Shallowtone: Difference between revisions

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-'''Shallowtone''' is a [[5-limit]] and higher [[temperament]] which tempers out the [[shallowtone comma]]. It is generated by a flat fifth, which is typically sharper than in [[mavila]] but flatter than in [[7edo]]. This gives it a high [[error]], although not as much as mavila. The ~[[5/4]] is reached by minus ten fifths octave-reduced, which is an augmented third (C-Ex) in melodic antidiatonic notation and a diminished third (C-Eb) in harmonic antidiatonic notation. The only reasonable extension to the 7-limit tempers out [[36/35]]. Additionally, there is a weak extension [[Greenwoodmic temperaments#Semishallowtone|semishallowtone]] with a half-octave period, tempering out [[405/392]].
+{{Infobox regtemp
+| Title = Shallowtone
+| Subgroups = 2.3.5, 2.3.5.7
+| Comma basis = [[295245/262144]] (5-limit)<br>[[36/35]], [[295245/262144]] (7-limit)
+| Mapping = 1; 1 -10 12
+| Edo join 1 = 7 | Edo join 2 = 30b
+| Generators = 3/2
+| Generators tuning = 681.2
+| Optimization method = CWE
+| MOS scales = [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[7L&nbsp;2s]]
+| Odd limit 1 = 5 | Mistuning 1 = 18.7 | Complexity 1 = 16
+| Odd limit 2 = 9 | Mistuning 2 = 45.3 | Complexity 2 = 23
+}}
+'''Shallowtone''' is a [[regular temperament|temperament]] where the [[shallowtone comma]] is [[tempering out|tempered out]]. It is generated by a flattened [[3/2|perfect fifth]], typically sharper than in [[mavila]] but flatter than in [[7edo]], and [[5/4]] is reached by minus ten fifths [[octave reduction|octave-reduced]], which is an augmented third (C–E𝄪) in melodic antidiatonic notation and a diminished third (C–E𝄫) in harmonic antidiatonic notation. This gives it a high [[error]], although not as much as mavila.
-The name was coined by [[User:CompactStar|CompactStar]] in 2024.
+The only reasonable [[extension]] to the [[7-limit]] tempers out [[36/35]]. Additionally, there is a weak extension [[Greenwoodmic temperaments #Semishallowtone|semishallowtone]] with a half-octave period, tempering out [[405/392]].
-See [[Syntonic-chromatic equivalence continuum#Shallowtone]] for technical data.
+The name was coined by [[User:CompactStar|CompactStar]] in 2024. It refers to how it is an opposite to [[deeptone]] (which is on the other side of 7edo), as well as because it is just below the diatonic "surface".
+See [[Mint temperaments #Shallowtone]] for technical data.
+== Interval chain ==
+In the following table, prime harmonics are labeled in '''bold'''.
+{| class="wikitable"
+|-
+! rowspan="2" | #
+! rowspan="2" | Cents*
+! colspan="2" | Approximate ratios
+|-
+! 5-limit
+! 13-limit extension
+|-
+| 0
+| 0.000
+| [[1/1]]
+|
+|-
+| 1
+| 681.801
+| '''[[3/2]]'''
+| [[13/9]]
+|-
+| 2
+| 163.602
+| [[9/8]]
+| [[11/10]], [[13/12]], [[35/32]], [[81/70]], [[8192/8019]]
+|-
+| 3
+| 845.403
+| [[27/16]]
+| '''[[13/8]]''', [[117/70]], [[4096/2673]]
+|-
+| 4
+| 327.204
+| [[81/64]]
+| [[39/32]], [[99/80]], [[1024/891]]
+|-
+| 5
+| 1009.0511
+| [[243/128]]
+| [[117/64]], [[512/297]]
+|-
+| 6
+| 490.806
+| [[512/405]], [[729/512]]
+| [[128/99]]
+|-
+| 7
+| 1172.607
+| [[256/135]]
+| [[64/33]]
+|-
+| 8
+| 654.408
+| [[64/45]]
+| '''[[16/11]]''', [[112/81]]
+|-
+| 9
+| 136.209
+| [[16/15]]
+| [[12/11]], [[28/27]]
+|-
+| 10
+| 818.010
+| '''[[8/5]]'''
+| [[14/9]], [[18/11]], [[64/39]]
+|-
+| 11
+| 299.811
+| [[6/5]]
+| [[7/6]], [[27/22]], [[63/52]]
+|-
+| 12
+| 981.612
+| [[9/5]]
+| '''[[7/4]]''', [[26/15]], [[81/44]]
+|-
+| 13
+| 463.413
+| [[27/20]]
+| [[13/10]], [[21/16]], [[243/176]]
+|-
+| 14
+| 1145.214
+| [[81/40]]
+| [[39/20]], [[63/32]], [[729/352]]
+|-
+| 15
+| 627.015
+| [[243/160]]
+| [[117/80]], [[189/128]]
+|}
+<nowiki/>* In 5-limit CTE tuning, octave reduced
-[[Category:Temperaments]]
 [[Category:Shallowtone| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
 [[Category:Mint temperaments]]
+[[Category:Exotemperaments]]
-{{stub}}