Xenharmonic Wiki

Deeptone: Difference between revisions

← Older edit

CompactStar (talk | contribs)

1 user group
5,879 edits
VisualWikitext
m Fix redirect target
Tag: Redirect target changed
CompactStar (talk | contribs)
autopatrolled
5,879 edits
No edit summary
 
(81 intermediate revisions by 7 users not shown)
Line 1: Line 1:
#REDIRECT [[Syntonic-chromatic equivalence continuum#Deeptone a.k.a. tragicomical]]
{{Infobox regtemp
| Title = Deeptone
| Subgroups = 2.3.5, 2.3.5.13
| Comma basis = [[177147/163840]] (5-limit); <br>[[1053/1024]], [[2187/2080]] (2.3.5.13)
| Edo join 1 = 7 | Edo join 2 = 33
| Mapping = 1; 1 11 -4
| Generators = 3/2
| Generators tuning = 689.6
| Optimization method = CWE
| MOS scales = [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[7L&nbsp;12s]]
| Pergen = (P8, P5)
| Odd limit 1 = 5 | Mistuning 1 = 12.3 | Complexity 1 = 12
}}
'''Deeptone''', a.k.a. '''tragicomical''', is a [[5-limit]] [[regular temperament|temperament]] that [[tempering out|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{{cent}}). 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 [[81/80|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 [[54edo|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 {{u|CompactStar}} in 2023, referring to the temperament featuring fifths even flatter than [[flattone]]. ''Tragicomical'' was allegedly given by [[groundfault]] no later than 2023.<ref>[https://en.xen.wiki/index.php?title=Talk:177147/163840&oldid=107105 Xenharmonic Wiki | Talk: 177147/163840]</ref>
 
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.
 
[[File:Deeptone7.mp3|none|thumb|Deeptone[7] major scale in 40edo.]]
 
== 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'''.
 
{| class="wikitable center-1 right-2"
|-
! #
! colspan="3" | Notation
! Cents*
! Approximate ratios<br>([[5-limit]])
! Additional ratios<br>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]], [[81/80|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]] ||
|}
<nowiki/>* In 5-limit CTE tuning
 
== Scales ==
When properly tuned, Deeptone features [[mos scale]]s of the families [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[7L&nbsp;12s]], and [[7L&nbsp;19s]]. Another important scale, intermediate between the [[7L&nbsp;5s]] and [[7L&nbsp;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 ===
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
| '''[[7edo|4\7]]'''
|
| 685.714
| '''Lower bound of 5-odd-limit diamond monotone'''
|-
|
| [[6/5]]
| 688.436
| 1/10-comma tuning
|-
| [[61edo|35\61]]
|
| 688.524
| 61bc val
|-
| [[54edo|31\54]]
|
| 688.889
| 54b val
|-
| [[47edo|27\47]]
|
| 689.362
|
|-
|
| [[5/4]]
| 689.665
| 1/11-comma tuning
|-
|
| [[20/13]]
| 689.719
|
|-
|
| [[16/13]]
| 689.868
|
|-
| [[40edo|23\40]]
|
| 690.000
|
|-
|
| [[15/13]]
| 690.484
|
|-
|
| [[15/8]]
| 690.689
| 1/12-comma tuning
|-
| [[33edo|19\33]]
|
| 690.909
|
|-
| '''[[12edo|7\12]]'''
|
| 700.000
| '''Upper bound of 5-odd-limit diamond monotone'''
|}
 
== References ==
<references/>
 
[[Category:Deeptone| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Deeptone"