39edt

← 38edt

39edt

40edt →

Prime factorization	3 × 13
Step size	48.7681 ¢
Octave	25\39edt (1219.2 ¢)
Consistency limit	3
Distinct consistency limit	3

39 equal divisions of the tritave (39edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 39 equal steps of approximately 48.8 cents each, or the 39th root of 3. It is also known as the Triple Bohlen-Pierce scale (Triple BP), since it divides each step of the equal-tempered Bohlen-Pierce scale (13edt) into three equal parts.

39edt can be described as approximately 24.606edo. This implies that each step of 39edt can be approximated by 5 steps of 123edo. 39edt contains within it a close approximation of 4ed11/5: every seventh step of 39edt equates to a step of 4ed11/5.

Theory

It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich; in fact it has a better no-twos 13-odd limit relative error than any other edt up to 914edt. Like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being contorted in the no-twos 7-limit, tempering out the same BP commas, 245/243 and 3125/3087, as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 1575/1573, and 847/845. An efficient traversal is therefore given by Mintra temperament, which in the 13-limit tempers out 275/273 and 1575/1573 alongside 245/243, and is generated by the interval of 11/7, which serves as a macrodiatonic "superpyth" fourth and splits the BPS generator of 9/7, up a tritave, in three.

If octaves are inserted, 39edt is related to the 49f&172f temperament in the full 13-limit, known as triboh, tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth no-twos zeta peak edt.

Approximation of prime harmonics in 39edt

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	+20.6	+23.1	-15.0	+22.6	+4.7
	Relative (%)	+39.4	+0.0	-13.4	-7.9	-12.4	-5.4	+42.3	+47.4	-30.8	+46.3	+9.6
Steps (reduced)		25 (25)	39 (0)	57 (18)	69 (30)	85 (7)	91 (13)	101 (23)	105 (27)	111 (33)	120 (3)	122 (5)

Intervals

All intervals shown are within the 91-throdd limit and are consistently represented.

Steps	Cents	Hekts	Enneatonic degree	Corresponding 3.5.7.11.13 subgroup intervals	Lambda (sLsLsLsLs, J = 1/1)	Mintaka[7] (E macro-Phrygian)
0	0	0	P1	1/1	J	E
1	48.8	33.3	SP1	77/75 (+3.2c); 65/63 (-5.3c)	^J	^E, vF
2	97.5	66.7	sA1/sm2	35/33 (-4.3c); 81/77 (+9.9c)	vK	F
3	146.3	100	A1/m2	99/91 (+0.4c); 49/45 (-1.1c); 27/25 (+13.1c)	K	^F, vGb, Dx
4	195.1	133.3	SA1/Sm2	55/49 (-4.9c); 91/81 (-6.5c); 39/35 (+7.7c)	^K	Gb, vE#
5	243.8	166.7	sM2/sd3	15/13 (-3.9c); 63/55 (+8.7c)	vK#, vLb	^Gb, E#
6	292.6	200	M2/d3	77/65 (-0.7c); 13/11 (+3.4c); 25/21 (-9.2c)	K#, Lb	vF#, ^E#
7	341.4	233.3	SM2/Sd3	11/9 (-6.0c); 91/75 (+6.6c)	^K#, ^Lb	F#
8	390.1	266.7	sA2/sP3/sd4	49/39 (-5.0c); 81/65 (+9.2c)	vL	vG, ^F#
9	438.9	300	A2/P3/d4	9/7 (+3.8c); 35/27 (-10.3c)	L	G
10	487.7	333.3	SA2/SP3/Sd4	65/49 (-1.5c); 33/25 (+7.0c)	^L	^G, vAb
11	536.4	366.7	sA3/sm4/sd5	15/11 (-0.5c)	vM	Ab
12	585.2	400	A3/m4/d5	7/5 (+2.7c)	M	^Ab, Fx
13	634.0	433.3	SA3/Sm4/Sd5	13/9 (-2.6c)	^M	vG#
14	682.7	466.7	sM4/sm5	135/91 (+0.07c); 49/33 (-1.6c); 81/55 (+12.6c)	vM#, vNb	G#
15	731.5	500	M4/m5	75/49 (-5.4c); 117/77 (+7.2c)	M#, Nb	vA, ^G#
16	780.3	533.3	SM4/Sm5	11/7 (-2.2c); 39/25 (+10.4c)	^M#, ^Nb	A
17	829.0	566.7	sA4/sM5	21/13 (-1.2c)	vN	^A, vBb
18	877.8	600	A4/M5	91/55 (+6.1c); 5/3 (-6.5c); 81/49 (+7.7c)	N	Bb
19	926.6	633.3	SA4/SM5	77/45 (-3.3c)	^N	^Bb, vCb, Gx
20	975.3	666.7	sA5/sm6/sd7	135/77 (+3.3c)	vO	vA#, Cb
21	1024.1	700	A5/m6/d7	165/91 (-6.1c); 9/5 (+6.5c); 49/27 (-7.7c)	O	A#, ^Cb
22	1072.9	733.3	SA5/Sm6/Sd7	13/7 (+1.2c)	^O	vB, ^A#
23	1121.6	766.7	sM6/sm7	21/11 (+2.2c); 25/13 (-10.4c)	vO#, vPb	B
24	1170.4	800	M6/m7	49/25 (+5.4c); 77/39 (-7.2c)	O#, Pb	^B, vC
25	1219.2	833.3	SM6/Sm7	91/45 (+0.07c); 99/49 (+1.6c); 55/27 (-12.6c)	^O#, ^Pb	C
26	1267.9	866.7	sA6/sM7/sd8	27/13 (+2.6c)	vP	^C, vDb
27	1316.7	900	A6/M7/d8	15/7 (-2.7c)	P	Db, vB#
28	1365.5	933.3	SA6/SM7/Sd8	11/5 (+0.5c)	^P	^Db, B#
29	1414.2	966.7	sP8/sd9	147/65 (+1.5c); 25/11 (-7.0c)	vQ	vC#, ^B#
30	1463.0	1000	P8/d9	7/3 (-3.8c); 81/35 (+10.3c)	Q	C#
31	1511.8	1033.3	SP8/Sd9	117/49 (+5.0c); 65/27 (-9.2c)	^Q	vD, ^C#
32	1560.5	1066.7	sA8/sm9	27/11 (+6.0c); 225/91 (+6.6c)	vQ#, vRb	D
33	1609.3	1100	A8/m9	195/77 (-0.7c); 33/13 (-3.4c); 63/25 (+9.2c)	Q#, Rb	^D, vEb
34	1658.1	1133.3	SA8/Sm9	13/5 (+3.9c); 55/21 (-8.7c)	^Q#, ^Rb	Eb
35	1706.9	1166.7	sM9/sd10	147/55 (+4.9c); 243/91 (+6.5c); 35/13 (-7.7c)	vR	^Eb, vFb, Cx
36	1755.7	1200	M9/d10	91/33 (+0.4c); 135/49 (+1.1c); 25/9 (-13.1c)	R	vD#, Fb
37	1804.5	1233.3	SM9/Sd10	99/35 (+4.3c); 77/27 (-9.9c)	^R	D#, ^Fb
38	1853.2	1266.7	sA9/sP10	225/77 (-3.2c); 189/65 (+5.3c)	vJ	vE, ^D#
39	1902.0	1300	A9/P10	3/1	J	E