43edt: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''[[Edt|Division of the third harmonic]] into 43 equal parts''' (43EDT) is related to [[27edo|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 [[9-odd-limit|10-integer-limit]].
{{ED intro}}


==Properties==
43EDT is related to [[27edo|27 EDO]], but with the 3/1 rather than the 2/1 being just. It has octaves compressed by about 5.7492{{c}} compressed and is consistent to the [[9-odd-limit|10-integer-limit]].
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|13EDT]] is not. The 4L+5s MOS has L=7 s=3.
== Properties ==
This tuning is related to 27EDO having 5.7{{c}} 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|13EDT]] is not. The {{sl|4L 5s}} MOS has {{nowrap|L {{=}} 7|s {{=}} 3}}.


== Harmonics ==
== Harmonics ==
Line 24: Line 26:


== Intervals ==
== Intervals ==
{| class="wikitable"
{{Interval table}}
|-
! | degrees
! | cents value
!hekts
! | corresponding <br>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
| | [[45/44|45/22]]
|-
| | 29
| | 1282.713
|876.744
| | ([[21/20|21/10]])
|-
| | 30
| | 1326.946
|906.977
| | ([[14/13|28/13]])
|-
| | 31
| | 1371.177
|937.209
| |(75/34)
|-
| | 32
| | 1415.408
|967.442
| | ([[17/15|34/15]])
|-
| | 33
| | 1459.640
|997.674
| | 7/3
|-
| | 34
| | 1503.871
|1027.907
| | 19/8
|-
| | 35
| | 1548.193
|1058.1395
| | [[11/9|22/9]]
|-
| | 36
| | 1592.334
|1088.372
| | 5/2
|-
| | 37
| | 1636.566
|1118.605
| | ([[9/7|18/7]])
|-
| | 38
| | 1680.797
|1148.837
| | [[33/25|66/25]]
|-
| | 39
| | 1725.029
|1179.069
| | 27/10
|-
| | 40
| | 1769.261
|1209.302
| | [[25/18|25/9]]
|-
| | 41
| | 1813.492
|1239.5345
| | 57/20
|-
| | 42
| | 1857.724
|1269.767
| | 117/40, [[19/13|38/13]]
|-
| | 43
| | 1901.955
|1300
| | '''exact [[3/1]]'''
|}


=43EDT as a regular temperament=
= 43EDT as a regular temperament =
43EDT tempers out a no-twos comma of |0 63 -43&gt;, leading the regular temperament supported by [[27edo|27]], [[190edo|190]], and [[217edo|217]] EDOs.
43EDT tempers out a no-twos comma of {{vector|0 63 -43}}, leading the regular temperament supported by [[27edo|27]], [[190edo|190]], and [[217edo|217]] EDOs.


==27&amp;190 temperament==
== {{nowrap|27 &amp; 190}} temperament ==
===5-limit===
=== 5-limit ===
Comma: |0 63 -43&gt;
Comma: {{vector|0 63 -43}}


POTE generator: ~|0 -41 28&gt; = 44.2294
POTE generator: ~{{vector|0 -41 28}} = 44.2294


Mapping: [&lt;1 0 0|, &lt;0 43 63|]
Mapping: [{{map|1 0 0}}, {{map|0 43 63}}]


EDOs: {{EDOs|27, 190, 217, 407, 597, 624, 841}}
EDOs: {{EDOs|27, 190, 217, 407, 597, 624, 841}}


===7-limit===
=== 7-limit ===
Commas: 4375/4374, 40353607/40000000
Commas: 4375/4374, 40353607/40000000


POTE generator: ~1029/1000 = 44.2288
POTE generator: ~1029/1000 = 44.2288


Mapping: [&lt;1 0 0 1|, &lt;0 43 63 49|]
Mapping: [{{map|1 0 0 1}}, {{map|0 43 63 49}}]


EDOs: {{EDOs|27, 190, 217}}
EDOs: {{EDOs|27, 190, 217}}
Line 271: Line 52:
Badness: 0.1659
Badness: 0.1659


==217&amp;407 temperament==
== {{nowrap|217 &amp; 407}} temperament ==
===7-limit===
=== 7-limit ===
Commas: 134217728/133984375, 512557306947/512000000000
Commas: 134217728/133984375, 512557306947/512000000000


POTE generator: ~525/512 = 44.2320
POTE generator: ~525/512 = 44.2320


Mapping: [&lt;1 0 0 9|, &lt;0 43 63 -168|]
Mapping: [{{map|1 0 0 9}}, {{map|0 43 63 -168}}]


EDOs: {{EDOs|217, 407, 624, 841, 1058, 1465}}
EDOs: {{EDOs|217, 407, 624, 841, 1058, 1465}}
Line 283: Line 64:
Badness: 0.3544
Badness: 0.3544


===11-limit===
=== 11-limit ===
Commas: 46656/46585, 131072/130977, 234375/234256
Commas: 46656/46585, 131072/130977, 234375/234256


POTE generator: ~525/512 = 44.2312
POTE generator: ~525/512 = 44.2312


Mapping: [&lt;1 0 0 9 -1|, &lt;0 43 63 -168 121|]
Mapping: [{{map|1 0 0 9 -1}}, {{map|0 43 63 -168 121}}]


EDOs: {{EDOs|217, 407, 624}}
EDOs: {{EDOs|217, 407, 624}}
Line 294: Line 75:
Badness: 0.1129
Badness: 0.1129


===13-limit===
=== 13-limit ===
Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375
Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375


POTE generator: ~40/39 = 44.2312
POTE generator: ~40/39 = 44.2312


Mapping: [&lt;1 0 0 9 -1 3|, &lt;0 43 63 -168 121 19|]
Mapping: [{{map|1 0 0 9 -1 3}}, {{map|0 43 63 -168 121 19}}]


EDOs: {{EDOs|217, 407, 624}}
EDOs: {{EDOs|217, 407, 624}}

Revision as of 14:56, 25 February 2025

← 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 31/43, 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

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