Deeptone: Difference between revisions

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'''Deeptone''' or '''tragicomical''' is a [[5-limit]] [[temperament]] generated by a fifth (~[[3/2]]), which is typically sharper than in 7edo but flatter than in [[flattone]] (approximately 686-691{{cent}}). The ~[[5/4]] is reached by eleven fifths octave-reduced, which is an augmented third (~~C-E#~~). A characteristic feature is that the [[81/80|syntonic comma]] is tuned negative, represented by a diminished unison (~~C-Cb~~). This means that ~[[81/64]] is a submajor third flatter than ~[[5/4]], and ~[[32/27]] is a supraminor third sharper than ~[[6/5]].

'''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]].

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

See [[Syntonic–chromatic equivalence continuum #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 ==

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]].

== 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.

== Interval chain ==

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

{| class="wikitable center-1 right-2"

|-

! #

! colspan="3" | 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 || G# || 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]] ||

|}

<nowiki/>* In 5-limit CTE tuning

== Notes ==

~~[[Category:Temperaments]]~~

[[Category:Deeptone| ]]

[[Category:Rank-2 temperaments]]

~~{{stub}}~~

@@ Line 1: / Line 1: @@
-'''Deeptone''' or '''tragicomical''' is a [[5-limit]] [[temperament]] generated by a fifth (~[[3/2]]), which is typically sharper than in 7edo but flatter than in [[flattone]] (approximately 686-691{{cent}}). The ~[[5/4]] is reached by eleven fifths octave-reduced, which is an augmented third (C-E#). A characteristic feature is that the [[81/80|syntonic comma]] is tuned negative, represented by a diminished unison (C-Cb). This means that ~[[81/64]] is a submajor third flatter than ~[[5/4]], and ~[[32/27]] is a supraminor third sharper than ~[[6/5]].
+'''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]].
-See [[Syntonic-chromatic equivalence continuum #Deeptone a.k.a. tragicomical]] for technical data.
+See [[Syntonic–chromatic equivalence continuum #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 ==
+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]].
+== 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.
+== Interval chain ==
+In the following table, prime harmonics and subharmonics are labeled in '''bold'''.
+{| class="wikitable center-1 right-2"
+|-
+! #
+! colspan="3" | 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 || G# || 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]] ||
+|}
+<nowiki/>* In 5-limit CTE tuning
+== Notes ==
+<references/>
-[[Category:Temperaments]]
 [[Category:Deeptone| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
-{{stub}}