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User:BudjarnLambeth/Sandbox2

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Revision as of 21:55, 21 August 2025 by BudjarnLambeth (talk | contribs) (→Octave stretch or compression)

7edo

Octave stretch or compression

What follows is a comparison of stretched- and compressed-octave 7edo tunings.

7edo
  • Step size: 171.429 ¢, octave size: 1200.0 ¢

Pure-octaves 7edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 7edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -16.2 +0.0 -43.5 -16.2 +59.7 +0.0 -32.5 -43.5 -37.0 -16.2
Relative (%) +0.0 -9.5 +0.0 -25.3 -9.5 +34.9 +0.0 -18.9 -25.3 -21.6 -9.5
Steps
(reduced) 		7
(0) 		11
(4) 		14
(0) 		16
(2) 		18
(4) 		20
(6) 		21
(0) 		22
(1) 		23
(2) 		24
(3) 		25
(4)
Approximation of harmonics in 7edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.6 +59.7 -59.7 +0.0 +66.5 -32.5 +45.3 -43.5 +43.5 -37.0 +57.4 -16.2
Relative (%) +9.7 +34.9 -34.8 +0.0 +38.8 -18.9 +26.5 -25.3 +25.4 -21.6 +33.5 -9.5
Steps
(reduced) 		26
(5) 		27
(6) 		27
(6) 		28
(0) 		29
(1) 		29
(1) 		30
(2) 		30
(2) 		31
(3) 		31
(3) 		32
(4) 		32
(4)
7et, 2.3.11.13 WE
  • Step size: 171.993 ¢, octave size: 1204.0 ¢

Stretching the octave of 7edo by around 4 ¢ results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 75.0 ¢. The 2.3.11.13 WE tuning and 2.3.11.13 TE tuning both do this.

Approximation of harmonics in 7et, 2.3.11.13 WE
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.0 -10.0 +7.9 -34.4 -6.1 +71.0 +11.9 -20.1 -30.5 -23.5 -2.1
Relative (%) +2.3 -5.8 +4.6 -20.0 -3.5 +41.3 +6.9 -11.7 -17.7 -13.7 -1.2
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 7et, 2.3.11.13 WE (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +31.3 +75.0 -44.5 +15.8 +82.8 -16.1 +62.3 -26.5 +61.0 -19.5 +75.5 +1.8
Relative (%) +18.2 +43.6 -25.8 +9.2 +48.2 -9.4 +36.2 -15.4 +35.5 -11.4 +43.9 +1.1
Step 26 27 27 28 29 29 30 30 31 31 32 32
18ed6
  • Step size: 172.331 ¢, octave size: 1206.3 ¢

Stretching the octave of 7edo by around 6 ¢ results in much improved primes 3, 5 and 7, but much worse primes 11 and 14. This approximates all harmonics up to 16 within 48.7 ¢. The tuning 18ed6 does this.

Approximation of harmonics in 18ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.3 -6.3 +12.6 -29.0 +0.0 +77.8 +18.9 -12.6 -22.7 -15.4 +6.3
Relative (%) +3.7 -3.7 +7.3 -16.8 +0.0 +45.1 +11.0 -7.3 -13.2 -8.9 +3.7
Steps
(reduced) 		7
(7) 		11
(11) 		14
(14) 		16
(16) 		18
(0) 		20
(2) 		21
(3) 		22
(4) 		23
(5) 		24
(6) 		25
(7)
Approximation of harmonics in 18ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +40.1 +84.1 -35.3 +25.3 -79.7 -6.3 +72.4 -16.4 +71.5 -9.1 -86.0 +12.6
Relative (%) +23.3 +48.8 -20.5 +14.7 -46.2 -3.7 +42.0 -9.5 +41.5 -5.3 -49.9 +7.3
Steps
(reduced) 		26
(8) 		27
(9) 		27
(9) 		28
(10) 		28
(10) 		29
(11) 		30
(12) 		30
(12) 		31
(13) 		31
(13) 		31
(13) 		32
(14)
7et, 2.3.5.11.13 WE
  • Step size: 172.390 ¢, octave size: 1206.7 ¢

Stretching the octave of 7edo by around 7 ¢ results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 85.7 ¢. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 TE tuning both do this.

Approximation of harmonics in 7et, 2.3.5.11.13 WE
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -5.7 +13.5 -28.1 +1.1 +79.0 +20.2 -11.3 -21.3 -14.0 +7.8
Relative (%) +3.9 -3.3 +7.8 -16.3 +0.6 +45.8 +11.7 -6.6 -12.4 -8.1 +4.5
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +41.6 +85.7 -33.7 +26.9 -78.0 -4.6 +74.2 -14.6 +73.3 -7.2 -84.2 +14.5
Relative (%) +24.1 +49.7 -19.6 +15.6 -45.3 -2.7 +43.0 -8.5 +42.5 -4.2 -48.8 +8.4
Step 26 27 27 28 28 29 30 30 31 31 31 32
15zpi
  • Step size: 172.495 ¢, octave size: 1207.5 ¢

Stretching the octave of 7edo by around 7.5 ¢ results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 84.0 ¢. The tuning 15zpi does this.

Approximation of harmonics in 15zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +7.5 -4.5 +14.9 -26.4 +3.0 +81.1 +22.4 -9.0 -18.9 -11.4 +10.4
Relative (%) +4.3 -2.6 +8.7 -15.3 +1.7 +47.0 +13.0 -5.2 -11.0 -6.6 +6.0
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 15zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.3 -84.0 -30.9 +29.9 -75.1 -1.6 +77.3 -11.5 +76.6 -4.0 -80.9 +17.9
Relative (%) +25.7 -48.7 -17.9 +17.3 -43.5 -0.9 +44.8 -6.6 +44.4 -2.3 -46.9 +10.4
Step 26 26 27 28 28 29 30 30 31 31 31 32
11edt
  • Step size: 172.905 ¢, octave size: 1210.3 ¢

Stretching the octave of 7edo by around NNN ¢ results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 83.6 ¢. The tuning 11edt does this.

Approximation of harmonics in 11edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced) 		7
(7) 		11
(0) 		14
(3) 		16
(5) 		18
(7) 		19
(8) 		21
(10) 		22
(0) 		23
(1) 		24
(2) 		25
(3)
Approximation of harmonics in 11edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +55.0 -73.3 -19.8 +41.3 -63.6 +10.3 -83.3 +0.8 -83.6 +8.7 -68.2 +31.0
Relative (%) +31.8 -42.4 -11.5 +23.9 -36.8 +6.0 -48.2 +0.5 -48.4 +5.1 -39.5 +17.9
Steps
(reduced) 		26
(4) 		26
(4) 		27
(5) 		28
(6) 		28
(6) 		29
(7) 		29
(7) 		30
(8) 		30
(8) 		31
(9) 		31
(9) 		32
(10)
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