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'''Deeptone''' or '''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. Four edos support deeptone in their [[patent val]] ([[7edo]], [[33edo]], [[40edo]], and [[47edo]]), and additionally there is the near-patent val [[54edo|54b-edo]].
{{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]].  


See [[Syntonic–chromatic equivalence continuum #Deeptone a.k.a. tragicomical]] for technical data.  
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.]]
[[File:Deeptone7.mp3|none|thumb|Deeptone[7] major scale in 40edo.]]
== Etymology ==
''Deeptone'' was given by [[user:CompactStar|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>


== Extensions ==
== 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 [[177147/163840]] 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]].
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]].


== Scales ==
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.
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.


== Interval chain ==
== Intervals ==
In the following table, prime harmonics and subharmonics are labeled in '''bold'''.  
In the following table, prime harmonics and subharmonics are labeled in '''bold'''.  


Line 22: Line 37:
! colspan="3" | Notation  
! colspan="3" | Notation  
! Cents*
! Cents*
! Approximate ratios (5-limit)
! Approximate ratios<br>([[5-limit]])
! Additional ratios in 2.3.5.13 extension
! Additional ratios<br>in 2.3.5.13 extension
|-
|-
| 0 || perfect unison || P1 || C || 0.0 || [[1/1]] ||
| 0 || perfect unison || P1 || C || 0.0 || [[1/1]] ||
Line 37: Line 52:
| 5 || major 7th || M7 || B || 1049.395 || [[243/128]] || [[24/13]]
| 5 || major 7th || M7 || B || 1049.395 || [[243/128]] || [[24/13]]
|-
|-
| 6 || aug 4th || A4 || G# || 539.274 || [[729/512]], [[320/243]] || [[18/13]]
| 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]]
| 7 || aug unison || A1 || C# || 29.153 || [[2187/2048]], [[81/80|80/81]] || [[27/26]]
Line 47: Line 62:
| 10 || aug 6th || A6 || A# || 898.79 || [[5/3]] ||
| 10 || aug 6th || A6 || A# || 898.79 || [[5/3]] ||
|-
|-
| 11 || aug 3rd || A3 || E# || 388.67 || '''[[5/4]]''' ||
| 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]]
|-
|-
| 12 || aug 7th || A7 || B# || 1078.55 || [[15/8]] ||
| 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'''
|}
|}
<nowiki/>* In 5-limit CTE tuning


== Notes ==
== References ==
<references/>
<references/>


[[Category:Temperaments]]
[[Category:Deeptone| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Deeptone| ]] <!-- main article -->
{{todo|inline=1|expand}}

Latest revision as of 01:01, 22 October 2026

Deeptone
Subgroups 2.3.5, 2.3.5.13
Comma basis 177147/163840 (5-limit);
1053/1024, 2187/2080 (2.3.5.13)
Reduced mapping ⟨1; 1 11 -4]
ET join 7 & 33
Generators (CWE) ~3/2 = 689.6 ¢
MOS scales 5L 2s, 7L 5s, 7L 12s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 5-odd-limit: 12.3 ¢
Target scale size 5-odd-limit: 12 notes

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.

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.

# 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

References

  1. ↑ Xenharmonic Wiki | Talk: 177147/163840
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