Deneb: Difference between revisions
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9:11:15 could be considered as this temperament's fundamental consonant chord in the place of 3:5:7 or of 4:5:6. | 9:11:15 could be considered as this temperament's fundamental consonant chord in the place of 3:5:7 or of 4:5:6. | ||
Due to the presence of a quasi-octave and quasi-fifth, you could actually use traditional tertian chords in this temperament if you want, but the fifth is 680 cents so basically it would be [[18edo]] | Due to the presence of a quasi-octave and quasi-fifth, you could actually use traditional tertian chords in this temperament if you want, but the fifth is 680 cents so basically it would be [[18edo]]-esque neutral triads. | ||
[[Category:Deneb| ]] <!-- main article --> | [[Category:Deneb| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category:Non-octave temperaments]] | [[Category:Non-octave temperaments]] | ||
Revision as of 00:44, 14 October 2026
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Todo: complete page
Add infobox. Rewrite, polish interval table. Add tunings section. |
Deneb is a nonoctave, tritave-repeating rank-2 temperament that tempers out the comma 6655/6561 in the 3.5.11 subgroup. It is generated by a neutral third that represents a flat 11/9, and three generators stacked represent 9/5 (and due to tritave-equivalence, -3 generators stacked represents 5/3). It possesses MOS scales of the families 1L 4s⟨3/1⟩, 5L 1s⟨3/1⟩, 6L 5s⟨3/1⟩, and 11L 6s⟨3/1⟩. It is one of the simplest temperaments in the 3.5.11 subgroup with decent accuracy.
This temperament was first discovered and named by CompactStar in 2023 after the star Deneb, following a series of non-octave temperaments that are named after stars.
For technical data, see No-twos subgroup temperaments #Deneb.
Interval chain
One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to 55/27 and placed 2 generators down. Unless you are using an unusually sharp generator, this quasi-octave will be very smeary, being almost 22 cents sharp in the CWE tuning, but it is probably still meriting of special treatment in terms of consonance and dissonance. (Incidentally, extending Deneb to actually equate the quasi-octave to 2/1 would result in the 2.3.5.11 version of dicot)
In the following table, prime harmonics and subharmonics are labeled in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 340.052 | 11/9 |
| 2 | 680.104 | 81/55, 121/81 |
| 3 | 1020.156 | 9/5 |
| 4 | 1360.208 | 11/5 |
| 5 | 1700.260 | 121/45 |
| 6 | 138.357 | 27/25 |
| 7 | 478.409 | 33/25 |
| 8 | 818.461 | 121/75 |
| 9 | 1158.513 | 243/125 |
| 10 | 1498.565 | 297/125 |
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| -1 | 1561.903 | 27/11 |
| -2 | 1221.851 | 55/27, 243/121 |
| -3 | 881.799 | 5/3 |
| -4 | 541.747 | 15/11 |
| -5 | 201.695 | 135/121 |
| -6 | 1763.598 | 25/9 |
| -7 | 1423.546 | 25/11 |
| -8 | 1083.494 | 225/121 |
| -9 | 743.442 | 125/81 |
| -10 | 403.390 | 125/99 |
* In 3.5.11-subgroup CWE tuning
Harmony
9:11:15 could be considered as this temperament's fundamental consonant chord in the place of 3:5:7 or of 4:5:6.
Due to the presence of a quasi-octave and quasi-fifth, you could actually use traditional tertian chords in this temperament if you want, but the fifth is 680 cents so basically it would be 18edo-esque neutral triads.