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The '''division of the 10th harmonic into 40 equal parts''' (40ED10) is related to [[12edo|12EDO]], but with 10/1 instead of 2/1 being just. The step size (99.657843 [[cent]]s) of this [[equal-step tuning]] is very close to 1\12 (one step of 12 EDO).
{{Infobox ET}}
{{ED intro}}


It is possible to call this division a form of '''decibel tuning''' or '''kilobyte tuning''', as
== Theory ==
40ed10 is related to [[12edo]], but with 10/1 instead of 2/1 being just. The octave is compressed from pure by 4.106{{c}}, a small but significant deviation.


<math>10^{\frac{1}{10}} \approx 2^{\frac{1}{3}} = 1.2589254 \approx 1.2599210</math>;
=== Harmonics ===
{{Harmonics in equal|40|10|1|intervals=integer}}
{{Harmonics in equal|40|10|1intervals=integer|start=12|columns=12|collapsed=1|title=Approximation of harmonics in 40ed10 (continued)}}


which lies in the basis of the definition of decibel. In addition, as a consequence of the previous formula,
=== Subsets and supersets ===
Since 40 factors into 2<sup>3</sup> × 5, 40ed10 has subset ed10's {{EDs|equave=10| 2, 4, 5, 8, 10, and 20 }}.  
 
=== Miscellany ===
It is possible to call this division a form of '''kilobyte tuning''', as


<math>2^{10} \approx 10^{3} = 1024 \approx 1000</math>;
<math>2^{10} \approx 10^{3} = 1024 \approx 1000</math>;


which lies in the basis of using a "decimal" prefix to an otherwise binary unit of information. The octave, which is 12\40 = 3\10, is compressed by about 4.1 cents.
which lies in the obsolete practice of using a decimal prefix to an otherwise binary unit of information.


== Theory ==
== Intervals ==
Since 40ED10 has relations to the proximity of 1024 to 1000, just like 12EDO it tempers out the lesser diesis of [[128/125]]. However in this situation the tempering has a different interpretation, namely that "in favor of 1000".
{| class="wikitable center-1 right-2"
 
=== Interval ===
{| class="wikitable"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 99.6578
| 99.7
| | [[18/17]]
| [[18/17]]
| |
|-
|-
| | 2
| 2
| | 199.3157
| 199.3
| |
| [[9/8]]
| |
|-
|-
| | 3
| 3
| | 298.9735
| 299.0
| | [[19/16]]
| [[6/5]]
| |
|-
|-
| | 4
| 4
| | 398.6314
| 398.6
| |
| [[5/4]]
| |
|-
|-
| | 5
| 5
| | 498.2892
| 498.3
| | [[4/3]]
| [[4/3]]
| |
|-
|-
| | 6
| 6
| | 597.9471
| 597.9
| | [[24/17]]
| [[7/5]]
| |
|-
|-
| | 7
| 7
| | 697.6049
| 697.6
| |
| [[3/2]]
| |
|-
|-
| | 8
| 8
| | 797.2627
| 797.3
| |
| [[8/5]]
| |
|-
|-
| | 9
| 9
| | 896.9206
| 896.9
| |
| [[5/3]]
| |
|-
|-
| | 10
| 10
| | 996.5784
| 996.6
| | [[16/9]]
| [[7/4]]
| |
|-
|-
| | 11
| 11
| | 1096.2363
| 1096.2
| | [[32/17]]
| [[15/8]]
| |
|-
|-
| | 12
| 12
| | 1195.8941
| 1195.9
| |
| [[2/1]]
| | compressed [[octave]]
|-
|-
| | 13
| 13
| | 1295.5520
| 1295.6
| |
| [[17/8]]
| |
|-
|-
| | 14
| 14
| | 1395.2098
| 1395.2
| | [[28/25|56/25]]
| [[9/4]]
| |
|-
|-
| | 15
| 15
| | 1494.8676
| 1494.9
| |
| [[12/5]]
| |
|-
|-
| | 16
| 16
| | 1594.5255
| 1594.5
| |
| [[5/2]]
| |
|-
|-
| | 17
| 17
| | 1694.1833
| 1694.2
| |
| [[8/3]]
| |
|-
|-
| | 18
| 18
| | 1793.8412
| 1793.8
| |
| [[14/5]]
| |
|-
|-
| | 19
| 19
| | 1893.4990
| 1893.5
| | [[112/75|224/75]]
| [[3/1]]
| |
|-
|-
| | 20
| 20
| | 1993.1569
| 1993.2
| |
| [[16/5]]
| |
|-
|-
| | 21
| 21
| | 2092.8147
| 2092.8
| | 375/112
| [[10/3]]
| |
|-
|-
| | 22
| 22
| | 2192.4725
| 2192.5
| |
| [[7/2]]
| |
|-
|-
| | 23
| 23
| | 2292.1304
| 2292.1
| |
| [[15/4]]
| |
|-
|-
| | 24
| 24
| | 2391.7882
| 2391.8
| |
| [[4/1]]
| |
|-
|-
| | 25
| 25
| | 2491.4461
| 2491.4
| |
| [[17/4]]
| |
|-
|-
| | 26
| 26
| | 2591.1039
| 2591.1
| | 125/28
| [[9/2]]
| |
|-
|-
| | 27
| 27
| | 2690.7618
| 2690.8
| |
| 19/4
| |
|-
|-
| | 28
| 28
| | 2790.4196
| 2790.4
| |
| [[5/1]]
| |
|-
|-
| | 29
| 29
| | 2890.0774
| 2890.1
| | 85/16
| [[16/3]]
| |
|-
|-
| | 30
| 30
| | 2989.7353
| 2989.7
| | [[45/32|45/8]]
| 17/3
| |
|-
|-
| | 31
| 31
| | 3089.3931
| 3089.4
| |
| [[6/1]]
| |
|-
|-
| | 32
| 32
| | 3189.0510
| 3189.1
| |
| 19/3
| |
|-
|-
| | 33
| 33
| | 3288.7088
| 3288.7
| |
| 20/3
| |
|-
|-
| | 34
| 34
| | 3388.3667
| 3388.4
| | 85/12
| [[7/1]]
| |
|-
|-
| | 35
| 35
| | 3488.0245
| 3488.0
| | [[15/2]]
| [[15/2]]
| |
|-
|-
| | 36
| 36
| | 3587.6823
| 3587.7
| |
| [[8/1]]
| |
|-
|-
| | 37
| 37
| | 3687.3402
| 3687.3
| |
| [[17/2]]
| |
|-
|-
| | 38
| 38
| | 3786.9980
| 3787.0
| |
| [[9/1]]
| |
|-
|-
| | 39
| 39
| | 3886.6559
| 3886.7
| | 85/9
| 19/2
| |
|-
|-
| | 40
| 40
| | 3986.3137
| 3986.3
| | '''exact [[10/1]]'''
| [[10/1]]
| |
|}
|}


== 40ed10 as a generator ==
== Regular temperaments ==
40ED10 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintilischis'' temperament). This temperament is supported by {{Val list|12, 253, 265, 277, 289, 301}}, and [[313edo|313]] EDOs.
40ed10 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[subgroup temperaments|subgroup temperament]] which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintilischis'' temperament). This temperament is supported by {{Optimal ET sequence| 12-, 253-, 265-, 277-, 289-, 301-, 313-, and 325edo }}.  
 
Tempering out 400/399 (equating 20/19 and 21/20) leads ''[[Schismatic family #Quintilipyth|quintilipyth]]'' (12&amp;253), and tempering out 476/475 (equating 19/17 with 28/25) leads ''[[Schismatic family #Quintaschis|quintaschis]]'' (12&amp;289).


Tempering out 400/399 (equating 20/19 and 21/20) leads to [[quintilipyth]] (12 & 253), and tempering out 476/475 (equating 19/17 with 28/25) leads to [[quintaschis]] (12 & 289).


; Quintilischis (12&amp;289)
== See also ==
'''2.3.5.17.19 subgroup'''<br>
* [[7edf]] – relative edf
Comma list: 4624/4617, 6144/6137, 6885/6859<br>
* [[12edo]] – relative edo
Gencom: [2 18/17; 4624/4617 6144/6137 6885/6859]<br>
* [[19edt]] – relative edt
Gencom mapping: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]<br>
* [[28ed5]] – relative ed5
POTE generator: ~18/17 = 99.652<br>
* [[31ed6]] – relative ed6
Vals: 12, 253, 265, 277, 289<br>
* [[34ed7]] – relative ed7
RMS error: 0.1636 cents<br><br>
* [[42ed11]] – relative ed11
; [[Schismatic family #Quintilipyth|Quintilipyth]] (12 &amp; 253)
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
'''7-limit'''<br>
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]
Comma list: 32805/32768, 9765625/9680832<br>
Mapping: [{{val|1 2 -1 -4}}, {{val|0 -5 40 82}}]<br>
POTE generator: ~625/588 = 99.625<br>
Vals: 12, 253, 265<br>
Badness: 0.253966<br><br>
'''11-limit'''<br>
Comma list: 1375/1372, 4375/4356, 32805/32768<br>
Mapping: [{{val|1 2 -1 -4 -7}}, {{val|0 -5 40 82 126}}]<br>
POTE generator: ~35/33 = 99.616<br>
Vals: 12, 253, 265, 518c, 783cc<br>
Badness: 0.113044<br><br>
'''13-limit'''<br>
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647<br>
Mapping: [{{val|1 2 -1 -4 -7 -9}}, {{val|0 -5 40 82 126 153}}]<br>
POTE generator: ~35/33 = 99.612<br>
Vals: 12f, 253, 518c, 771cc<br>
Badness: 0.069127<br><br>
'''17-limit'''<br>
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619<br>
Mapping: [{{val|1 2 -1 -4 -7 -9 5}}, {{val|0 -5 40 82 126 153 -11}}]<br>
POTE generator: ~18/17 = 99.612<br>
Vals: 12f, 253, 518c, 771cc<br>
Badness: 0.045992<br><br>
'''19-limit'''<br>
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971<br>
Mapping: [{{val|1 2 -1 -4 -7 -9 5 4}}, {{val|0 -5 40 82 126 153 -11 3}}]<br>
POTE generator: ~18/17 = 99.615<br>
Vals: 12f, 253, 265, 518ch<br>
Badness: 0.038155<br><br>
; [[Schismatic family #Quintaschis|Quintaschis]] (12 &amp; 289)
'''7-limit'''<br>
Comma list: 32805/32768, 49009212/48828125<br>
Mapping: [{{val|1 2 -1 -5}}, {{val|0 -5 40 94}}]<br>
POTE generator: ~200/189 = 99.664<br>
Vals: 12, 277d, 289<br>
Badness: 0.132890<br><br>
'''11-limit'''<br>
Comma list: 441/440, 32805/32768, 1953125/1951488<br>
Mapping: [{{val|1 2 -1 -5 -8}}, {{val|0 -5 40 94 138}}]<br>
POTE generator: ~35/33 = 99.653<br>
Vals: 12, 277d, 289<br>
Badness: 0.111477<br><br>
'''13-limit'''<br>
Comma list: 364/363, 441/440, 32805/32768, 109512/109375<br>
Mapping: [{{val|1 2 -1 -5 -8 -11}}, {{val|0 -5 40 94 138 177}}]<br>
POTE generator: ~35/33 = 99.658<br>
Vals: 12f, 277df, 289<br>
Badness: 0.074218<br><br>
'''17-limit'''<br>
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768<br>
Mapping: [{{val|1 2 -1 -5 -8 -11 5}}, {{val|0 -5 40 94 138 177 -11}}]<br>
POTE generator: ~18/17 = 99.656<br>
Vals: 12f, 277df, 289<br>
Badness: 0.050571<br><br>
'''19-limit'''<br>
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859<br>
Mapping: [{{val|1 2 -1 -5 -8 -11 5 4}}, {{val|0 -5 40 94 138 177 -11 3}}]<br>
POTE generator: ~18/17 = 99.659<br>
Vals: 12f, 277df, 289<br>
Badness: 0.042120<br><br>
; [[Schismatic family #Quintaschis|Quintahelenic]] (12 &amp; 301)
'''11-limit'''<br>
Comma list: 5632/5625, 8019/8000, 151263/151250<br>
Mapping: [{{val|1 2 -1 -5 -9}}, {{val|0 -5 40 94 150}}]<br>
POTE generator: ~200/189 = 99.671<br>
Vals: 12, 289e, 301, 915, 1216ce<br>
Badness: 0.082225<br><br>
'''13-limit'''<br>
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000<br>
Mapping: [{{val|1 2 -1 -5 -9 -11}}, {{val|0 -5 40 94 150 177}}]<br>
POTE generator: ~200/189 = 99.661<br>
Vals: 12f, 289e, 301<br>
Badness: 0.055570<br><br>
'''17-limit'''<br>
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750<br>
Mapping: [{{val|1 2 -1 -5 -9 -11 5}}, {{val|0 -5 40 94 150 177 -11}}]<br>
POTE generator: ~18/17 = 99.665<br>
Vals: 12f, 289e, 301<br>
Badness: 0.040412<br><br>
'''19-limit'''<br>
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700<br>
Mapping: [{{val|1 2 -1 -5 -9 -11 5 4}}, {{val|0 -5 40 94 150 177 -11 3}}]<br>
POTE generator: ~18/17 = 99.668<br>
Vals: 12f, 289e, 301<br>
Badness: 0.036840<br><br>
; [[Schismatic family #Quintaschis|Quintahelenoid]] (12 &amp; 301)
'''13-limit'''<br>
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436<br>
Mapping: [{{val|1 2 -1 -5 -9 14}}, {{val|0 -5 40 94 150 -124}}]<br>
POTE generator: ~200/189 = 99.672<br>
Vals: 12, 301, 614, 915<br>
Badness: 0.066108<br><br>
'''17-limit'''<br>
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157<br>
Mapping: [{{val|1 2 -1 -5 -9 14 5}}, {{val|0 -5 40 94 150 -124 -11}}]<br>
POTE generator: ~18/17 = 99.671<br>
Vals: 12, 301, 915gg, 1216cegg<br>
Badness: 0.047908<br><br>
'''19-limit'''<br>
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137<br>
Mapping: [{{val|1 2 -1 -5 -9 14 5 4}}, {{val|0 -5 40 94 150 -124 -11 3}}]<br>
POTE generator: ~18/17 = 99.672<br>
Vals: 12, 301, 614gh, 915gghh<br>
Badness: 0.039542<br><br>


[[Category:Equal-step tuning]]
[[Category:12edo]]
[[Category:Ed10]]
[[Category:Sonifications]]
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