Deeptone
| Deeptone |
1053/1024, 2187/2080 (2.3.5.13)
Deeptone, a.k.a. tragicomical, is a 5-limit temperament that tempers out the comma 177147/163840. It is generated by a fifth (~3/2), which is typically sharper than in 7edo but flatter than in flattone or flattertone (approximately 686-691 ¢). The ~5/4 is reached by eleven fifths octave-reduced, which is an augmented third (C–E♯). A characteristic feature of deeptone is that the syntonic comma is tuned negative, represented by a diminished unison (C–C♭). This means that ~81/64 is a submajor third, tuned flatter than ~5/4, and ~32/27 is a supraminor third, tuned sharper than ~6/5. It has high error because of its flat tuning of 3/2, although the well-known mavila temperament has a 3/2 that is much flatter still. It is rated as high-badness by most badness metrics due to the combination of high error and high complexity, but it is still of a low enough badness to be usable. Four edos support deeptone in their patent val (7edo, 33edo, 40edo, and 47edo), and additionally there is the near-patent val 54b-edo.
It can be regarded as an unusual example of a cluster temperament with 7 clusters of notes in an octave. The chroma or "super-comma" in this case represents 80/81~2187/2048 (in the 2.3.5.13 extension, 80/81~2187/2048~27/26).
The name deeptone was given by CompactStar in 2023, referring to the temperament featuring fifths even flatter than flattone. Tragicomical was allegedly given by groundfault no later than 2023.[1]
A related temperament is shallowtone, which is generated by a fifth slightly flat of 7edo's fifth instead of slightly sharp (one example of a shallowtone generator would be 37edo's flat fifth), and its mapping of 6/5 is the same as deeptone's mapping of 5/4. Basically, shallowtone is to deeptone as mavila is to meantone. It is of even higher badness than deeptone.
See Syntonic–chromatic equivalence continuum #Deeptone a.k.a. tragicomical and No-sevens subgroup temperaments#Deeptone a.k.a. tragicomical for technical data.
Extensions
Deeptone has an accurate 13/8, represented by the interval notated as a minor sixth, so it is very plausible to have a 2.3.5.13-subgroup extension tempering out both 2187/2080 and 1053/1024 (the tridecimal quartertone). This also gives a more simple interpretation of deeptone's major and minor thirds as tridecimal neutral thirds 16/13 and 39/32 respectively. Tuning this deeptone extension in 40edo results in essentially perfect tuning of 13/8 and 16/13, which is inherited from 10edo.
It is possible to map 8/7 to the double-augmented 2nd, but this necessitates the usage of a 19-note or larger MOS, and 33edo does not support it in the patent val, only 40edo and 47edo do.
Intervals
In the following table, prime harmonics and subharmonics are labeled in bold.
| # | Notation | Cents* | Approximate ratios (5-limit) |
Additional ratios in 2.3.5.13 extension | ||
|---|---|---|---|---|---|---|
| 0 | perfect unison | P1 | C | 0.0 | 1/1 | |
| 1 | perfect 5th | P5 | G | 689.879 | 3/2 | |
| 2 | major 2nd | M2 | D | 179.758 | 9/8 | 128/117 |
| 3 | major 6th | M6 | A | 869.637 | 27/16 | 64/39 |
| 4 | major 3rd | M3 | E | 359.516 | 81/64 | 16/13 |
| 5 | major 7th | M7 | B | 1049.395 | 243/128 | 24/13 |
| 6 | aug 4th | A4 | F# | 539.274 | 729/512, 320/243 | 18/13 |
| 7 | aug unison | A1 | C# | 29.153 | 2187/2048, 80/81 | 27/26 |
| 8 | aug 5th | A5 | G# | 719.032 | 40/27 | |
| 9 | aug 2nd | A2 | D# | 208.911 | 10/9 | |
| 10 | aug 6th | A6 | A# | 898.79 | 5/3 | |
| 11 | aug 3rd | A3 | E# | 388.669 | 5/4 | |
| 12 | aug 7th | A7 | B# | 1078.548 | 15/8 | |
| 13 | doubly-aug 4th | AA4 | Fx | 568.427 | 45/32 | |
| 14 | doubly-aug unison | AA1 | Cx | 58.306 | 135/128 | |
| 15 | doubly-aug 5th | AA5 | Gx | 748.185 | 405/256 | 20/13 |
| 16 | doubly-aug 2nd | AA2 | Dx | 238.064 | 800/729, 1215/1024 | 15/13 |
| 17 | doubly-aug 6th | AA6 | Ax | 927.943 | 400/243 | |
| 18 | doubly-aug 3rd | AA3 | Ex | 417.822 | 100/81 | |
| 19 | doubly-aug 7th | AA7 | Bx | 1107.701 | 50/27 | |
* In 5-limit CTE tuning
Scales
When properly tuned, Deeptone features mos scales of the families 2L 3s, 5L 2s, 7L 5s, 7L 12s, and 7L 19s. Another important scale, intermediate between the 7L 5s and 7L 12s scales, is the "Deeptone[14]" non-mos scale. This scale is notable for being the smallest deeptone scale that is generally "complete" for 5-limit purposes (as noted by groundfault) since Deeptone[12] has a paucity of 5-limit triads. Deeptone[14] is essentially just two interlaced Deeptone[7] diatonic scales with one of them being offset by an augmented unison above the other.
Tunings
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo) |
Generator (¢) | Comments |
|---|---|---|---|
| 4\7 | 685.714 | Lower bound of 5-odd-limit diamond monotone | |
| 6/5 | 688.436 | 1/10-comma tuning | |
| 35\61 | 688.524 | 61bc val | |
| 31\54 | 688.889 | 54b val | |
| 27\47 | 689.362 | ||
| 5/4 | 689.665 | 1/11-comma tuning | |
| 20/13 | 689.719 | ||
| 16/13 | 689.868 | ||
| 23\40 | 690.000 | ||
| 15/13 | 690.484 | ||
| 15/8 | 690.689 | 1/12-comma tuning | |
| 19\33 | 690.909 | ||
| 7\12 | 700.000 | Upper bound of 5-odd-limit diamond monotone |