16edo: Difference between revisions

{{EDO intro|16}}

16edo is not especially good at representing most low-odd-limit musical intervals, but it has a [[7/4]] which is only six cents sharp, and a [[5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.

 

Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]

! Degree

! [[Cent]]s

! Approximate <br> Ratios*

! colspan="2" | Melodic names, <br> major wider than minor

! colspan="2" | Harmonic names, <br> major narrower <br> than minor

! Interval Names <br> Just

! Interval <br> Names <br> Simplified

| D#, Eb

| Db, E#

| E#

| Eb

| Fb

| F#

| F#, Gb

| Fb, G#

| 19/14, 27/20, 52/35, 256/189

| G#, Ab

| Gb, A#

| 28/19, 40/27, 35/26, 189/128

| A#, Bb

| Ab, B#

| B#

| Bb

| Cb

| C#

| C#, Db

| Cb, D#

<nowiki>*</nowiki> based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

 

 

 

 

16edo 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 16edo.

Mos scales like mavila[7] (or "inverse/anti-diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A-G #/b notation as described above. Alternatively, one can utilize the Mavila[9] MOS, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. [[Armodue_theory|Armodue notation]] of 16-EDO "Mavila-[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16edo keyboard. If the 9-note "Enneatonic" MOS is adopted as a notational basis for 16edo, then we need an entirely different set of interval classes than any of the heptatonic classes described above; perhaps it even makes sense to refer to octaves as 2/1, "[[decave]]".

! colspan="2" | Mavila[9] Notation

| 1#, 2b

| 2#, 3b

| 3, 4bb

| 4b

| 4, 5b

| 4#, 5

| 5#, 6b

| 6, 7bb

| 6#, 7b

| 7#, 8b

| 8, 9bb

| 9#, 1b

=== Armodue theory (4-line staff) ===

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

For translations of parts of the Armodue pages see the [[Armodue]] on this wiki

=== Chord names ===

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).

! | chord

! | JI ratios

! colspan="3" | harmonic name

! colspan="3" | melodic name

| 0-5-9

| 0-4-9

| D F# A

| D Fb A

| 0-4-8

| D F# A#

| D Fb Ab

| 0-5-10

| D F Ab

| D F A#

| 0-5-9-13

| D F A C#

| D F A Cb

| 0-5-9-12

| Dm(b6)

| D F A B#

| 0-5-9-14

| 0-4-9-13

| D F# A C#

| D Fb A Cb

Using melodic names, interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:

{| class="wikitable" style="text-align:center;"

|-

|-

 

16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).

From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174 cents, and take the 300-cent minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5 cents), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .

The interval between the 28th &amp; 19th harmonics, 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

{{Uniform map|13|15.5|16.5}}

16et [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)

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

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

![[Monzo]]

![[Cent]]s

![[Color name]]

|[[135/128]]

|{{monzo| -7 3 1 }}

| Major chroma

|[[648/625]]

|{{monzo| 3 4 -4 }}

| Major diesis

|[[3125/3072]]

|{{monzo| -10 -1 5 }}

|[[6115295232/6103515625|(20 digits)]]

|{{monzo| 23 6 -14 }}

|[[Vishnuzma]]

|[[36/35]]

|{{monzo| 2 2 -1 -1 }}

| Septimal quartertone

|[[525/512]]

|{{monzo| -9 1 2 1 }}

|[[50/49]]

|{{monzo| 1 0 2 -2 }}

|[[64827/64000]]

|{{monzo| -9 3 -3 4 }}

| Squalentine

|[[3125/3087]]

|{{monzo| 0 -2 5 -3 }}

| Gariboh

|[[126/125]]

|{{monzo| 1 2 -3 1 }}

|[[1029/1024]]

|{{monzo| -10 1 0 3 }}

|[[6144/6125]]

|{{monzo| 11 1 -3 -2 }}

|[[121/120]]

|{{monzo| -3 -1 -1 0 2 }}

|[[176/175]]

|{{monzo| 4 0 -2 -1 1 }}

|[[385/384]]

|{{monzo| -7 -1 1 1 1 }}

|[[441/440]]

|{{monzo| -3 2 -1 2 -1 }}

|[[3025/3024]]

|{{monzo| -4 -3 2 -1 2 }}

*[[List of 16et rank two temperaments by badness]]

 

 

! Periods<br>per octave

|[[Valentine]], [[slurpee]]

|[[Gorgo]]

|[[magic]]/muggles

|[[Mavila]]/armodue

|[[Bipelog]]

|[[Lemba]], [[astrology]]

|[[Diminished]]/demolished

|  

Because 16edo does not approximate 3/2 well at all, triadic harmony based on heptatonic thirds is not a great option for typical harmonic timbres.

 

 

Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".

The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.

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".

'''Un-annotated diagram'''

Please explain this image.

'''Lumatone mapping'''

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