|
|
| (198 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| = Title1 = | | == Approximations of odd harmonics == |
| {{Harmonics in equal|40|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| {{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|2|intervals=odd|columns=7}} |
| {{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|4|intervals=odd|columns=7}} |
| | | {{harmonics in equal|5|intervals=odd|columns=7}} |
| = Title2 = | | {{harmonics in equal|6|intervals=odd|columns=7}} |
| == Octave stretch or compression == | | {{harmonics in equal|7|intervals=odd|columns=7}} |
| Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]].
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 41edo tunings.
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[147ed12]] / [[106ed6]] / [[65edt]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| * 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
| | {{harmonics in equal|19|intervals=odd|columns=7}} |
| Compressing the octave of 41edo by around 0.2{{c}} results in just slightly improved primes 3, 11 and 13, but just slightly worse primes , 5 and 7. This approximates all harmonics up to 16 within 7.6{{c}}. The tunings 147ed12, 106ed6 and 65edt each do this.
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| {{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}} | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| {{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}} | | {{harmonics in equal|22|intervals=odd|columns=7}} |
| | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| ; 41edo
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| * Step size: 29.268{{c}}, octave size: 1200.00{{c}}
| | {{harmonics in equal|25|intervals=odd|columns=7}} |
| Pure-octaves 41edo approximates all harmonics up to 16 within 8.3{{c}}. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| ; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| * Step size: 29.277{{c}}, octave size: 1200.35{{c}}
| | {{harmonics in equal|31|intervals=odd|columns=7}} |
| Stretching the octave of 41edo by around 0.5{{c}} results in just slightly improved primes 5 and 7, but just slightly worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}.
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| {{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}} | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| {{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}} | | {{harmonics in equal|34|intervals=odd|columns=7}} |
| | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| ; [[WE|41et, 7-limit WE tuning]]
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| * Step size: 29.288{{c}}, octave size: 1200.81{{c}}
| | {{harmonics in equal|37|intervals=odd|columns=7}} |
| Stretching the octave of 41edo by just under 1{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| {{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}} | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| {{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}} | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| | {{harmonics in equal|43|intervals=odd|columns=7}} |
| | {{harmonics in equal|44|intervals=odd|columns=7}} |
| | {{harmonics in equal|45|intervals=odd|columns=7}} |
| | {{harmonics in equal|46|intervals=odd|columns=7}} |
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| | {{harmonics in equal|49|intervals=odd|columns=7}} |
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| | {{harmonics in equal|51|intervals=odd|columns=7}} |
| | {{harmonics in equal|52|intervals=odd|columns=7}} |
| | {{harmonics in equal|53|intervals=odd|columns=7}} |