Xenharmonic Wiki

43edt: Difference between revisions

← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,312 edits
m →Properties: Split wide harmonics table into 2 short tables
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
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==
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}}
Retrieved from "https://en.xen.wiki/w/43edt"