16edo

A comparison of 15edo and 16edo, showing their similarities (see Carlos Alpha) and 16edo's antidiatonic scale.

16edo is the tuning which divides the octave into 16 equal parts of 75 cents each. It is notable for its antidiatonic scale.

Tuning theory

Edostep interpretations

In the 2.5.7.13.19 subgroup, 16edo's edostep, the eka, represents the following intervals:

• 20/19, the difference between 19/16 and 5/4
• 133/128, the difference between 8/7 and 19/16
• 26/25, the difference between 5/4 and 13/10
• 169/160, the difference between 13/10 and 16/13
• 256/245, the difference between 35/32 and 8/7
• 133/128, the difference between 20/19 and 35/32

JI approximation

16edo does not approximate prime 3 very well, with its fifth being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to 12edo in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is antidiatonic, with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).

16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for erac theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale.

Approximation of prime harmonics in 16edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-27.0	-11.3	+6.2	-26.3	-15.5	-30.0	+2.5	-28.3	+20.4	-20.0
	Relative (%)	0.0	-35.9	-15.1	+8.2	-35.1	-20.7	-39.9	+3.3	-37.7	+27.2	-26.7
Steps (reduced)		16 (0)	25 (9)	37 (5)	45 (13)	55 (7)	59 (11)	65 (1)	68 (4)	72 (8)	78 (14)	79 (15)

Scales and chords

The antidiatonic scale can be understood as somewhat of an "inverted" counterpart of the standard major scale. In antidiatonic, minor thirds are evenly divisible in two while major thirds are not, which is backwards from 12edo. Additionally, there are perfect and augmented fifths rather than perfect and diminished.

Due to 16edo dividing the fifth into three parts, 16edo supports slendric arto and tendo triads alongside its major and minor triads. These may be treated as forms of suspended triads or as augmented and diminished thirds, depending on the framework being used.

16edo shares its mapping of 5/4, 8/7, and 3/2 in terms of edosteps with 15edo and with the Carlos Alpha scale.

Practicality

Many brass instruments rely on both binary keying patterns and the harmonic series to play the full range of notes; since 16edo finds its first good odd harmonic at 7/1, it is then fortuitous that, as a power of two, 16edo fits perfectly within 4 keys on such an instrument. Also, being a power of two, 16edo is easy to tune by taking lots of square roots, which are the only form of root that is constructible geometrically.

Regular temperaments

16edo shares Mabilic with 25edo and Jubilic with 26edo.

Notation

The most common notation system for 16edo is melodic antidiatonic notation, where the seven nominals are the antidiatonic scale, and # and b raise and lower by one eka respectively. Armodue notation can also be used, where the numbers 1-9 are used for notes of the 9-note armotonic scale, and each staff corresponds to a single octave (indicated by a numeric clef).

Armodue theory

Due to 16edo's tuning inaccuracy but structural similarity to standard diatonic tunings, Armodue theory (the xenharmonic community's interpretation of a theory by Luca Attanasio) chooses to separate the notions of functional consonance and proper concordance, allowing the fifth to become fundamental in the former, but 5/4 and 7/4 to become fundamental in the latter. This is done to emphasize the 7th harmonic over the 3rd and 5th harmonics, and has precedent with the treatment of the fourth as a functional dissonance despite its proper concordance in counterpoint. One large step of the 16edo antidiatonic scale represents the 7th subharmonic, adding a small step results in the 5th harmonic, and adding another small step results in the 3rd subharmonic.

Functional consonance is assigned linearly based on interval size, from the closest to the unison/octave being the most dissonant to the closest to the tritone being the most consonant, with the exception of the tritone and octave themselves, which serve as the most dissonant and consonant intervals respectively.

Interval		Type	Notes
0	16	Stable consonance
7	9	Open consonance	Has the same basic structural role in Armodue theory as its counterpart does in 12edo.
6	10	Neutral consonance	10\16 can be seen as an alternative to 9\16 for bounding chords, and 6\16 to 5\16 for being their internal intervals.
5, 4	11, 12	Sweet consonance	The Armodue system emphasizes the familiarity of 4\16 and 12\16 with an audience accustomed to 12edo.
3	13	Sweet dissonance
2	14	Neutral dissonance
1	15	Harsh dissonance
8		Unstable dissonance	Any chord containing this interval is considered unstable, meaning that the standard 4:5:6:7 harmonic tetrad is an unstable chord, and omitting the 5 or 7 can lead to a stabilized sound.

Note that the 7th harmonic, the interval explicitly emphasized by Armodue theory, is a "sweet dissonance" - this implies a more functional, expressive role of dissonance compared to the conventional perception, and a distinct sound for Armodue's central genre. This octave-centric sense of consonance and dissonance also favors octave-heavy timbres like organs.

Armodue theory also de-emphasizes a chord's position with respect to the scale (mainly using it for the smoothness of voice movement), focusing primarily on its quality. Progressions involve chords which gradate in overall quality from one extreme of consonance or dissonance to the other. This suggests that the sense of "dissonance" relevant to Armodue theory is closer to the role of the dominant chord in conventional theory.

The armotonic scale

The armotonic scale is the 9-note extension of the 7-note antidiatonic scale. It is seen as the standard counterpart to diatonic in Armodue theory. One notable characteristic of the armotonic scale is the division of the category of "third" into two separate scale degrees, so that each note has either a tendo or major third over it, and either an arto or minor third over it.

Table of intervals

Interval	Cents	ADIN	Armodue
0	0	Unison	Unison
1	75	Subminor second	Minor second
2	150	Nearminor second	Major second
3	225	Nearmajor second	Minor third
4	300	Nearminor third	Major third
5	375	Nearmajor third	Minor fourth
6	450	Supermajor third	Major fourth
7	525	Perfect fourth	Perfect fifth
8	600	(Tritone)	Augmented fifth / diminished sixth
9	675	Perfect fifth	Perfect sixth
10	750	Subminor sixth	Minor seventh
11	825	Nearminor sixth	Major seventh
12	900	Nearmajor sixth	Minor eighth
13	975	Nearminor seventh	Major eighth
14	1050	Nearmajor seventh	Minor ninth
15	1125	Supermajor seventh	Major ninth
16	1200	Octave	Tenth