Deeptone: Difference between revisions

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| Odd limit 1 = 5 | Mistuning 1 = 12.3 | Complexity 1 = 12

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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. 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]].

'''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 example of a [[cluster temperament]].

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).

See [[Syntonic–chromatic equivalence continuum #Deeptone a.k.a. tragicomical]] for technical data.

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.]]

~~== 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 ==

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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.

== ~~Scales ==~~

== Intervals ==

When properly tuned, Deeptone features [[mos scale]]s 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.

~~== Interval chain~~ ==

In the following table, prime harmonics and subharmonics are labeled in '''bold'''.

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! colspan="3" | Notation

! 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]] ||

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<nowiki/>* In 5-limit CTE tuning

== ~~Notes~~ ==

== Scales ==

When properly tuned, Deeptone features [[mos scale]]s 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 ===

{| 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 ==

[[Category:Deeptone| ]]

[[Category:Rank-2 temperaments]]

@@ Line 12: / Line 12: @@
 | Odd limit 1 = 5 | Mistuning 1 = 12.3 | Complexity 1 = 12
 }}
-'''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. 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]].
+'''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 example of a [[cluster temperament]].
+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).
-See [[Syntonic–chromatic equivalence continuum #Deeptone a.k.a. tragicomical]] for technical data.
+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.]]
-== 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 ==
@@ Line 28: / Line 29: @@
 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.
-== Scales ==
+== Intervals ==
-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 ==
 In the following table, prime harmonics and subharmonics are labeled in '''bold'''.
@@ Line 39: / Line 37: @@
 ! colspan="3" | Notation
 ! 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]] ||
@@ Line 84: / Line 82: @@
 <nowiki/>* In 5-limit CTE tuning
-== Notes ==
+== 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]]

Deeptone: Difference between revisions

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