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User:BudjarnLambeth/Sandbox2

< User:BudjarnLambeth
Revision as of 23:34, 21 August 2025 by BudjarnLambeth (talk | contribs) (→Octave stretch or compression)

Title1

Approximation of harmonics in ZPINAME
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 ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 +21.5 +6.7 +40.7 +10.1 +6.7 +24.9 -39.9 +10.1
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 +0.0 +2.5 +16.6 +1.2 +34.7 +3.7 +0.0 +17.8 -47.1 +2.5
Relative (%) +1.2 +0.0 +2.5 +16.6 +1.2 +34.6 +3.7 +0.0 +17.8 -47.1 +2.5
Steps
(reduced) 		12
(12) 		19
(0) 		24
(5) 		28
(9) 		31
(12) 		34
(15) 		36
(17) 		38
(0) 		40
(2) 		41
(3) 		43
(5)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -0.8 +1.5 +15.5 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Relative (%) +0.8 -0.8 +1.5 +15.4 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(0) 		34
(3) 		36
(5) 		38
(7) 		40
(9) 		41
(10) 		43
(12)

Title2

Octave stretch or compression

What follows is a comparison of stretched- and compressed-octave 12edo tunings.

12et, 7-limit WE tuning
  • Step size: 99.664 ¢, octave size: 1196.0 ¢

Compressing the octave of 12edo by 4 ¢ results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit TE tuning both do this. 40ed10 does this as well. An argument could be made that such tunings harmonies involving the 7th harmonic to regular old 12edo without even needing to add any new notes to the octave.

Approximation of harmonics in 12et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.0 -8.3 -8.1 +4.3 -12.4 +19.8 -12.1 -16.7 +0.2 +34.6 -16.4
Relative (%) -4.0 -8.4 -8.1 +4.3 -12.4 +19.8 -12.1 -16.7 +0.2 +34.7 -16.5
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 12et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.4 +15.7 -4.1 -16.1 -21.4 -20.7 -14.6 -3.8 +11.4 +30.5 -46.4 -20.4
Relative (%) +44.5 +15.8 -4.1 -16.2 -21.5 -20.8 -14.7 -3.8 +11.4 +30.6 -46.6 -20.5
Step 45 46 47 48 49 50 51 52 53 54 54 55
34zpi
  • Step size: 99.807 ¢, octave size: 1197.7 ¢

Compressing the octave of 12edo by around 2 ¢ results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit meantone, for composers seeking more pure thirds and sixths than regular 12edo.

Approximation of harmonics in 34zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.3 -5.6 -4.6 +8.3 -7.9 +24.6 -6.9 -11.2 +6.0 +40.6 -10.3
Relative (%) -2.3 -5.6 -4.6 +8.3 -8.0 +24.7 -7.0 -11.3 +6.0 +40.7 -10.3
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 34zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -49.0 +22.3 +2.7 -9.3 -14.4 -13.6 -7.4 +3.7 +19.0 +38.3 -38.7 -12.6
Relative (%) -49.1 +22.3 +2.7 -9.3 -14.4 -13.6 -7.4 +3.7 +19.0 +38.3 -38.8 -12.6
Step 44 46 47 48 49 50 51 52 53 54 54 55
12et, 5-limit WE tuning
  • Step size: 99.868 ¢, octave size: 1198.4 ¢

Compressing the octave of 12edo by around 1 ¢ results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit TE tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.

Approximation of harmonics in 12et, 5-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.6 -4.5 -3.2 +10.0 -6.0 +26.7 -4.8 -8.9 +8.4 +43.1 -7.6
Relative (%) -1.6 -4.5 -3.2 +10.0 -6.1 +26.7 -4.8 -8.9 +8.4 +43.2 -7.6
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 12et, 5-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -46.3 +25.1 +5.5 -6.3 -11.4 -10.5 -4.2 +6.8 +22.2 +41.6 -35.4 -9.2
Relative (%) -46.4 +25.1 +5.5 -6.3 -11.4 -10.5 -4.3 +6.8 +22.3 +41.6 -35.4 -9.2
Step 44 46 47 48 49 50 51 52 53 54 54 55
12edo
  • Step size: 100.000 ¢, octave size: 1200.0 ¢

Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.

Approximation of harmonics in 12edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in 12edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
31ed6
  • Step size: 100.063 ¢, octave size: 1200.8 ¢

Stretching the octave of 12edo by a little less than 1 ¢ results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. The tuning 31ed6 does this.

Approximation of harmonics in 31ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -0.8 +1.5 +15.5 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Relative (%) +0.8 -0.8 +1.5 +15.4 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(0) 		34
(3) 		36
(5) 		38
(7) 		40
(9) 		41
(10) 		43
(12)
Approximation of harmonics in 31ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -37.8 +34.1 +14.7 +3.0 -1.9 -0.8 +5.7 +17.0 +32.6 -48.0 -24.9 +1.5
Relative (%) -37.7 +34.1 +14.7 +3.0 -1.9 -0.8 +5.7 +17.0 +32.5 -47.9 -24.9 +1.5
Steps
(reduced) 		44
(13) 		46
(15) 		47
(16) 		48
(17) 		49
(18) 		50
(19) 		51
(20) 		52
(21) 		53
(22) 		53
(22) 		54
(23) 		55
(24)
19edt
  • Step size: 101.103 ¢, octave size: 1201.2 ¢

Stretching the octave of 12edo by a little more than 1 ¢ results in an improved prime 3, but worse prime 5. The tuning 19edt does this.

Approximation of harmonics in 19edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 +0.0 +2.5 +16.6 +1.2 +34.7 +3.7 +0.0 +17.8 -47.1 +2.5
Relative (%) +1.2 +0.0 +2.5 +16.6 +1.2 +34.6 +3.7 +0.0 +17.8 -47.1 +2.5
Steps
(reduced) 		12
(12) 		19
(0) 		24
(5) 		28
(9) 		31
(12) 		34
(15) 		36
(17) 		38
(0) 		40
(2) 		41
(3) 		43
(5)
Approximation of harmonics in 19edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -36.0 +35.9 +16.6 +4.9 +0.1 +1.2 +7.7 +19.0 +34.7 -45.9 -22.7 +3.7
Relative (%) -36.0 +35.9 +16.6 +4.9 +0.1 +1.2 +7.7 +19.0 +34.6 -45.8 -22.7 +3.7
Steps
(reduced) 		44
(6) 		46
(8) 		47
(9) 		48
(10) 		49
(11) 		50
(12) 		51
(13) 		52
(14) 		53
(15) 		53
(15) 		54
(16) 		55
(17)
7edf
  • Step size: 100.3 ¢, octave size: 1203.35 ¢

Stretching the octave of 12edo by around 3 ¢ results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to Pythagorean tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.

Approximation of harmonics in 7edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 +21.5 +6.7 +40.7 +10.1 +6.7 +24.9 -39.9 +10.1
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1)
Approximation of harmonics in 7edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -28.2 +44.0 +24.9 +13.4 +8.7 +10.1 +16.7 +28.2 +44.0 -36.5 -13.2 +13.4
Relative (%) -28.2 +43.9 +24.8 +13.4 +8.7 +10.0 +16.7 +28.1 +43.9 -36.4 -13.2 +13.4
Steps
(reduced) 		44
(2) 		46
(4) 		47
(5) 		48
(6) 		49
(0) 		50
(1) 		51
(2) 		52
(3) 		53
(4) 		53
(4) 		54
(5) 		55
(6)
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