Deeptone: Difference between revisions

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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 2s]], [[7L 5s]], [[7L 12s]]

| Pergen = (P8, P5)

| 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]].

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

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== 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 ==

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| 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]]

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| 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# || 1078.55 || [[15/8]] ||

| 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

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[[Category:~~Temperaments~~]]

[[Category:Deeptone| ]]

[[Category:Rank-2 temperaments]]

~~[[Category:Deeptone| ]] ~~

~~{{todo|inline=1|expand}}~~

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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''' 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.
@@ Line 9: / Line 22: @@
 == 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 ==
@@ Line 37: / Line 50: @@
 | 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]]
@@ Line 47: / Line 60: @@
 | 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# || 1078.55 || [[15/8]] ||
+| 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
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 <references/>
-[[Category:Temperaments]]
+[[Category:Deeptone| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]
-[[Category:Deeptone| ]] <!-- main article -->
-{{todo|inline=1|expand}}