Deneb
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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
Since it has the tritave-reduced 5th and 11th harmonics, 9:11:15 could be considered as this temperament's fundamental consonant chord. Interestingly, 9:11:15 is somewhat close to a first-inversion minor triad, so it may sound very vaguely familiar to some.
There is also the simpler but much wider chord 5:9:11, a tritave-equivalent inversion of 9:11:15.
Due to the presence of a quasi-octave and quasi-fifth, there are actually some "neutral triads" available in this temperament, but the quasi-fifth is Mavila-sized so it's not very accurate.