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| = 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}} |
| What follows is a comparison of compressed-octave 27edo tunings.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| ; 27edo
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * Step size: 44.444{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| {{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}} | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| {{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}} | | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[WE|27et, 13-limit WE tuning]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * Step size: 44.375{{c}}, octave size: 1198.9{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| Compressing the octave of 27edo by around 2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| {{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}} | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| {{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|20|intervals=odd|columns=7}} |
| | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| ; [[97ed12]]
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: 1197.5{{c}}
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| Compressing the octave of 27edo by around 2.5{{c}} has the same benefits and drawbacks as the 13-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 97ed12 does this.
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| ; [[zpi|106zpi]] / [[70ed6]] / [[WE|27et, 7-limit WE tuning]]
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| * Step size: ~44.306{{c}}, octave size: ~1196.2{{c}}
| | {{harmonics in equal|29|intervals=odd|columns=7}} |
| Compressing the octave of 27edo by around 3.5{{c}} results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| {{Harmonics in cet|44.306|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{Harmonics in cet|44.306|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi (continued)}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| ; [[90ed10]]
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: 1195.9{{c}}
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| Compressing the octave of 27edo by around 5.5{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| {{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| ; [[43edt]]
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: 1204.3{{c}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| Compressing the octave of 27edo by around 5.5{{c}} results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2{{c}}. The tuning 43edt does this.
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| {{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}} | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}} | | {{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}} |