43edt

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43edt

44edt →

Prime factorization	43 (prime)
Step size	44.2315¢
Octave	27\43edt (1194.25¢)
Consistency limit	10
Distinct consistency limit	7

Division of the third harmonic into 43 equal parts (43EDT) is related to 27 EDO, but with the 3/1 rather than the 2/1 being just. The octave is about 5.7492 cents compressed and the step size is about 44.2315 cents. It is consistent to the 10-integer-limit.

Properties

This tuning is related to 27EDO having ~5.7 cent octave compression, a small but significant deviation. This is particularly relevant because 27EDO tunes the 3rd, 5th, 7th and 13th harmonics sharp, thus 43EDT improves those approximations.

However, in addition to its rich octave-based harmony, the 43EDT is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, Bohlen-Pierce harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that 13EDT is not. The 4L+5s MOS has L=7 s=3.

Harmonics

Approximation of prime harmonics in 43edt

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-5.7	+0.0	+0.3	-7.2	+6.4	-17.4	+4.7	-10.9	+12.2	+9.0	-18.0
	Relative (%)	-13.0	+0.0	+0.6	-16.3	+14.6	-39.3	+10.7	-24.6	+27.6	+20.3	-40.7
Steps (reduced)		27 (27)	43 (0)	63 (20)	76 (33)	94 (8)	100 (14)	111 (25)	115 (29)	123 (37)	132 (3)	134 (5)

Approximation of prime harmonics in 43edt

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-14.7	-15.5	-9.5	+13.5	-17.6	+17.9	+4.4	+18.9	+7.0	+3.1	-0.9
	Relative (%)	-33.2	-35.0	-21.4	+30.4	-39.8	+40.4	+9.9	+42.7	+15.7	+7.0	-2.1
Steps (reduced)		141 (12)	145 (16)	147 (18)	151 (22)	155 (26)	160 (31)	161 (32)	165 (36)	167 (38)	168 (39)	171 (42)

Intervals

degrees	cents value	hekts	corresponding JI intervals
1	44.232	30.233	40/39, 39/38
2	88.463	60.465	20/19
3	132.695	90.698	27/25
4	176.926	120.93	10/9
5	221.158	151.163	25/22
6	265.389	181.395	(7/6)
7	309.621	211.628	6/5
8	353.852	241.8605	27/22
9	398.084	272.093	24/19
10	442.315	302.326	9/7
11	486.547	332.558	(45/34)
12	530.778	362.791	(34/25)
13	575.01	393.023	(39/28)
14	619.241	423.256	10/7
15	663.473	453.488	22/15
16	707.704	483.721	3/2
17	751.936	513.9535	105/68, 20/13
18	796.167	544.186	19/12
19	840.399	574.419	13/8
20	884.63	604.651	5/3
21	928.862	634.883	12/7
22	973.093	665.116	7/4
23	1017.325	695.349	9/5
24	1061.556	725.581	24/13
25	1105.788	755.814	36/19
26	1150.019	786.0465	68/35, 39/20
27	1194.251	816.279	2/1
28	1238.482	846.511	45/22
29	1282.713	876.744	(21/10)
30	1326.946	906.977	(28/13)
31	1371.177	937.209	(75/34)
32	1415.408	967.442	(34/15)
33	1459.640	997.674	7/3
34	1503.871	1027.907	19/8
35	1548.193	1058.1395	22/9
36	1592.334	1088.372	5/2
37	1636.566	1118.605	(18/7)
38	1680.797	1148.837	66/25
39	1725.029	1179.069	27/10
40	1769.261	1209.302	25/9
41	1813.492	1239.5345	57/20
42	1857.724	1269.767	117/40, 38/13
43	1901.955	1300	exact 3/1

43EDT as a regular temperament

43EDT tempers out a no-twos comma of |0 63 -43>, leading the regular temperament supported by 27, 190, and 217 EDOs.

27&190 temperament

5-limit

Comma: |0 63 -43>

POTE generator: ~|0 -41 28> = 44.2294

Mapping: [<1 0 0|, <0 43 63|]

EDOs: 27, 190, 217, 407, 597, 624, 841

7-limit

Commas: 4375/4374, 40353607/40000000

POTE generator: ~1029/1000 = 44.2288

Mapping: [<1 0 0 1|, <0 43 63 49|]

EDOs: 27, 190, 217

Badness: 0.1659

217&407 temperament

7-limit

Commas: 134217728/133984375, 512557306947/512000000000

POTE generator: ~525/512 = 44.2320

Mapping: [<1 0 0 9|, <0 43 63 -168|]

EDOs: 217, 407, 624, 841, 1058, 1465

Badness: 0.3544

11-limit

Commas: 46656/46585, 131072/130977, 234375/234256

POTE generator: ~525/512 = 44.2312

Mapping: [<1 0 0 9 -1|, <0 43 63 -168 121|]

EDOs: 217, 407, 624

Badness: 0.1129

13-limit

Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375

POTE generator: ~40/39 = 44.2312

Mapping: [<1 0 0 9 -1 3|, <0 43 63 -168 121 19|]

EDOs: 217, 407, 624

Badness: 0.0503