10edt: Difference between revisions

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<span style="font-size: 18px; line-height: 27px;">'''10 Equal Divisions of the Tritave'''</span>
{{Infobox ET}}
{{ED intro}}


== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord.
10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.
=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|intervals=prime}}
=== Interval table ===
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Degrees
! Degrees
| | Cents
! [[Cent]]s
| | Approximate Ratios
! [[Hekt]]s
! Approximate Ratios
|-
|-
| | 0
| colspan="3" | 0
| | 0
| <span style="color: #660000;">[[1/1]]</span>
| | <span style="color: #660000;">[[1/1|1/1]]</span>
|-
|-
| | 1
| 1
| | 190.196
| 190.196
| | [[10/9|10/9]], [[28/25|28/25]]
| 130
| [[10/9]], [[28/25]]
|-
|-
| | 2
| 2
| | 380.391
| 380.391
| | <span style="color: #660000;">[[5/4|5/4]]</span>
| 260
| <span style="color: #660000;">[[5/4]]</span>
|-
|-
| | 3
| 3
| | 570.587
| 570.587
| | [[7/5|7/5]]
| 390
| [[7/5]]
|-
|-
| | 4
| 4
| | 760.782
| 760.782
| | <span style="color: #660000;">[[14/9|14/9]]</span>
| 520
| <span style="color: #660000;">[[14/9]]</span>
|-
|-
| | 5
| 5
| | 950.978
| 950.978
| | [[19/11|19/11]]?
| 650
| 45/26, [[26/15]]
|-
|-
| | 6
| 6
| | 1141.173
| 1141.173
| | <span style="color: #660000;">[[27/14|27/14]]</span>
| 780
| <span style="color: #660000;">[[27/14]]</span>
|-
|-
| | 7
| 7
| | 1331.369
| 1331.369
| | [[15/7|15/7]] ([[15/14|15/14]] plus an octave)
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
|-
| | 8
| 8
| | 1521.564
| 1521.564
| | [[12/5]] (<span style="color: #660000;">[[6/5|6/5]]</span> plus an octave)
| 1040
| [[12/5]] (<span style="color: #660000;">[[6/5]]</span> plus an octave)
|-
|-
| | 9
| 9
| | 1711.760
| 1711.760
| | [[27/10|27/10]]
| 1170
| [[27/20|27/10]]
|-
|-
| | 10
| 10
| | 1901.955
| 1901.955
| | [[3/1|3/1]]
| 1300
| [[3/1]]
|}
|}


10edt, like [[5edt|5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:Macrotonal]]
[[Category:edt]]
[[category:macrotonal]]

Latest revision as of 15:31, 31 July 2025

← 9edt 10edt 11edt →
Prime factorization 2 × 5
Step size 190.196 ¢ 
Octave 6\10edt (1141.17 ¢) (→ 3\5edt)
Consistency limit 3
Distinct consistency limit 3

10 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 10edt or 10ed3), is a nonoctave tuning system that divides the interval of 3/1 into 10 equal parts of about 190 ¢ each. Each step represents a frequency ratio of 31/10, or the 10th root of 3.

Theory

10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the 5/4 inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as 7/5. We can use this to readily construct chords such as 4:5:7:12, although the 7/4, being 18 cents flat, does considerable damage to the consonance of this chord.

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.

One step of 10edt can serve as the generator for pocus temperament, a merge of sensamagic and 2.3.5.7.13 catakleismic, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

Harmonics

Approximation of harmonics in 10edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -58.8 +0.0 +72.5 +66.6 -58.8 +54.7 +13.7 +0.0 +7.8 +33.0 +72.5
Relative (%) -30.9 +0.0 +38.1 +35.0 -30.9 +28.8 +7.2 +0.0 +4.1 +17.3 +38.1
Steps
(reduced) 		6
(6) 		10
(0) 		13
(3) 		15
(5) 		16
(6) 		18
(8) 		19
(9) 		20
(0) 		21
(1) 		22
(2) 		23
(3)
Approximation of prime harmonics in 10edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -58.8 +0.0 +66.6 +54.7 +33.0 -66.0 +40.1 +37.8 +87.4 +66.5 -49.0
Relative (%) -30.9 +0.0 +35.0 +28.8 +17.3 -34.7 +21.1 +19.9 +46.0 +35.0 -25.7
Steps
(reduced) 		6
(6) 		10
(0) 		15
(5) 		18
(8) 		22
(2) 		23
(3) 		26
(6) 		27
(7) 		29
(9) 		31
(1) 		31
(1)

Interval table

Degrees Cents Hekts Approximate Ratios
0 1/1
1 190.196 130 10/9, 28/25
2 380.391 260 5/4
3 570.587 390 7/5
4 760.782 520 14/9
5 950.978 650 45/26, 26/15
6 1141.173 780 27/14
7 1331.369 910 15/7 (15/14 plus an octave)
8 1521.564 1040 12/5 (6/5 plus an octave)
9 1711.760 1170 27/10
10 1901.955 1300 3/1
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