10ed7/3: Difference between revisions

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{{ED intro}}
{{ED intro}}


== Theory==
== Theory ==
This tuning tempers out [[16/15]] in the [[5-limit]]; [[28/27]], [[50/49]], and [[36/35]] in the [[7-limit]]; [[22/21]], [[56/55]], 77/75, and [[55/54]] in the [[11-limit]]; [[26/25]], [[52/49]], 65/63, [[66/65|66/65,]] [[78/77]], and [[40/39]] in the [[13-limit]]; [[34/33]], 51/49, [[52/51]], 75/68, 77/68, and 51/50 in the [[17-limit]]; [[20/19]], [[64/57]], [[76/75]], [[77/76]], and [[39/38]] in the [[19-limit]]; [[24/23]], [[70/69|70/69,]] and [[46/45]] in the [[23-limit]]; [[30/29]], 58/57, and [[32/29]] in the [[29-limit]]; 33/31, [[63/62]], 65/62,  and 34/31 in the [[31-limit]]; 38/37, 39/37, 75/74, 77/74, and 40/37 in the [[37-limit]]; [[42/41]] and 44/41 in the [[41-limit]]; 43/42, 43/41, and 44/43 in the [[43-limit]]; [[47/46]], [[48/47]], and 47/45 in the [[47-limit]]; 54/53, 56/53, and 55/53 in the [[53-limit]]; 59/58, 60/59, [[64/59]], and 59/57 in the [[59-limit]]; 62/61, [[65/61]], 66/61, and 63/61 in the [[61-limit]]; 69/67, 72/67, and 70/67 in the [[67-limit]]; [[71/67]], 71/69, 72/71, and [[71/70]] in the [[71-limit]]; 73/68, 75/73, 76/73, 77/73, 78/73; and 74/73 in the [[73-limit]]; and 81/79 in the [[79-limit]].
10ed7/3 is essentially a tritave stretch of [[13edt]], the equalized [[Bohlen-Pierce]] scale, and as a result tempers out [[245/243]] and [[3125/3087]] in the [[3.5.7 subgroup]], as well as [[529/525]] and [[1127/1125]] when prime 23 is introduced. It fails to represent any other primes of note within 20 cents. Making this stretch makes [[5/1]] and [[23/1]] closer to just compared to 13edt, while both [[3/1]] and [[7/1]] are about 5 [[cents]] sharp of just.  


==Intervals==
=== Harmonics ===
{{Interval table}}
 
==Harmonics==
{{Harmonics in equal
{{Harmonics in equal
| steps = 10
| steps = 10
| num = 7
| num = 7
| denom = 3
| denom = 3
| intervals = prime
}}
}}
{{Harmonics in equal
{{Harmonics in equal
Line 21: Line 17:
| start = 12
| start = 12
| collapsed = 1
| collapsed = 1
| intervals = prime
}}
}}
== Intervals ==
{{Interval table}}

Revision as of 12:51, 7 August 2025

← 9ed7/3 10ed7/3 11ed7/3 →
Prime factorization 2 × 5
Step size 146.687 ¢ 
Octave 8\10ed7/3 (1173.5 ¢) (→ 4\5ed7/3)
Twelfth 13\10ed7/3 (1906.93 ¢)
(convergent)
Consistency limit 7
Distinct consistency limit 4

10 equal divisions of 7/3 (abbreviated 10ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 10 equal parts of about 147 ¢ each. Each step represents a frequency ratio of (7/3)1/10, or the 10th root of 7/3.

Theory

10ed7/3 is essentially a tritave stretch of 13edt, the equalized Bohlen-Pierce scale, and as a result tempers out 245/243 and 3125/3087 in the 3.5.7 subgroup, as well as 529/525 and 1127/1125 when prime 23 is introduced. It fails to represent any other primes of note within 20 cents. Making this stretch makes 5/1 and 23/1 closer to just compared to 13edt, while both 3/1 and 7/1 are about 5 cents sharp of just.

Harmonics

Approximation of harmonics in 10ed7/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -26.5 +5.0 -53.0 +0.7 -21.5 +5.0 +67.2 +10.0 -25.8 -44.1 -48.0
Relative (%) -18.1 +3.4 -36.1 +0.5 -14.7 +3.4 +45.8 +6.8 -17.6 -30.0 -32.7
Steps
(reduced) 		8
(8) 		13
(3) 		16
(6) 		19
(9) 		21
(1) 		23
(3) 		25
(5) 		26
(6) 		27
(7) 		28
(8) 		29
(9)
Approximation of harmonics in 10ed7/3
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -39.9 -21.5 +5.7 +40.7 -64.3 -16.5 +36.5 -52.3 +10.0 -70.6 -0.9
Relative (%) -27.2 -14.7 +3.9 +27.7 -43.8 -11.3 +24.9 -35.6 +6.8 -48.1 -0.6
Steps
(reduced) 		30
(0) 		31
(1) 		32
(2) 		33
(3) 		33
(3) 		34
(4) 		35
(5) 		35
(5) 		36
(6) 		36
(6) 		37
(7)

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 146.7 11/10, 12/11, 13/12, 14/13, 15/14, 21/19
2 293.4 6/5, 7/6, 13/11, 20/17
3 440.1 9/7, 13/10, 14/11, 17/13, 19/15, 22/17
4 586.7 7/5, 10/7, 17/12, 18/13
5 733.4 3/2, 14/9, 17/11, 20/13
6 880.1 5/3, 18/11, 22/13
7 1026.8 9/5, 11/6, 20/11
8 1173.5 2/1
9 1320.2 13/6, 15/7, 17/8, 19/9
10 1466.9 7/3
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