Deneb
| Deneb |
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Todo: complete page
Add minimax error and target scale size. |
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 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 | |
|---|---|---|---|
| 3.5.11 subgroup | 3.5.8.11 extension | ||
| 0 | 0.000 | 1/1 | |
| 1 | 340.052 | 11/9 | 40/33 |
| 2 | 680.104 | 81/55, 121/81 | 40/27 |
| 3 | 1020.156 | 9/5 | |
| 4 | 1360.208 | 11/5 | 24/11 |
| 5 | 1700.260 | 121/45 | 8/3 |
| 6 | 138.357 | 27/25 | 88/81 |
| 7 | 478.409 | 33/25 | 72/55, 320/243 |
| 8 | 818.461 | 121/75 | 8/5 |
| 9 | 1158.513 | 243/125 | 64/33, 88/45 |
| 10 | 1498.565 | 297/125 | 64/27 |
| 11 | 1838.617 | 363/125 | 72/25 |
| 12 | 276.714 | 729/625 | 64/55 |
| 13 | 616.766 | 891/625 | 64/45 |
| 14 | 956.818 | 1089/625 | 512/297 |
| 15 | 1296.870 | 1331/625, 6561/3125 | 512/243, 576/275 |
| 16 | 1636.922 | 8019/3125, 14641/5625 | 64/25 |
| # | Cents* | Approximate ratios | |
|---|---|---|---|
| 3.5.11 subgroup | 3.5.8.11 extension | ||
| 0 | 0.000 | 1/1 | |
| -1 | 1561.903 | 27/11 | 99/40 |
| -2 | 1221.851 | 55/27, 243/121 | 81/40 |
| -3 | 881.799 | 5/3 | |
| -4 | 541.747 | 15/11 | 11/8 |
| -5 | 201.695 | 135/121 | 9/8 |
| -6 | 1763.598 | 25/9 | 243/88 |
| -7 | 1423.546 | 25/11 | 55/24, 729/320 |
| -8 | 1083.494 | 225/121 | 15/8 |
| -9 | 743.442 | 125/81 | 99/64, 135/88 |
| -10 | 403.390 | 125/99 | 81/64 |
| -11 | 63.338 | 125/121 | 25/24 |
| -12 | 1625.241 | 625/243 | 165/64 |
| -13 | 1285.189 | 625/297 | 135/64 |
| -14 | 945.137 | 625/363 | 891/512 |
| -15 | 605.085 | 1875/1331, 3125/2187 | 275/192, 729/512 |
| -16 | 265.033 | 3125/2673, 16875/14641 | 75/64 |
* In 3.5.11-subgroup CWE tuning
Chords and harmony
9:11:15 could serve as the fundamental consonant chord for Deneb analogous to 4:5:6 for meantone and 3:5:7 for BPS. 9:11:15 is somewhat close to a first-inversion minor triad, so it may sound very vaguely familiar to some. 9:11:15 also has a simpler (but very wide) tritave-equivalent inversion, 5:9:11. The 3.5.8.11 extension has 3:5:8 and 9:11:15:24. The 3.5.11.13 extensions allow for the isoharmonic chord 9:11:13:15 (although, for high-complexity extensions, it will appear infrequently).
Due to the presence of a quasi-octave and quasi-fifth, there are Mavila-esque neutral triads splitting the quasi-fifth in half (1-11/9-81/55). Technically, Deneb makes 1-11/9-81/55 an essentially tempered dyadic chord.
Scales
Reasonable tunings of Deneb possess MOS scales of the families 1L 4s⟨3/1⟩, 5L 1s⟨3/1⟩, 6L 5s⟨3/1⟩, and 11L 6s⟨3/1⟩. Interestingly, the 6L 5s⟨3/1⟩ Deneb[11] scale is melodically similar to softer diatonic scales because both approximate 7edo/11edt.
Extensions
Add 13
There are several possible extensions to 3.5.11.13 and higher no-2s no-7s limits, but it's mainly weak extensions. The only practical strong extension of Deneb to 3.5.11.13, b28 & b39, maps 13/9 to +13 generators up. This only really works for generators sharp of 5\28edt. 3.5.11 Deneb can suffice using the diatonic-like 11-note MOS 6L 5s⟨3/1⟩, but this extension requires at least the 17-note MOS 11L 6s⟨3/1⟩ to properly use the ratios of 13.
Fomalhaut is a weak extension of Deneb to higher limits that splits the interval of 11/3 in three. Fomalhaut was considered in the no-2s no-7s 23-limit (3.5.11.13.17.19.23) from the start. It is considerably more complex than Deneb, and requires at least an 18-note MOS to properly use (technically, all prime harmonics 3-23 are accessible in the 13-note MOS, but the 13-note MOS is very hard, and said prime harmonics are in different directions).
Another possible extension of Deneb to 3.5.11.13 is Threneb, which temper out the threedie, 2197/2187. This splits the tritave into 3 13/9s, and 13/9 becomes the new period while the generator remains the same. The EDT join of this extension is b39&b45 (only EDTs that are a multiple of 3 are supported). Unfortunately, the smaller MOS scales generated by this are once again very hard.
Also, the 3.5.7.11 temperament Mintra, although it's mainly known as a strong extension of Mintaka, is additionally a weak extension of Deneb and a weak extension of BPS, combining all three temperaments into an efficient no-2s 11-limit temperament. The way it is a weak extension of Deneb is that 27/11 (the tritave-complement of 11/9) is split into two 11/7s. Mintra can then be further extended to 3.5.7.11.13.
Add 8
5 generators stacked actually is a very good approximation of 8/3, so if you're okay with introducing limited factors of 2, Deneb can be easily extended to the 3.5.8.11 subgroup (or using the technically "canonical" ordering, 8.3.5.11, but using 8/1 as an equivalence would be rather impractical). This extension tempers out 121/120 and 2200/2187 and is related to the octave-repeating amity/hitchcock temperament. Interestingly, in the 3.5.8.11 extension, ~8/1 = ~55/272 * ~64/33, with ~55/27 being the aforementioned sharp quasi-octave, and ~64/33 being the undecimal infraoctave. Furthermore, ~55/27 and ~64/33 share the same scale degree in Deneb[11].
Tunings
Tuning spectrum
| Edt generator |
Unchanged interval (eigenmonzo) |
Generator (¢) | Comments |
|---|---|---|---|
| 1\6 | 316.993 | Lower bound of 3.5.11 tonality diamond monotone | |
| 3\17 | 335.639 | Lower bound to generate 11L 6s⟨3/1⟩ MOS scale | |
| 8\45 | 338.125 | ||
| 5/3 | 339.199 | ||
| 5\28 | 339.635 | ||
| 33/25 | 340.372 | ||
| 12\67 | 340.649 | ||
| 11/5 | 341.251 | Equal error on 11/9 and 5/3 | |
| 7\39 | 341.377 | ||
| 9\50 | 342.352 | ||
| 2\11 | 345.810 | Upper bound to generate 11L 6s⟨3/1⟩ MOS scale | |
| 11/9 | 347.408 | Untempered tuning | |
| 1\5 | 380.391 | Upper bound of 3.5.11 tonality diamond monotone |