Non-over-1 temperament
A non-over-2 temperament, or non-/2 temperament, is a regular temperament that tempers a subgroup corresponding to a harmonic series chord r:n1:...:nk where r ≠ 2, but is not meant to approximate a chord of the form 2:m1:...:mk. Non-over-2 temperaments give regular-temperament interpretations to edos that approximate over-2 chords such as 4:5:6:7:11 poorly, such as 14edo, 18edo, 23edo and 29edo.
Examples
Tridec
In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a 4/3 perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the 29edo and 37edo tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a non-over-2 temperament that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between 7edo's fifth and 5edo's fifth.
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; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-2 temperament.
The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.
| 21edo | 29edo | 37edo | Optimal (POTE) tuning | JI intervals represented (2.3.7/5.11/5.13/5 subgroup) | |
|---|---|---|---|---|---|
| generator (g) | 8\21, 457.14 | 11\29, 455.17 | 14\37, 454.05 | 455.22 | 13/10 |
| L (3g - octave) | 3\21, 171.43 | 4\29, 165.52 | 5\37, 162.16 | 165.65 | 11/10 |
| s (-5g + 2 octaves) | 2\21, 114.29 | 3\29, 124.14 | 4\37, 129.73 | 123.91 | 14/13, 15/14 |
Sensi
Sensi is effectively a non-over-2 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