Non-over-1 temperament: Difference between revisions
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== Examples == | == Examples == | ||
=== Petrtri === | === Petrtri === | ||
Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by 13edo and 21edo. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the [[oneirotonic]] MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. | Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by 13edo and 21edo. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the [[oneirotonic]] MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. Thus it is fair to say that approximations of 5:9:11:13 in the 8-note MOS use this non-over-2 temperament. | ||
=== Sensi === | === Sensi === | ||
[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].) | [[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].) | ||
Revision as of 19:13, 3 April 2021
A non-over-1 temperament is a regular temperament that tempers a subgroup corresponding to a harmonic series chord r:n1:...:nk where r ≠ 1, but is not meant to approximate a chord of the form 1:m1:...:mk. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate over-1 chords such as 4:5:6:7:11 poorly, such as 14edo, 18edo, 23edo and 29edo.
Examples
Petrtri
Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by 13edo and 21edo. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the oneirotonic MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. Thus it is fair to say that approximations of 5:9:11:13 in the 8-note MOS use this non-over-2 temperament.
Sensi
Sensi is effectively a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m1:...:mk (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See x31eq data page.)
| Generators | Cents* | Approximate ratios† |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 443.322 | 13/10~9/7 |
| 2 | 886.644 | 42/25~5/3 |
| 3 | 129.966 | 13/12~14/13~15/14~27/25 |
| 4 | 573.288 | 7/5~25/18~18/13 |
| 5 | 1016.610 | 9/5~70/39 |
| 6 | 259.932 | 7/6~15/13 |
| 7 | 703.253 | 3/2 |
- * in 2.3.5.7.13 POTE tuning
- † 2.3.5.7.13 ratio interpretations
Tridec
Tridec is a temperament generated by a generator near 455.2178¢ (for example, 11\29 or 14\37) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5.
Tridec essentially contains all the notes of 2.3.5 porcupine temperament and satisfies all its relations; hence it is essentially the same as 13-limit Ammonite.