13edt

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13edt

14edt →

Prime factorization	13 (prime)
Step size	146.304 ¢
Octave	8\13edt (1170.43 ¢)
Consistency limit	7
Distinct consistency limit	4

A plot of the no-twos Z-function, in terms of which 13edt is the fourth no-twos zeta peak EDT.

13 equal divisions of the tritave (13edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 13 equal steps of 146.3 cents each, or the thirteenth root of 3. It is best known as the equal-tempered version of the Bohlen-Pierce scale, and therefore has received by far the most attention among equal divisions of the tritave.

It provides an excellent approximation to the 3.5.7 subgroup, especially for its size, being comparable to 34edo's accuracy in the 5-limit. In this subgroup, it tempers out 245/243 and 3125/3087, the same commas as bohpier temperament. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 (26edt, 39edt and 52edt) come to the fore.

13edt can be described as approximately 8.202edo. This implies that each step of 13edt can be approximated by 5 steps of 41edo.

In the no-2 3/1-equave-7-limit, 13edt maintains the smallest relative error of any EDT until 258edt and 271edt, and the smallest absolute error until 56edt.

Theory

Approximation of odd harmonics in 13edt

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.00	-6.53	-3.83	+0.00	-54.80	-51.40	-6.53	+69.39	+23.14	-3.83	-15.02
	Relative (%)	+0.0	-4.5	-2.6	+0.0	-37.5	-35.1	-4.5	+47.4	+15.8	-2.6	-10.3
Steps (reduced)		13 (0)	19 (6)	23 (10)	26 (0)	28 (2)	30 (4)	32 (6)	34 (8)	35 (9)	36 (10)	37 (11)

Approximation of odd harmonics in 13edt

Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	-13.07	+0.00	+22.59	+53.44	-54.80	-10.36	+39.74	-51.40	+8.32	+72.17	-6.53
	Relative (%)	-8.9	+0.0	+15.4	+36.5	-37.5	-7.1	+27.2	-35.1	+5.7	+49.3	-4.5
Steps (reduced)		38 (12)	39 (0)	40 (1)	41 (2)	41 (2)	42 (3)	43 (4)	43 (4)	44 (5)	45 (6)	45 (6)

• Relationship between Bohlen-Pierce and octave-ful temperaments

Intervals

Main article: Intervals of BP

Steps	Cents	Hekts	Enneatonic degree	Corresponding 3.5.7 subgroup intervals	Lambda (sLsLsLsLs, E = 1/1)
0	0	0	P1	1/1	E
1	146.3	100	A1/m2	49/45 (-1.1c); 27/25 (+13.1c)	F
2	292.6	200	M2/d3	25/21 (-9.2c)	F#, Gb
3	438.9	300	A2/P3/d4	9/7 (+3.8c)	G
4	585.2	400	A3/m4/d5	7/5 (+2.7c)	H
5	731.5	500	M4/m5	75/49 (-5.4c)	H#, Jb
6	877.8	600	A4/M5	5/3 (-6.5c)	J
7	1024.1	700	A5/m6/d7	9/5 (+6.5c)	A
8	1170.4	800	M6/m7	49/25 (+5.4c)	A#, Bb
9	1316.7	900	A6/M7/d8	15/7 (-2.7c)	B
10	1463.0	1000	P8/d9	7/3 (-3.8c)	C
11	1609.3	1100	A8/m9	63/25 (+9.2c)	C#, Db
12	1755.7	1200	M9/d10	135/49 (+1.1c); 25/9 (-13.1c)	D
13	1902.0	1300	A9/P10	3/1	E

JI approximation

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal Equave stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
3.5.7	245/243, 3125/3087	[⟨13 19 23]] (b13)	+1.393	1.150	0.79

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per tritave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperament
1	1\13	146.30	49/45	Procyon
1	2\13	292.61	25/21	Sirius
1	3\13	438.91	9/7	BPS
1	4\13	585.22	7/5	Canopus
1	5\13	731.63	75/49
1	6\13	877.83	5/3	Arcturus

See also

• Bohlen-p_et
• Catalog of 3.5.7 subgroup rank two temperaments
• No-twos subgroup temperaments#3.5.7 subgroup temperaments
• 19ED5: relative ED5
• 23ED7: relative ED7