16edo

Prime factorization

2⁴

Step size

75¢

Fifth

9\16 (675¢)

Semitones (A1:m2)

-1:3 (-75¢ : 225¢)

Dual sharp fifth

10\16 (750¢) (→5\8)

Dual flat fifth

9\16 (675¢)

Dual major 2nd

3\16 (225¢)

Consistency limit

7

Distinct consistency limit

5

16 equal divisions of the octave (abbreviated 16edo or 16ed2), also called 16-tone equal temperament (16tet) or 16 equal temperament (16et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 16 equal parts of exactly 75 ¢ each. Each step represents a frequency ratio of 2^1/16, or the 16th root of 2.

Theory

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.

Odd harmonics

Approximation of odd harmonics in 16edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-27.0	-11.3	+6.2	+21.1	-26.3	-15.5	+36.7	-30.0	+2.5	-20.8	-28.3
Error	relative (%)	-36	-15	+8	+28	-35	-21	+49	-40	+3	-28	-38
Steps (reduced)		25 (9)	37 (5)	45 (13)	51 (3)	55 (7)	59 (11)	63 (15)	65 (1)	68 (4)	70 (6)	72 (8)

Intervals

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic 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 is not P1 - M3 - P5. (But see below in "Chord Names".)

The second approach is to preserve the *harmonic* meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".

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

Degree	Cents	Approximate Ratios*	Melodic names, major wider than minor		Harmonic names, major narrower than minor		Interval Names Just	Interval Names Simplified
0	0	1/1	unison	D	unison	D	unison	unison
1	75	28/27, 27/26	aug 1, dim 2nd	D#, Eb	dim 1, aug 2nd	Db, E#	subminor 2nd	min 2nd
2	150	35/32	minor 2nd	E	major 2nd	E	neutral 2nd	maj 2nd
3	225	8/7	major 2nd	E#	minor 2nd	Eb	supermajor 2nd, septimal whole-tone	perf 2nd
4	300	19/16, 32/27	minor 3rd	Fb	major 3rd	F#	minor 3rd	min 3rd
5	375	5/4, 16/13, 26/21	major 3rd	F	minor 3rd	F	major 3rd	maj 3rd
6	450	13/10, 35/27	aug 3rd, dim 4th	F#, Gb	dim 3rd, aug 4th	Fb, G#	sub-4th, supermajor 3rd	min 4th
7	525	19/14, 27/20, 52/35, 256/189	perfect 4th	G	perfect 4th	G	wide 4th	maj 4th
8	600	7/5, 10/7	aug 4th, dim 5th	G#, Ab	dim 4th, aug 5th	Gb, A#	tritone	aug 4th, dim 5th
9	675	28/19, 40/27, 35/26, 189/128	perfect 5th	A	perfect 5th	A	narrow 5th	min 5th
10	750	20/13, 54/35	aug 5th, dim 6th	A#, Bb	dim 5th, aug 6th	Ab, B#	super-5th, subminor 6th	maj 5th
11	825	8/5, 13/8, 21/13	minor 6th	B	major 6th	B	minor 6th	min 6th
12	900	27/16, 32/19	major 6th	B#	minor 6th	Bb	major 6th	maj 6th
13	975	7/4	minor 7th	Cb	major 7th	C#	subminor 7th, septimal minor 7th	perf 7th
14	1050	64/35	major 7th	C	minor 7th	C	neutral 7th	min 7th
15	1125	27/14, 52/27	aug 7th, dim 8ve	C#, Db	dim 7th, aug 8ve	Cb, D#	supermajor 7th	maj 7th
16	1200	2/1	8ve	D	8ve	D	octave	octave

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

Approximation to JI

Selected just intervals by error

The following table shows how 15-odd-limit intervals are represented in 16edo (ordered by absolute error).

15-odd-limit intervals by direct approximation (even if inconsistent)
Interval, complement	Error (abs., in cents)
12/11, 11/6	0.637
13/10, 20/13	4.214
8/7, 7/4	6.174
13/11, 22/13	10.790
5/4, 8/5	11.314
13/12, 24/13	11.427
15/11, 22/15	11.951
9/7, 14/9	14.916
11/10, 20/11	15.004
16/13, 13/8	15.528
6/5, 5/3	15.641
7/5, 10/7	17.488
9/8, 16/9	21.090
14/13, 13/7	21.702
15/13, 26/15	22.741
11/8, 16/11	26.318
4/3, 3/2	26.955
11/9, 18/11	27.592
15/14, 28/15	30.557
10/9, 9/5	32.404
14/11, 11/7	32.492
7/6, 12/7	33.129
18/13, 13/9	36.618
16/15, 15/8	36.731

15-odd-limit intervals by patent val mapping
Interval, complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/6, 12/11	0.637	0.8
13/10, 20/13	4.214	5.6
7/4, 8/7	6.174	8.2
13/11, 22/13	10.790	14.4
5/4, 8/5	11.314	15.1
13/12, 24/13	11.427	15.2
15/11, 22/15	11.951	15.9
11/10, 20/11	15.004	20.0
13/8, 16/13	15.528	20.7
5/3, 6/5	15.641	20.9
7/5, 10/7	17.488	23.3
13/7, 14/13	21.702	28.9
15/13, 26/15	22.741	30.3
11/8, 16/11	26.318	35.1
3/2, 4/3	26.955	35.9
11/9, 18/11	27.592	36.8
11/7, 14/11	32.492	43.3
7/6, 12/7	33.129	44.2
15/8, 16/15	38.269	51.0
13/9, 18/13	38.382	51.2
9/5, 10/9	42.596	56.8
15/14, 28/15	44.443	59.3
9/8, 16/9	53.910	71.9
9/7, 14/9	60.084	80.1

It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it very discordant, although playing this in the context of a larger chord, and with specialized timbres, can make this less noticeable.

16ed2-001.svg

Notation

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 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".

Degree	Cents	Mavila[9] Notation
0	0	unison	1
1	75	aug unison, minor 2nd	1#, 2b
2	150	major 2nd	2
3	225	aug 2nd, minor 3rd	2#, 3b
4	300	major 3rd, dim 4th	3, 4bb
5	375	minor 4th	4b
6	450	major 4th, dim 5th	4, 5b
7	525	aug 4th, minor 5th	4#, 5
8	600	aug 5th, dim 6th	5#, 6b
9	675	perfect 6th, dim 7th	6, 7bb
10	750	aug 6th, minor 7th	6#, 7b
11	825	major 7th	7
12	900	aug 7th, minor 8th	7#, 8b
13	975	major 8th, dim 9th	8, 9bb
14	1050	minor 9th	9
15	1125	major 9th, dim 10ve	9#, 1b
16	1200	10ve (Decave)	1

Armodue theory (4-line staff)

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	harmonic name			melodic name
0-5-9	4:5:6	D F A	Dm	D minor	D F A	D	D major
0-4-9	10:12:15	D F# A	D	D major	D Fb A	Dm	D minor
0-4-8	5:6:7	D F# A#	Daug	D augmented	D Fb Ab	Ddim	D diminished
0-5-10		D F Ab	Ddim	D diminished	D F A#	Daug	D augmented
0-5-9-13	4:5:6:7	D F A C#	Dm(M7)	D minor-major	D F A Cb	D7	D seven
0-5-9-12		D F A Bb	Dm(b6)	D minor flat-six	D F A B#	D6	D six
0-5-9-14		D F A C	Dm7	D minor seven	D F A C	DM7	D major seven
0-4-9-13		D F# A C#	DM7	D major seven	D Fb A Cb	DM7	D minor seven

Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). See Ups and Downs Notation #Chords and Chord Progressions for more examples.

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:

initial question	reverse everything	do the math	reverse again
M2 + M2	m2 + m2	dim3	aug3
D to F#	D to Fb	dim3	aug3
D to F	D to F	m3	M3
Eb + m3	E# + M3	G##	Gbb
Eb + P5	E# + P5	B#	Bb
A minor chord	A major chord	A C# E	A Cb E
Eb major chord	E# minor chord	E# G# B#	Eb Gb Db
Gm7 = G + m3 + P5 + m7	G + M3 + P5 + M7	G B D F#	G B D Fb
Ab7aug = Ab + M3 + A5 + m7	A# + m3 + d5 + M7	A# C# E G##	Ab Cb E Gbb
what chord is D F A#?	D F Ab	D + m3 + d5	D + M3 + A5 = Daug
what chord is C E Gb Bb?	C E G# B#	C + M3 + A5 + A7	C + m3 + d5 + d7 = Cdim7
C major scale = C + M2 + M3 + P4 + P5 + M6 + M7 + P8	C + m2 + m3 + P4 + P5 + m6 + m7 + P8	C Db Eb F G Ab Bb C	C D# E# F G A# B# C
C minor scale = C + M2 + m3 + P4 + P5 + m6 + m7 + P8	C + m2 + M3 + P4 + P5 + M6 + M7 + P8	C Db E F G A B C	C D# E F G A B C
what scale is A B# Cb D E F Gb A?	A Bb C# D E F G# A	A + m2 + M3 + P4 + P5 + m6 + M7	A + M2 + m3 + P4 + P5 + M6 + m7 = A dorian

Octave theory

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.

16edo is also a tuning for the 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).

16edo is also a tuning for the no-threes 7-limit temperament tempering out 546875:524288, which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "Magic family of scales".

Easley Blackwood Jr writes of 16edo:

"16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh."

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 & 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

Regular temperament properties

Uniform maps

13-limit uniform maps between 15.5 and 16.5
Min. size	Max. size	Wart notation	Map
15.5000	15.5387	16ccdeeffff	⟨16 25 36 44 54 57]
15.5387	15.7197	16ccdeeff	⟨16 25 36 44 54 58]
15.7197	15.7540	16deeff	⟨16 25 37 44 54 58]
15.7540	15.8089	16dff	⟨16 25 37 44 55 58]
15.8089	15.8512	16d	⟨16 25 37 44 55 59]
15.8512	16.0431	16	⟨16 25 37 45 55 59]
16.0431	16.0792	16e	⟨16 25 37 45 56 59]
16.0792	16.0887	16ef	⟨16 25 37 45 56 60]
16.0887	16.1504	16bef	⟨16 26 37 45 56 60]
16.1504	16.2074	16bcef	⟨16 26 38 45 56 60]
16.2074	16.3322	16bcddef	⟨16 26 38 46 56 60]
16.3322	16.3494	16bcddeeef	⟨16 26 38 46 57 60]
16.3494	16.5000	16bcddeeefff	⟨16 26 38 46 57 61]

Commas

16et tempers out the following commas. (Note: This assumes val ⟨16 25 37 45 55 59].)

Prime Limit	Ratio^[1]	Monzo	Cents	Color name	Name
5	135/128	[-7 3 1⟩	92.18	Layobi	Major chroma
5	648/625	[3 4 -4⟩	62.57	Quadgu	Major diesis
5	3125/3072	[-10 -1 5⟩	29.61	Laquinyo	Magic comma
5	(20 digits)	[23 6 -14⟩	3.34	Sasepbiru	Vishnuzma
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Septimal quartertone
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicennma
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Jubilisma
7	64827/64000	[-9 3 -3 4⟩	22.23	Laquadzo-atrigu	Squalentine
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Starling comma
7	1029/1024	[-10 1 0 3⟩	8.43	Latrizo	Gamelisma
7	6144/6125	[11 1 -3 -2⟩	5.36	Sarurutrigu	Porwell comma
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
11	3025/3024	[-4 -3 2 -1 2⟩	0.57	Loloruyoyo	Lehmerisma

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Rank-2 temperaments

List of 16et rank two temperaments by badness

Important mosses include:

magic anti-diatonic 3L4s 1414141 (5\16, 1\1)
magic superdiatonic 3L7s 1311311311 (5\16, 1\1)
Pathological magic chromatic 11121121112 3L10s (5\16, 1\1)
mavila anti-diatonic 2L5s 2223223 (9\16, 1\1)
mavila superdiatonic 7L2s 222212221 (9\16, 1\1)
gorgo 5L1s 333331 (3\16, 1\1)
lemba 4L2s 332332 (3\16, 1\2)
Pathological 1L 12s 4 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
Pathological 1L 13s 3 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
Pathological 2L 12s 2 1 1 1 1 1 1 2 1 1 1 1 1 1 (1\16, 1\2)
Pathological 1L 14s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)

Temperaments listed by generator size:

Periods per octave	Generator	Temperaments
1	1\16	Valentine, slurpee
1	3\16	Gorgo
1	5\16	magic/muggles
1	7\16	Mavila/armodue
2	1\16	Bipelog
2	3\16	Lemba, astrology
4	1\16	Diminished/demolished
8	1\16

Mavila

[5]:	5 2 5 2 2
[7]:	3 2 2 3 2 2 2
[9]:	1 2 2 2 1 2 2 2 2

Diminished

[8]:	1 3 1 3 1 3 1 3
[12]:	1 1 2 1 1 2 1 1 2 1 1 2

Magic

[7]: 1 4 1 4 1 4 1

[10]: 1 3 1 1 3 1 1 1 3 1

[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1

Cynder/Gorgo

[5]: 3 3 4 3 3

[6]: 3 3 1 3 3 3

[11]: 1 2 1 2 1 2 1 2 1 2 1

Lemba/Astrology

[4]: 3 5 3 5

[6]: 3 2 3 3 2 3

[10]: 2 1 2 1 2 2 1 2 1 2

Metallic harmony

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.

However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use

it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant.

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".

MOS scales supporting metallic harmony in 16edo

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.

Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.

See: Metallic Harmony.

Diagrams

16-tone piano layout based on the mavila[7]/antidiatonic scale

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".