40ed10: Difference between revisions

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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]]

Latest revision as of 13:27, 10 June 2025

← 39ed10 40ed10 41ed10 →
Prime factorization 23 × 5
Step size 99.6578 ¢ 
Octave 12\40ed10 (1195.89 ¢) (→ 3\10ed10)
Twelfth 19\40ed10 (1893.5 ¢)
Consistency limit 10
Distinct consistency limit 6

40 equal divisions of the 10th harmonic (abbreviated 40ed10) is a nonoctave tuning system that divides the interval of 10/1 into 40 equal parts of about 99.7 ¢ each. Each step represents a frequency ratio of 101/40, or the 40th root of 10.

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 ¢, a small but significant deviation.

Harmonics

Approximation of harmonics in 40ed10
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.1 -8.5 -8.2 +4.1 -12.6 +19.5 -12.3 -16.9 +0.0 +34.3 -16.7
Relative (%) -4.1 -8.5 -8.2 +4.1 -12.6 +19.6 -12.4 -17.0 +0.0 +34.4 -16.7
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(31) 		34
(34) 		36
(36) 		38
(38) 		40
(0) 		42
(2) 		43
(3)
Approximation of harmonics in 40ed10 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.1 +15.4 -4.4 -16.4 -21.7 -21.0 -15.0 -4.1 +11.1 +30.2 -46.8 -20.8
Relative (%) +44.2 +15.5 -4.4 -16.5 -21.8 -21.1 -15.0 -4.1 +11.1 +30.3 -46.9 -20.8
Steps
(reduced) 		45
(5) 		46
(6) 		47
(7) 		48
(8) 		49
(9) 		50
(10) 		51
(11) 		52
(12) 		53
(13) 		54
(14) 		54
(14) 		55
(15)

Subsets and supersets

Since 40 factors into 23 × 5, 40ed10 has subset ed10's 2, 4, 5, 8, 10, and 20.

Miscellany

It is possible to call this division a form of kilobyte tuning, as

[math]\displaystyle{ 2^{10} \approx 10^{3} = 1024 \approx 1000 }[/math];

which lies in the obsolete practice of using a decimal prefix to an otherwise binary unit of information.

Intervals

# 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 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 12-, 253-, 265-, 277-, 289-, 301-, 313-, and 325edo.

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).

See also

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