3ed4/3: Difference between revisions

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-The Cube Root of the [[Perfect_fourth|Perfect fourth]] ([[4/3|4:3]]) is a nonoctave scale which divides the just perfect fourth (frequency ratio 4:3) into three steps of approximately 166.015[[cent|¢]] each.
+{{Infobox ET}}
+{{ED intro}}
-==Intervals==
+== Theory ==
+=== Harmonics ===
+{{Harmonics in equal|3|4|3|intervals=integer|columns=11}}
+{{Harmonics in equal|3|4|3|intervals=integer|columns=11|start=12|collapsed=true|title=Approximation of harmonics in 3ed4/3 (continued)}}
+== Intervals ==
 {| class="wikitable"
 |-
-| | degrees of CRP4
+! #
-| | cents value
+! Cents
-| | cents value [[octave-reduce|octave-reduce]]d
+! Approximate ratios
 |-
-| | 0
+| 0
-| | 0.00
+| 0.000
-| |
+| [[1/1]]
 |-
-| | 1
+| 1
-| | 166.01
+| 166.015
-| |
+| [[11/10]]
 |-
-| | 2
+| 2
-| | 332.03
+| 332.030
-| |
+|
 |-
-| | 3
+| 3
-| | 498.04
+| 498.045
-| |
+| [[4/3]]
 |-
-| | 4
+| 4
-| | 664.06
+| 664.060
-| |
+| [[22/15]]
 |-
-| | 5
+| 5
-| | 830.07
+| 830.075
-| |
+| [[13/8]]
 |-
-| | 6
+| 6
-| | 996.09
+| 996.090
-| |
+| [[16/9]]
 |-
-| | 7
+| 7
-| | 1162.10
+| 1162.105
-| |
+| 88/45
 |-
-| | 8
+| 8
-| | 1328.12
+| 1328.120
-| | 128.12
+| [[13/6]]
 |-
-| | 9
+| 9
-| | 1494.13
+| 1494.135
-| | 294.13
+| [[64/27]]
 |-
-| | 10
+| 10
-| | 1660.15
+| 1660.150
-| | 460.15
+|
 |-
-| | 11
+| 11
-| | 1826.16
+| 1826.165
-| | 626.16
+| [[13/9]]
 |-
-| | 12
+| 12
-| | 1992.18
+| 1992.180
-| | 792.18
+|
 |-
-| | 13
+| 13
-| | 2158.19
+| 2158.195
-| | 958.19
+|
 |-
-| | 14
+| 14
-| | 2324.21
+| 2324.210
-| | 1124.21
+|
 |-
-| | 15
+| 15
-| | 2490.22
+| 2490.225
-| | 90.22
+| [[135/32]]
 |-
-| | 16
+| 16
-| | 2656.24
+| 2656.240
-| | 256.24
+|
 |-
-| | 17
+| 17
-| | 2822.25
+| 2822.255
-| | 422.25
+|
 |-
-| | 18
+| 18
-| | 2988.27
+| 2988.270
-| | 588.27
+| [[45/8]]
 |-
-| | 19
+| 19
-| | 3154.28
+| 3154.285
-| | 754.28
+|
 |-
-| | 20
+| 20
-| | 3320.30
+| 3320.300
-| | 920.30
+| [[17/5]]
 |-
-| | 21
+| 21
-| | 3486.31
+| 3486.315
-| | 1086.31
+| [[15/2]]
 |-
-| | 22
+| 22
-| | 3652.33
+| 3652.330
-| | 52.33
+|
 |-
-| | 23
+| 23
-| | 3818.34
+| 3818.345
-| | 218.34
+| [[68/15]]
+|-
+| 24
+| 3984.360
+| [[10/1]]
 |}
-[[Category:edonoi]]
-[[Category:edp4]]
+== Regular temperaments ==
-[[Category:equal]]
+ed4/3 tuning is related to temperaments which temper out [[4000/3993]] (wizardharry temperament). The unit step of 3ed4/3 is approximately a cent sharp of [[11/10]]. Tempering out 4000/3993 leads equating three 11/10s with 4/3, hence wizardharry temperaments split the fourth in three.
-[[Category:nonoctave]]
-[[Category:p4]]
+Tempering out both [[55/54]] and [[100/99]] (equating 10/9 with 11/10 and 12/11) leads to [[porcupine]] (2.3.5.11 subgroup) or [[sonic]] (full 11-limit). Sonic temperaments include [[porcupine]], [[hystrix]], [[porky]], [[coendou]], [[hedgehog]], [[nautilus]], [[ammonite]], [[ceratitid]], and [[opossum]].
+Other wizardharry temperaments include [[octoid]], [[harry]], [[tritikleismic]], [[wizard]], [[Porwell temperaments #Septisuperfourth|septisuperfourth]], [[unthirds]], [[supers]], [[alphaquarter]], [[quincy]], [[stearnscape]], [[pogo]], [[marvolo]], [[cotritone]], [[echidna]], [[marvo]], [[mystery]], [[zarvo]], [[escaped]], [[thuja]], and [[escapade]].
+[[Category:Equal-step tuning]]
+[[Category:Nonoctave]]
+[[Category:Perfect fourth]]