16ed5/3

← 15ed5/3

16ed5/3

17ed5/3 →

Prime factorization	24
Step size	55.2724 ¢
Octave	22\16ed5/3 (1215.99 ¢) (→ 11\8ed5/3)
Twelfth	34\16ed5/3 (1879.26 ¢) (→ 17\8ed5/3)
Consistency limit	2
Distinct consistency limit	2

16ed5/3 is the equal division of the just major sixth into sixteen parts of 55.2724 cents each, corresponding to 21.7106edo. It is very closely related to the escapade temperament. It is vaguely equivalent to 22edo.

It very accurately approximates a number of low complexity just intervals, such as: 4/3 (<1¢), 5/4 (<1¢), 11/8 (<2¢), 11/10 (<1¢), 16/15 (<2¢), and 25/16 (<2¢). It also approximates the just fifth and octave to within 17¢, making it a flexible non-octave scale. Notably, having a period of 5/3, the diatonic minor third (6/5) is the period-reduced diatonic octave. This means both are approximated identically (16¢ sharp).

Harmonics

Approximation of harmonics in 16ed5/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+16.0	-22.7	-23.3	-22.7	-6.7	+2.8	-7.3	+9.9	-6.7	-5.9	+9.3
	Relative (%)	+28.9	-41.1	-42.1	-41.1	-12.1	+5.1	-13.2	+17.9	-12.1	-10.6	+16.8
Steps (reduced)		22 (6)	34 (2)	43 (11)	50 (2)	56 (8)	61 (13)	65 (1)	69 (5)	72 (8)	75 (11)	78 (14)

Intervals

16ed5/3 can be notated using steps 7 (~5/4) and 9 (~4/3) as generators, as these are accurate to within 0.6¢. The resulting scale is a heptatonic 2L 5s (similar to the octave repeating antidiatonic). It can also be notated using the fifth-generated Blackcomb temperament as discussed in #Temperaments, which lines up quite nicely with diatonic notation, aside from the "minor second" being in neutral second range and "perfect fourth" being in superfourth range.

Degree	Cents	5/3.4/3.11/6.31/18 subgroup interval	Other interpretations	2L 5s<5/3> mos-interval	2L 5s<5/3> notation	1L 4s<5/3> (Blackcomb[5]) interval	1L 4s<5/3> (Blackcomb[5]) notation	Diatonic interval
0	0.0000	1/1		unison	E	unison	C	unison
1	55.2724	31/30, 32/31, 33/32	36/35	aug unison	E#	aug unison	C#	quartertone
2	110.5448	16/15, 33/31	21/20	min mos2nd	Fb	double-aug unison, dim second	Cx, Dbb	minor second
3	165.8173	11/10		maj mos2nd	F	minor second	Db	neutral second
4	221.0897	25/22	8/7, 17/15	min mos3rd	F#/Gb	major second	D	major second
5	276.3621	75/64, 88/75	7/6, 20/17	maj mos3rd	G	aug second	D#	subminor third
6	331.6345	40/33, 75/62	6/5, 17/14	dim mos4th	G#/Ab	minor third	Eb	minor third
7	386.9069	5/4		perf mos4th	A	major third	E	major third
8	442.1794	31/24, 40/31	9/7, 35/27, 22/17	aug mos4th	A#/Bb	aug third	E#	supermajor third
9	497.4517	4/3		perf mos5th	B	dim fourth	Fb	just fourth
10	552.7242	11/8, 62/45	25/18, 18/13	aug mos5th	B#	perfect fourth	F	wide fourth
11	607.9966	44/31, 64/45	10/7, 17/12	min mos6th	Cb	aug fourth	F#	large tritone
12	663.2690	22/15	72/49	maj mos6th	C	dim fifth	Gb	narrow fifth
13	718.5415	50/33	3/2	min mos7th	C#/Db	perfect fifth	G	acute fifth
14	773.8129	25/16		maj mos7th	D	aug fifth	G#	subminor sixth
15	829.0863	50/31	8/5, 13/8	dim mos8ave	D#/Eb	dim sixth	Cb	minor sixth
16	884.3587	5/3		mosoctave	E	perfect sixth	C	major sixth
17	939.6311	31/18, 55/32	12/7, 19/11	aug mos8ave	E#	aug sixth	C#	supermajor sixth
18	994.9035	16/9, 55/31	7/4	min mos9th	Fb	double-aug sixth, dim seventh	Cx, Dbb	minor seventh
19	1050.1760	11/6		maj mos9th	F	minor seventh	Db	neutral seventh
20	1105.4484	176/93, 125/66, 256/135	40/21, (27/14), 17/9	min mos10th	F#/Gb	major seventh	D	major seventh
21	1160.7208	88/45, 125/64	35/18, 43/22	maj mos10th	G	aug seventh	D#	narrow octave
22	1215.9932	200/99, 121/60, 125/62	2/1	dim mos11th	G#/Ab	minor octave	Eb	octave

These intervals are close to a few other related scales:

	22edo	7ed5/4	23ed18\17	16ed5/3	9ed4/3 (Noleta)	43ed4	34edt	21edo
1	54.54545	55.188	55.2429	55.2724	55.338	55.8140	55.9399	57.1429
2	109.0909	110.375	110.4859	110.5448	110.677	111.6729	111.8797	114.2857
3	163.6364	165.563	165.7288	165.8173	166.015	167.4419	167.8196	171.4286
4	218.1818	220.751	220.9718	221.0897	221.353	223.2558	223.7594	228.5714
5	272.7273	275.938	276.2147	276.3621	276.692	279.0698	279.6993	285.7143
6	327.2727	331.126	331.4576	331.6345	332.030	334.8837	335.6391	342.8571
7	381.8182	386.314	386.7006	386.9069	387.368	390.6977	391.5790	400
8	436.3636	441.501	441.9435	442.1794	442.707	446.5116	447.5188	457.1429
9	490.9091	496.689	497.1865	497.4517	498.045	502.3256	503.4587	514.2857
10	545.5455	551.877	552.4294	552.7242	553.383	558.1395	559.3985	571.4286
11	600	607.064	607.6723	607.9966	608.722	613.9535	615.3384	628.5714
12	654.5455	662.252	662.9153	663.269	664.060	669.7674	671.2782	685.7143
13	709.0909	717.440	718.1582	718.5415	719.398	725.5814	727.2181	742.8571
14	763.6364	772.627	773.4011	773.8129	774.737	781.3954	783.1579	800
15	818.1818	827.815	828.6441	829.0863	830.075	837.7209	839.0978	857.1429
16	872.7273	883.003	883.8870	884.3587	885.413	893.0233	895.0376	914.2857

MOS Scales

16edVI supports the same MOS scales as 16edo, as such it contains the following scales:

Periods per octave	Generator	Pattern
1	1\16	1L Ns
1	3\16	1L 4s, 5L 1s
1	5\16	3L 4s, 3L 7s
1	7\16	2L 5s, 7L 2s
2	1\16	2L 8s, 2L 10s, 2L 12s
2	3\16	4L 2s, 6L 4s
4	1\16	4L 4s, 4L 8s

For the 2L 5s scale, the genchain is this:

F#

C#

G#

D#

A#

E#

B#

F

C

G

D

A

E

B

Fb

Cb

Gb

Db

Ab

Eb

Bb

Fbb

Cbb

Gbb

Dbb

A2

A6

A3

A7

A4

A1

A5

M2

M6

M3

M7

P4

P1

P5

m2

m6

m3

m7

d4

d1

d5

d2

d6

d3

d7

Commas

Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the diaschisma, the marvel comma, Archytas' comma, and the jubilisma, all discussed in the temperaments section. In addition, being an even division of the 5/3, it tempers the sensamagic comma, as the half mosoctave is midway between 9/7 and 35/27. This is analogous to the tritone in 2n edo systems. The keema is tempered due to the septimal interpretation of the diatonic sevenths, and the mothwellsma is tempered by two major mos3rds (7/6) resulting in an augmented mos5th (11/8).

Temperaments

The 2L 5s scale is generated by a very accurate 4/3, such that two of them wind up on a near exact 16/9, which period-reduces to 16/15 (the minor mossecond). This interval taken 2 times is approximated by an 8/7, and taken 3 times is approximated by a 6/5 (or 2/1 in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - 225/224 ((¹⁶/₁₅)²=(⁸/₇)), and the diaschisma - 2048/2025 ((¹⁶/₁₅)³=(⁶/₅)).

The diaschisma can also be tempered by taking 5 generators to mean a 3/2 ((⁴/₃)⁵=(³/₂)·(⁵/₃)²), while the marvel comma can also be tempered with a stack of 3 generators, making a 10/7 ((⁴/₃)³=(¹⁰/₇)·(⁵/₃)).

The tempered marvel comma also means that the two large tritones (pental and septimal) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step.

Both of the 7-limit approaches also temper Archytas' comma as a result of equating the 16/9 with 7/4, and the jubilisma (50/49) due to tritone equivalence. These are relatively large commas, given the step size (about half, and 7/11ths respectively).

This shows the close relationships with srutal and pajara octave temperaments. In 16ed5/3's case, there is a close equivalence to 22edo's pajara tuning.

16ed5/3 primes can be mapped on the 31-limit to the val ⟨65 103 151 183 225 241 266 276 294 316 322], using every 3 steps of a shrinked 65edo (-2.431¢ per octave). It differs from the patent val of 65edo in the mapping of prime 7 (val 65d).

As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, Ayceman proposes the name tristone for the basic diaschismic temperament, based on the 16/15 to 6/5 relationship, as well as the following variants and extensions:

Tristone

Subgroup: 5/3.20/9.10/3

Comma list: 2048/2025

POL2 generator: ~5/4 = 389.8224

Mapping: [⟨1 2 5], ⟨0 -1 -6]]

TE complexity: 1.988720

RMS error: 2.228679 cents

Optimal ET sequence: 9ed5/3, 16ed5/3, 25ed5/3

Tridistone

Subgroup: 5/3.20/9.10/3.1000/189

Comma list: 2048/2025, 225/224, 64/63, 50/49

POL2 generator: ~5/4 = 389.6140

Mapping: [⟨1 2 5 5], ⟨0 -1 -6 -4]]

TE complexity: 1.724923

RMS error: 8.489179 cents

Optimal ET sequence: 9ed5/3, 16ed5/3

Metatristone

Subgroup: 5/3.20/9.5/2

Comma list: 2048/2025

POL2 generator: ~5/4 = 390.5180

Mapping: [⟨1 2 4], ⟨0 -1 -5]]

TE complexity: 2.192193

RMS error: 2.021819 cents

Optimal ET sequence: 9ed5/3, 16ed5/3, 25ed5/3

Metatridistone

Subgroup: 5/3.20/9.5/2.250/63

Comma list: 2048/2025, 225/224, 64/63, 50/49

POL2 generator: ~5/4 = 390.5430

Mapping: [⟨1 2 4 4], ⟨0 -1 -5 -3]]

TE complexity: 1.895168

RMS error: 7.910273 cents

Optimal ET sequence: 9ed5/3, 16ed5/3

16ed5/3 also supports Blackcomb temperament which is built on 5/4 and 3/2 in a very similar way to octave-repeating meantone but is less accurate. Blackcomb tempers out the comma 250/243, the amount by which 3 3/2's exceed 5/4 sixth-reduced, in the 5/3.2.3 subgroup (equal to the 5-limit).

See also

• Alpha, beta, and gamma family of equal divisions