Bohlen-Pierce

The Bohlen-Pierce system is a non-octave tuning system of (exact or well-tempered) 13 equal divisions of the perfect twelfth (or tritave). In the Bohlen-Pierce system, the tritave is generally seen as the interval of equivalence, and harmony emphasizes the odd harmonics (such as the chord 3:5:7:9). It is the smallest EDT that has a tuning of lambda, the 9-note scale of the sensamagic temperament (which serves an analogous role to meantone in tritave-based harmony).

General theory

JI approximation

13edt's 2/1 is flat about 30 cents, making it most prominently a 3.5.7 subgroup temperament, although it also contains approximations of 19, 23, and 29. 7/3 (the tritave-reduced 7th harmonic) is flattened by a small amount, so that a sharpened 9/7 stacks twice to a flattened 5/3, as in sensamagic temperament. As a full 7-limit temperament, it supports sensi, albeit with a severely flattened octave.

Approximation of prime harmonics in 8.202edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-29.6	+0.0	-6.5	-3.8	-54.8	-51.4	+69.4	+23.2	-15.0	+22.7	+53.5
	Relative (%)	-20.2	+0.0	-4.4	-2.6	-37.4	-35.1	+47.5	+15.8	-10.2	+15.5	+36.6
Steps (reduced)		8 (-0.202)	13 (4.798)	19 (2.596)	23 (6.596)	28 (3.394)	30 (5.394)	34 (1.192)	35 (2.192)	37 (4.192)	40 (7.192)	41 (8.192)

Note: Due to a bug with the template, the step counts are octave-reduced instead of tritave-reduced.

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

Edtstep interpretations

The step of 13edt can be interpreted as the following ratios in the 3.5.7 subgroup.

• 49/45
• 27/25

Intervals and notation

The "Intervals represented" column reflects the just (well-tempered) tuning of Bohlen-Pierce.

Steps	Cents	Intervals represented	Note name (Lambda)	Name (Lambda)	Interval category (ADIN*)
0	0	1/1	A	Unison	Unison
1	146.3	27/25	A#, Bb	Minor second	Neutral second
2	292.6	25/21	B	Major second	Farminor third
3	438.9	9/7	C	Perfect third	Supermajor third
4	585.2	7/5	C#, Db	Minor fourth	Nearaugmented fourth
5	731.5	75/49	D	Major fourth, minor fifth	Superfifth
6	877.8	5/3	E	Major fifth	Nearmajor sixth
7	1024.1	9/5	F	Minor sixth	Nearminor seventh
8	1170.4	49/25	F#, Gb	Major sixth, minor seventh	Suboctave
9	1316.7	15/7	G	Major seventh	Nearminor ninth
10	1463.0	7/3	H	Perfect eighth	Subminor tenth
11	1609.3	63/25	H#, Jb	Minor ninth	Farmajor tenth
12	1755.7	25/9	J	Major ninth	Neutral eleventh
13	1902.0	3/1	A	Tritave	Perfect twelfth

*As a subset of 41edo

Multiples

39edt

39edt, sometimes known as Triple Bohlen-Pierce, additionally adds approximations of the 11th and 13th harmonics to 13edt.

Approximation of prime harmonics in 24.606edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+19.2	+0.0	-6.5	-3.8	-6.0	-2.6	+20.7	+23.2	-15.0	+22.7	+4.7
	Relative (%)	+39.4	+0.0	-13.3	-7.8	-12.3	-5.3	+42.4	+47.5	-30.7	+46.5	+9.7
Steps (reduced)		25 (0.394)	39 (14.394)	57 (7.788)	69 (19.788)	85 (11.182)	91 (17.182)	101 (2.576)	105 (6.576)	111 (12.576)	120 (21.576)	122 (23.576)

65edt

13edt is approximately 8.2edo. Dividing each step into fifths to obtain a more accurate octave results in a slightly detuned version of 41edo.

Bohpier temperament

Bohpier temperament is somewhat analogous to blackwood or compton temperament, but based on 13edt, adding the octave as a separate generator (in this case, the period). It is supported by 41edo.