7th-octave temperaments: Difference between revisions
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{{Fractional-octave navigation|7}} | {{Fractional-octave navigation|7}} | ||
7th octave temperament | a 7th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 7. The most notable 7th-octave family is the [[whitewood family]] – tempering out [[2187/2048]] and associating 4\7 to [[3/2]]. | ||
A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\7. | |||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
* ''Septant →'' [[Schismatic family #Septant|Schismatic family]] | |||
* ''Brahmagupta →'' [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]] | |||
* ''Absurdity'' ''→'' [[Syntonic-chromatic equivalence continuum #Absurdity|Syntonic chromatic equivalence continuum]] | |||
== Jamesbond == | |||
This temperament uses exactly the same 5-limit as 7et, but the harmonic 7 is mapped to an independent generator. It is so named because its [[wedgie]] starts with {{multival| 0 0 7 … }}. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 25/24, 81/80 | |||
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }} | |||
{{Multival|legend=1| 0 0 7 0 11 16 }} | |||
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861 | |||
{{Optimal ET sequence|legend=1| 7, 14c }} | |||
[[Badness]]: 0.041714 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 25/24, 33/32, 45/44 | |||
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }} | |||
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090 | |||
{{Optimal ET sequence|legend=1| 7, 14c }} | |||
Badness: 0.023524 | |||
==Akjaysmic (rank-3)== | |||
{{Main|Akjaysma}} | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 25/24, 27/26, 33/32, 40/39 | |||
Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }} | |||
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 949.236 | |||
{{Optimal ET sequence|legend=1| 7, 14c }} | |||
Badness: 0.023003 | |||
==== Septimal ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 25/24, 33/32, 45/44, 65/63 | |||
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }} | |||
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 952.555 | |||
{{Optimal ET sequence|legend=1| 7, 14cf }} | |||
Badness: 0.022569 | |||
== Akjaysmic (rank-3) == | |||
{{Main| Akjaysma }} | |||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 20: | Line 78: | ||
{{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }} | {{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }} | ||
=== 11-limit === | === 11-limit === | ||
| Line 33: | Line 92: | ||
{{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }} | {{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }} | ||
==Nitrogen== | |||
== Nitrogen == | |||
Described as 140 & 1407 temperament in the 7-limit, named after the 7th element for being period-7 and also because 140 and 1407 only contain numbers 7 and 14, atomic number and atomic weight of nitrogen respectively. On top of this connection to the number 7, it also reaches 7th harmonic 7 generators down. | Described as 140 & 1407 temperament in the 7-limit, named after the 7th element for being period-7 and also because 140 and 1407 only contain numbers 7 and 14, atomic number and atomic weight of nitrogen respectively. On top of this connection to the number 7, it also reaches 7th harmonic 7 generators down. | ||