43edt

← 42edt

43edt

44edt →

Prime factorization	43 (prime)
Step size	44.2315 ¢
Octave	27\43edt (1194.25 ¢)
Consistency limit	10
Distinct consistency limit	8

43 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 43edt or 43ed3), is a nonoctave tuning system that divides the interval of 3/1 into 43 equal parts of about 44.2 ¢ each. Each step represents a frequency ratio of 3^1/43, or the 43rd root of 3.

43EDT is related to 27 EDO, but with the 3/1 rather than the 2/1 being just. It has octaves compressed by about 5.7492 ¢ compressed and is consistent to the 10-integer-limit.

Properties

This tuning is related to 27EDO having 5.7 ¢ octave compression, a small but significant deviation. This is particularly relevant because the harmonics 27EDO approximates well—3, 5, 7, and 13—are all tuned sharp, so 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 Template:Sl 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

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	44.2	30.2
2	88.5	60.5	19/18, 20/19, 21/20
3	132.7	90.7	13/12, 14/13, 27/25
4	176.9	120.9	10/9, 21/19
5	221.2	151.2	17/15, 25/22
6	265.4	181.4	7/6
7	309.6	211.6	6/5
8	353.9	241.9	11/9, 16/13, 27/22
9	398.1	272.1	24/19, 29/23
10	442.3	302.3	22/17
11	486.5	332.6
12	530.8	362.8	15/11, 19/14
13	575	393	25/18
14	619.2	423.3	10/7
15	663.5	453.5	19/13, 22/15, 25/17
16	707.7	483.7	3/2
17	751.9	514	17/11, 20/13
18	796.2	544.2	19/12, 27/17, 30/19
19	840.4	574.4	13/8
20	884.6	604.7	5/3
21	928.9	634.9	12/7, 29/17
22	973.1	665.1	7/4
23	1017.3	695.3	9/5
24	1061.6	725.6	24/13
25	1105.8	755.8	17/9, 19/10
26	1150	786
27	1194.3	816.3	2/1
28	1238.5	846.5
29	1282.7	876.7	21/10, 23/11
30	1326.9	907	28/13
31	1371.2	937.2	11/5
32	1415.4	967.4	25/11
33	1459.6	997.7
34	1503.9	1027.9	19/8
35	1548.1	1058.1	22/9, 27/11
36	1592.3	1088.4	5/2
37	1636.6	1118.6	18/7
38	1680.8	1148.8	29/11
39	1725	1179.1	19/7, 27/10
40	1769.3	1209.3	25/9
41	1813.5	1239.5	20/7
42	1857.7	1269.8
43	1902	1300	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