.5;">13/8, 21/13</span>

|-

| style="text-align:center;" | 14

| style="text-align:center;" | la

| style="text-align:center;" | M10

| style="text-align:center;" | 5/3

|-

| style="text-align:center;" | 15

| style="text-align:center;" | li/ta

| style="text-align:center;" | m11, A10

| style="text-align:center;" | 7/4, 12/7, 26/15

|-

| style="text-align:center;" | 16

| style="text-align:center;" | te

| style="text-align:center;" | M11, m12

| style="text-align:center;" | 9/5, 16/9

|-

| style="text-align:center;" | 17

| style="text-align:center;" | ti

| style="text-align:center;" | M12

| style="text-align:center;" | 15/8, 13/7, <span style="line-height: 1.5;">28/15, 24/13</span>

|-

| style="text-align:center;" | 18

| style="text-align:center;" | da

| style="text-align:center;" | A12, d13

| style="text-align:center;" | <span style="line-height: 1.5;">48/25, </span>40/21, 27/14, <span style="line-height: 1.5;">25/13, 52/27</span>

|-

| style="text-align:center;" | 19

| style="text-align:center;" | do

| style="text-align:center;" | P13

| style="text-align:center;" | 2/1

|}

<nowiki>*</nowiki> based on treating 19-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.

Using [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:

| style="text-align:center;" | {a, b, 0, -1}

| style="text-align:center;" | 9/7, 12/7

==Chord Names==

|}

== Instruments ==