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The '''division of the 10th harmonic into 40 equal parts''' is related to [[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.
 
== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 99.7
| [[18/17]]
|-
| 2
| 199.3
| [[9/8]]
|-
| 3
| 299.0
| [[6/5]]
|-
| 4
| 398.6
| [[5/4]]
|-
| 5
| 498.3
| [[4/3]]
|-
| 6
| 597.9
| [[7/5]]
|-
| 7
| 697.6
| [[3/2]]
|-
| 8
| 797.3
| [[8/5]]
|-
| 9
| 896.9
| [[5/3]]
|-
| 10
| 996.6
| [[7/4]]
|-
| 11
| 1096.2
| [[15/8]]
|-
| 12
| 1195.9
| [[2/1]]
|-
| 13
| 1295.6
| [[17/8]]
|-
| 14
| 1395.2
| [[9/4]]
|-
| 15
| 1494.9
| [[12/5]]
|-
| 16
| 1594.5
| [[5/2]]
|-
| 17
| 1694.2
| [[8/3]]
|-
| 18
| 1793.8
| [[14/5]]
|-
| 19
| 1893.5
| [[3/1]]
|-
| 20
| 1993.2
| [[16/5]]
|-
| 21
| 2092.8
| [[10/3]]
|-
| 22
| 2192.5
| [[7/2]]
|-
| 23
| 2292.1
| [[15/4]]
|-
| 24
| 2391.8
| [[4/1]]
|-
| 25
| 2491.4
| [[17/4]]
|-
| 26
| 2591.1
| [[9/2]]
|-
| 27
| 2690.8
| 19/4
|-
| 28
| 2790.4
| [[5/1]]
|-
| 29
| 2890.1
| [[16/3]]
|-
| 30
| 2989.7
| 17/3
|-
| 31
| 3089.4
| [[6/1]]
|-
| 32
| 3189.1
| 19/3
|-
| 33
| 3288.7
| 20/3
|-
| 34
| 3388.4
| [[7/1]]
|-
| 35
| 3488.0
| [[15/2]]
|-
| 36
| 3587.7
| [[8/1]]
|-
| 37
| 3687.3
| [[17/2]]
|-
| 38
| 3787.0
| [[9/1]]
|-
| 39
| 3886.7
| 19/2
|-
| 40
| 3986.3
| [[10/1]]
|}
 
== 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 {{Optimal ET sequence| 12-, 253-, 265-, 277-, 289-, 301-, 313-, and 325edo }}.  


== Theory ==
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).
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".
 
== See also ==
* [[7edf]] – relative edf
* [[12edo]] – relative edo
* [[19edt]] – relative edt
* [[28ed5]] – relative ed5
* [[31ed6]] – relative ed6
* [[34ed7]] – relative ed7
* [[42ed11]] – relative ed11
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]


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