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]]. | '''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. | ||
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== 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 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 == | == Scales == | ||
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== Interval chain == | == Interval chain == | ||
In the following table, prime harmonics are labeled in '''bold'''. | In the following table, prime harmonics and subharmonics are labeled in '''bold'''. | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
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| 0 || perfect unison || P1 || C || 0.0 || [[1/1]] || | | 0 || perfect unison || P1 || C || 0.0 || [[1/1]] || | ||
|- | |- | ||
| 1 || perfect 5th || P5 || G || 689.879 || [[3/2]] || | | 1 || perfect 5th || P5 || G || 689.879 || '''[[3/2]]''' || | ||
|- | |- | ||
| 2 || major 2nd || M2 || D || 179.758 || [[9/8]] || [[128/117]] | | 2 || major 2nd || M2 || D || 179.758 || [[9/8]] || [[128/117]] | ||
| Line 33: | Line 33: | ||
| 3 || major 6th || M6 || A || 869.637 || [[27/16]] || [[64/39]] | | 3 || major 6th || M6 || A || 869.637 || [[27/16]] || [[64/39]] | ||
|- | |- | ||
| 4 || major 3rd || M3 || E || 359.516 || [[81/64]] || [[16/13]] | | 4 || major 3rd || M3 || E || 359.516 || [[81/64]] || '''[[16/13]]''' | ||
|- | |- | ||
| 5 || major 7th || M7 || B || 1049.395 || [[243/128]] | | 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 || G# || 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]] | ||
|- | |||
| 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 | <nowiki/>* In 5-limit CTE tuning | ||
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<references/> | <references/> | ||
[[Category:Deeptone| ]] <!-- main article --> | [[Category:Deeptone| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | |||