Xenharmonic Wiki

43edt: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
14,512 edits
No edit summary
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(9 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{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==
== Theory ==
{{Harmonics in equal|43|3|1|prec=2|columns=15}}
43edt is related to [[27edo]], but with the 3/1 rather than the 2/1 being just. Like 27edo, it is consistent to the [[9-odd-limit|10-integer-limit]]. It has octaves compressed by about 5.7492{{c}}, 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.
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.
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 {{mos scalesig|4L 5s<3/1>|link=1}} [[mos]] has {{nowrap|L {{=}} 7|s {{=}} 3}}.


{| class="wikitable"
=== Harmonics ===
{{Harmonics in equal|43|3|1}}
{{Harmonics in equal|43|3|1|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 43edt (continued)}}
 
=== Subsets and supersets ===
43edt is the 14th [[prime equal division|prime edt]], following [[41edt]] and coming before [[47edt]].
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
|-
! | degrees
! #
! | cents value
! Cents
!hekts
! [[Hekt]]s
! | corresponding <br>JI intervals
! Approximate ratios
|-
|-
| | 1
| 1
| | 44.232
| 44.2
|30.233
| 30.2
| | 40/39, 39/38
| 39/38, 40/39
|-
|-
| | 2
| 2
| | 88.463
| 88.5
|60.465
| 60.5
| | [[20/19]]
| [[20/19]]
|-
|-
| | 3
| 3
| | 132.695
| 132.7
|90.698
| 90.7
| | [[27/25]]
| [[27/25]]
|-
|-
| | 4
| 4
| | 176.926
| 176.9
|120.93
| 120.9
| |[[10/9]]
| [[10/9]]
|-
|-
| | 5
| 5
| | 221.158
| 221.2
|151.163
| 151.2
| | [[25/22]]
| [[25/22]]
|-
|-
| | 6
| 6
| | 265.389
| 265.4
|181.395
| 181.4
| | ([[7/6]])
| [[7/6]]
|-
|-
| | 7
| 7
| | 309.621
| 309.6
|211.628
| 211.6
| | [[6/5]]
| [[6/5]]
|-
|-
| | 8
| 8
| | 353.852
| 353.9
|241.8605
| 241.9
| | [[27/22]]
| [[27/22]]
|-
|-
| | 9
| 9
| | 398.084
| 398.1
|272.093
| 272.1
| |24/19
| [[24/19]]
|-
|-
| | 10
| 10
| | 442.315
| 442.3
|302.326
| 302.3
| |9/7
| [[9/7]]
|-
|-
| | 11
| 11
| | 486.547
| 486.5
|332.558
| 332.6
| | (45/34)
| [[45/34]]
|-
|-
| | 12
| 12
| | 530.778
| 530.8
|362.791
| 362.8
| | (34/25)
| [[34/25]]
|-
|-
| | 13
| 13
| | 575.01
| 575.0
|393.023
| 393.0
| | (39/28)
| [[39/28]]
|-
|-
| | 14
| 14
| | 619.241
| 619.2
|423.256
| 423.3
| | [[10/7]]
| [[10/7]]
|-
|-
| | 15
| 15
| | 663.473
| 663.5
|453.488
| 453.5
| | [[22/15]]
| [[22/15]]
|-
|-
| | 16
| 16
| | 707.704
| 707.7
|483.721
| 483.7
| |[[3/2]]
| [[3/2]]
|-
|-
| | 17
| 17
| | 751.936
| 751.9
|513.9535
| 514.0
| |105/68, [[20/13]]
| [[20/13]], 105/68
|-
|-
| | 18
| 18
| | 796.167
| 796.2
|544.186
| 544.2
| | [[19/12]]
| [[19/12]]
|-
|-
| | 19
| 19
| | 840.399
| 840.4
|574.419
| 574.4
| | [[13/8]]
| [[13/8]]
|-
|-
| | 20
| 20
| | 884.63
| 884.6
|604.651
| 604.7
| | [[5/3]]
| [[5/3]]
|-
|-
| | 21
| 21
| | 928.862
| 928.9
|634.883
| 634.9
| |[[12/7]]
| [[12/7]]
|-
|-
| | 22
| 22
| | 973.093
| 973.1
|665.116
| 665.1
| | 7/4
| [[7/4]]
|-
|-
| | 23
| 23
| | 1017.325
| 1017.3
|695.349
| 695.3
| | [[9/5]]
| [[9/5]]
|-
|-
| | 24
| 24
| | 1061.556
| 1061.6
|725.581
| 725.6
| | [[24/13]]
| [[24/13]]
|-
|-
| | 25
| 25
| | 1105.788
| 1105.8
|755.814
| 755.8
| | [[36/19]]
| [[36/19]]
|-
|-
| | 26
| 26
| | 1150.019
| 1150.0
|786.0465
| 786.0
| | 68/35, 39/20
| [[39/20]], [[68/35]]
|-
|-
| | 27
| 27
| | 1194.251
| 1194.3
|816.279
| 816.3
| |[[2/1]]
| [[2/1]]
|-
|-
| | 28
| 28
| | 1238.482
| 1238.5
|846.511
| 846.5
| | [[45/44|45/22]]
| [[45/22]]
|-
|-
| | 29
| 29
| | 1282.713
| 1282.7
|876.744
| 876.7
| | ([[21/20|21/10]])
| [[21/10]]
|-
|-
| | 30
| 30
| | 1326.946
| 1326.9
|906.977
| 907.0
| | ([[14/13|28/13]])
| [[28/13]]
|-
|-
| | 31
| 31
| | 1371.177
| 1371.2
|937.209
| 937.2
| |(75/34)
| 75/34
|-
|-
| | 32
| 32
| | 1415.408
| 1415.4
|967.442
| 967.4
| | ([[17/15|34/15]])
| [[34/15]]
|-
|-
| | 33
| 33
| | 1459.640
| 1459.6
|997.674
| 997.7
| | 7/3
| [[7/3]]
|-
|-
| | 34
| 34
| | 1503.871
| 1503.9
|1027.907
| 1027.9
| | 19/8
| [[19/8]]
|-
|-
| | 35
| 35
| | 1548.193
| 1548.1
|1058.1395
| 1058.1
| | [[11/9|22/9]]
| [[22/9]]
|-
|-
| | 36
| 36
| | 1592.334
| 1592.3
|1088.372
| 1088.3
| | 5/2
| [[5/2]]
|-
|-
| | 37
| 37
| | 1636.566
| 1636.6
|1118.605
| 1118.6
| | ([[9/7|18/7]])
| [[18/7]]
|-
|-
| | 38
| 38
| | 1680.797
| 1680.8
|1148.837
| 1148.8
| | [[33/25|66/25]]
| [[66/25]]
|-
|-
| | 39
| 39
| | 1725.029
| 1725.0
|1179.069
| 1179.1
| | 27/10
| [[27/10]]
|-
|-
| | 40
| 40
| | 1769.261
| 1769.3
|1209.302
| 1209.3
| | [[25/18|25/9]]
| [[25/9]]
|-
|-
| | 41
| 41
| | 1813.492
| 1813.5
|1239.5345
| 1239.5
| | 57/20
| 57/20
|-
|-
| | 42
| 42
| | 1857.724
| 1857.7
|1269.767
| 1269.8
| | 117/40, [[19/13|38/13]]
| [[38/13]], 117/40
|-
|-
| | 43
| 43
| | 1901.955
| 1902.0
|1300
| 1300.0
| | '''exact [[3/1]]'''
| [[3/1]]
|}
|}


=43EDT as a regular temperament=
== Related regular temperaments ==
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 the no-twos comma of {{monzo| 0 63 -43 }}, leading to the regular temperament [[support]]ed by [[27edo|27-]], [[190edo|190-]], and [[217edo]].
 
=== 27 &amp; 190 temperament ===
==== 5-limit ====
Subgroup: 2.3.5
 
Comma list: {{monzo| 0 63 -43 }}


==27&amp;190 temperament==
Mapping: {{mapping| 1 0 0 | 0 43 63 }}
===5-limit===
Comma: |0 63 -43&gt;


POTE generator: ~|0 -41 28&gt; = 44.2294
Optimal tuning (POTE): ~{{monzo| 0 -41 28 }} = 44.2294


Mapping: [&lt;1 0 0|, &lt;0 43 63|]
{{Optimal ET sequence|legend=0| 27, 190, 217, 407, 597, 624, 841 }}


EDOs: {{EDOs|27, 190, 217, 407, 597, 624, 841}}
==== 7-limit ====
Subgroup: 2.3.5.7


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


POTE generator: ~1029/1000 = 44.2288
Mapping: {{mapping| 1 0 0 1 | 0 43 63 49 }}


Mapping: [&lt;1 0 0 1|, &lt;0 43 63 49|]
Optimal tuning (POTE): ~1029/1000 = 44.2288


EDOs: {{EDOs|27, 190, 217}}
{{Optimal ET sequence|legend=0| 27, 190, 217 }}


Badness: 0.1659
Badness: 0.1659


==217&amp;407 temperament==
=== 217 &amp; 407 temperament ===
===7-limit===
==== 7-limit ====
Commas: 134217728/133984375, 512557306947/512000000000
Subgroup: 2.3.5.7
 
Comma list: 134217728/133984375, 512557306947/512000000000


POTE generator: ~525/512 = 44.2320
Mapping: {{mapping| 1 0 0 9 | 0 43 63 -168 }}


Mapping: [&lt;1 0 0 9|, &lt;0 43 63 -168|]
Optimal tuning (POTE): ~525/512 = 44.2320


EDOs: {{EDOs|217, 407, 624, 841, 1058, 1465}}
{{Optimal ET sequence|legend=0| 217, 407, 624, 841, 1058, 1465 }}


Badness: 0.3544
Badness: 0.3544


===11-limit===
==== 11-limit ====
Commas: 46656/46585, 131072/130977, 234375/234256
Subgroup: 2.3.5.7.11
 
Comma list: 46656/46585, 131072/130977, 234375/234256


POTE generator: ~525/512 = 44.2312
Mapping: {{mapping| 1 0 0 9 -1 | 0 43 63 -168 121 }}


Mapping: [&lt;1 0 0 9 -1|, &lt;0 43 63 -168 121|]
Optimal tuning (POTE): ~525/512 = 44.2312


EDOs: {{EDOs|217, 407, 624}}
{{Optimal ET sequence|legend=0| 217, 407, 624 }}


Badness: 0.1129
Badness: 0.1129


===13-limit===
==== 13-limit ====
Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375


POTE generator: ~40/39 = 44.2312
Mapping: {{mapping| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}


Mapping: [&lt;1 0 0 9 -1 3|, &lt;0 43 63 -168 121 19|]
Optimal tuning (POTE): ~40/39 = 44.2312


EDOs: {{EDOs|217, 407, 624}}
{{Optimal ET sequence|legend=0| 217, 407, 624 }}


Badness: 0.0503
Badness: 0.0503


[[Category:Edt]]
== See also ==
[[Category:Edonoi]]
* [[16edf]] – relative edf
* [[27edo]] – relative edo
* [[70ed6]] – relative ed6
* [[90ed10]] – relative ed10
* [[97ed12]] – relative ed12
 
[[Category:27edo]]
Retrieved from "https://en.xen.wiki/w/43edt"