Non-over-1 temperament: Difference between revisions
mNo edit summary |
→Sensi: power |
||
| (19 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
A | A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Assuming octave equivalence, this means a subgroup of the form 2.m<sub>1</sub>/f.[...].m<sub>r</sub>/f, where f:m<sub>1</sub>:...:m<sub>r</sub> is a JI chord none of whose notes is a power of 2. | ||
Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]]. | |||
== Examples == | == Examples == | ||
=== | === Greeley === | ||
{{See also|Subgroup temperaments #Greeley|Chromatic pairs #Greeley}} | |||
Greeley is the [[23edo|23]]&[[31edo|31]] temperament on the 2.5/3.7/3.11/3 subgroup, with a [[MOS]] [[generator]] size close to [[porcupine]] but smaller (the POL2 generator is around 155.7756¢). | |||
* One generator represents 12/11 and 11/10. | |||
* Two generators represents 6/5. | |||
* Four generators represents 10/7. | |||
* Six generators represents 12/7. | |||
=== Petrtri === | |||
{{See also|Subgroup temperaments #Petrtri|Chromatic pairs #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-1 temperament. | |||
=== Sensi === | === Sensi === | ||
[[Sensi]] is ''effectively'' a non-over- | {{See also|Sensipent family #Sensi|Chromatic pairs #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<sup>n</sup>: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].) | |||
{| class="wikitable" | {| class="wikitable" | ||
| Line 46: | Line 56: | ||
| | 7 | | | 7 | ||
| style="text-align:right;" | 703.253 | | style="text-align:right;" | 703.253 | ||
| | | | | 3/2 | ||
|} | |} | ||
| Line 52: | Line 62: | ||
: <sup>†</sup> 2.3.5.7.13 ratio interpretations | : <sup>†</sup> 2.3.5.7.13 ratio interpretations | ||
=== Tridec === | |||
{{See also|Subgroup temperaments #Tridec|Chromatic pairs #Tridec}} | |||
Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo|11\29]] or [[37edo|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]]. | |||
[[Category: | [[Category:Regular temperament theory]] | ||
Latest revision as of 00:47, 24 April 2026
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. Assuming octave equivalence, this means a subgroup of the form 2.m1/f.[...].mr/f, where f:m1:...:mr is a JI chord none of whose notes is a power of 2.
Non-over-1 temperaments give regular-temperament interpretations to edos that approximate over-1 chords (assuming octave equivalence) such as 4:5:6:7:11 poorly, such as 14edo, 18edo, 23edo and 29edo, thus may have much xenharmonic potential. Many of these temperaments have an octatonic structure, as 8edo represents non-over-1 intervals well for its size as far as the 17-limit.
Examples
Greeley
Greeley is the 23&31 temperament on the 2.5/3.7/3.11/3 subgroup, with a MOS generator size close to porcupine but smaller (the POL2 generator is around 155.7756¢).
- One generator represents 12/11 and 11/10.
- Two generators represents 6/5.
- Four generators represents 10/7.
- Six generators represents 12/7.
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-1 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 2n: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.