16edo: Difference between revisions

| ja = 16平均律

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'''16-EDO''' is the [[Equal_division_of_the_octave|equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent|cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7/4|7/4]] which is only six cents sharp, and a [[5/4|5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step.

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. (But see below in "Chord Names".)

*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

16edo chords can be named using ups and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated (see below).

| style="text-align:center;" | 4.214

|-

|-

| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]

| style="text-align:center;" | 10.790

|-

|-

| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]

| style="text-align:center;" | 15.004

|-

|-

| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]

| style="text-align:center;" | 22.741

|-

|-

|-

| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]

| style="text-align:center;" | 4.214

|-

|-

| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]

| style="text-align:center;" | 10.790

|-

|-

| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]

| style="text-align:center;" | 15.004

|-

|-

| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]

| style="text-align:center;" | 22.741

|-

|-

|-

| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]

[[:File:16ed2-001.svg|16ed2-001.svg]]

The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.

The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.

16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. The Armodue model uses a 4-line staff for 16-EDO.

This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower, as per the "harmonic notation" above. Simply swap sharps with flats for "melodic notation".

[[List_of_16et_rank_two_temperaments_by_badness|List of 16et rank two temperaments by badness]]

{| class="wikitable"

See [[Metallic_Harmony|Metallic Harmony]].

16 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes [[val|val]] &lt; 16 25 37 45 55 59 |.)

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[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.

Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki.

[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]

Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011

Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)

[https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve Huckleberry Regional Preserve] by City of the Asleep