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| = Title1 = | | == Approximations of odd harmonics == |
| == Octave stretch or compression == | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| 99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
| | {{harmonics in equal|2|intervals=odd|columns=7}} |
| | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 99edo tunings.
| | {{harmonics in equal|4|intervals=odd|columns=7}} |
| | | {{harmonics in equal|5|intervals=odd|columns=7}} |
| ; [[zpi|567zpi]]
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| * Step size: 12.138{{c}}, octave size: 1201.66{{c}}
| | {{harmonics in equal|7|intervals=odd|columns=7}} |
| Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}} | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}} | | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| ; [[WE|99et, 13-limit WE tuning]]
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| * Step size: 12.123{{c}}, octave size: 1200.18{{c}}
| | {{harmonics in equal|13|intervals=odd|columns=7}} |
| Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}} | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|16|intervals=odd|columns=7}} |
| | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| ; 99edo
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * Step size: 12.121{{c}}, octave size: 1200.00{{c}}
| | {{harmonics in equal|19|intervals=odd|columns=7}} |
| Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| {{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}} | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| {{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}} | | {{harmonics in equal|22|intervals=odd|columns=7}} |
| | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| ; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| * Step size: 12.117{{c}}, octave size: 1199.58{{c}}
| | {{harmonics in equal|25|intervals=odd|columns=7}} |
| Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| ; [[zpi|568zpi]]
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| * Step size: 12.115{{c}}, octave size: 1199.39{{c}}
| | {{harmonics in equal|31|intervals=odd|columns=7}} |
| Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}} | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}} | | {{harmonics in equal|34|intervals=odd|columns=7}} |
| | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| ; [[157edt]] / [[ed5|230ed5]]
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| * Step size: 12.114{{c}}, octave size: 1199.32{{c}}
| | {{harmonics in equal|37|intervals=odd|columns=7}} |
| Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| {{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}} | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| {{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}} | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| | | {{harmonics in equal|41|intervals=odd|columns=7}} |
| = Title2 = | | {{harmonics in equal|42|intervals=odd|columns=7}} |
| === Placeholder === | | {{harmonics in equal|43|intervals=odd|columns=7}} |
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