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Octave (interval region): Difference between revisions

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{{About|the interval region|the octave as a just ratio|2/1}}
{{About|the interval region|the octave as a just ratio|2/1}}
{{Wikipedia}}
{{Wikipedia}}
A '''perfect octave''' ('''P8''') or '''octave''' ('''8ve''') is an [[interval]] that is approximately 1200 [[cent]]s in [[interval size measure|size]]. While a rough tuning range for octaves is sharper than 1170 cents according to [[Margo Schulter]]'s theory of interval regions, the term ''octave'' tends to imply a function within music that only works with intervals that corresponding to a [[just]] [[ratio]] of [[2/1]] or a close approximation thereof, usually preferred to be sharp-tempered if tempered. Other intervals are also classified as octaves, sometimes called '''wolf octaves''' or '''imperfect octaves''', if they are reasonably mapped to 7\7 and [[24edo|24\24]] (precisely seven steps of the diatonic scale and twelve steps of the chromatic scale). The use of 24edo's 24\24 as the mapping criteria here rather than [[12edo]]'s 12\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].  
A '''perfect octave''' ('''P8''') or '''octave''' ('''8ve''') is an [[interval]] that is approximately 1200 [[cent]]s in [[interval size measure|size]]. While a rough tuning range for octaves is sharper than 1170 cents according to [[Margo Schulter]]'s theory of interval regions, the term ''octave'' tends to imply a function within music that only works with intervals that corresponding to a [[just]] [[ratio]] of [[2/1]] or a close approximation thereof, usually preferred to be sharp-tempered if tempered. Other intervals are also classified as octaves, sometimes called '''wolf octaves''' or '''imperfect octaves''', if they are reasonably mapped to 7\7 and [[24edo|24＼24]] (precisely seven steps of the diatonic scale and twelve steps of the chromatic scale). The use of 24edo's 24＼24 as the mapping criteria here rather than [[12edo]]'s 12＼12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].  


The aforementioned function is the interval of equivalence, or [[equave]], because tones separated by an octave are perceived to have the same or similar [[pitch class]] to the average human listener. The reason for this phenomenon is probably due to the strong concordance of the octave or the strong amplitude of the second [[harmonic]] in most harmonic instruments.  As such, it is common practice to [[octave-reduce]] intervals so that they lie within the octave.
The aforementioned function is the interval of equivalence, or [[equave]], because tones separated by an octave are perceived to have the same or similar [[pitch class]] to the average human listener. The reason for this phenomenon is probably due to the strong concordance of the octave or the strong amplitude of the second [[harmonic]] in most harmonic instruments.  As such, it is common practice to [[octave-reduce]] intervals so that they lie within the octave.
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Several notable ones are:
Several notable ones are:
{| class="wikitable"
 
{| class="wikitable sortable center-all right-3"
|-
! class="unsortable" | Interval
! Prime <br>limit
! Distance <br>from 2/1
! Comma
|-
| [[1048576/531441]]
| 3
| 23.4600
| [[Pythagorean comma|531441/524288]]
|-
| [[160/81]]
| 5
| 21.5063
| [[81/80]]
|-
| [[125/64]]
| 5
| 41.0589
| [[128/125]]
|-
|-
! Interval
| [[243/125]]
! Size <br>(cents)
| 5
! Prime limit
| 49.1661
| [[250/243]]
|-
|-
| [[2/1]]
| [[448/225]]
| 1200
| 7
| 2
| 7.7115
| [[225/224]]
|-
|-
| [[1048576/531441]]
| [[125/63]]
| 1176.54
| 7
| rowspan="2" | 3
| 13.7948
| [[126/125]]
|-
|-
| [[531441/262144]]
| [[63/32]]
| 1223.46
| 7
| 27.2641
| [[64/63]]
|-
|-
| [[160/81]]
| [[49/25]]
| 1178.49
| 7
| rowspan="2" | 5
| 34.9756
| [[50/49]]
|-
|-
| [[81/40]]
| [[96/49]]
| 1221.51
| 7
| 35.6968
| [[49/48]]
|-
|-
| [[35/18]]
| [[35/18]]
| 1151.23
| 7
| rowspan="10" | 7
| 48.7704
| [[36/35]]
|-
| [[484/243]]
| 11
| 7.1391
| [[243/242]]
|-
| [[175/88]]
| 11
| 9.8646
| [[176/175]]
|-
| [[240/121]]
| 11
| 14.3672
| [[121/120]]
|-
| [[99/50]]
| 11
| 17.3995
| [[100/99]]
|-
| [[196/99]]
| 11
| 17.5761
| [[99/98]]
|-
| [[55/28]]
| 11
| 31.1943
| [[56/55]]
|-
| [[108/55]]
| 11
| 31.7667
| [[55/54]]
|-
| [[88/45]]
| 11
| 38.9058
| [[45/44]]
|-
| [[64/33]]
| 11
| 53.2729
| [[33/32]]
|-
| [[195/98]]
| 13
| 8.8554
| [[196/195]]
|-
| [[336/169]]
| 13
| 10.2744
| [[169/168]]
|-
|-
| [[96/49]]
| [[143/72]]
| 1164.30
| 13
| 12.0644
| [[144/143]]
|-
|-
| [[49/25]]
| [[208/105]]
| 1165.02
| 13
| 16.5670
| [[105/104]]
|-
|-
| [[63/32]]
| [[180/91]]
| 1172.74
| 13
| 19.1299
| [[91/90]]
|-
|-
| [[125/63]]
| [[77/39]]
| 1186.21
| 13
| 22.3388
| [[78/77]]
|-
|-
| [[252/125]]
| [[65/33]]
| 1213.79
| 13
| 26.4316
| [[66/65]]
|-
|-
| [[128/63]]
| [[128/65]]
| 1227.26
| 13
| 26.8414
| [[65/64]]
|-
|-
| [[100/49]]
| [[39/20]]
| 1234.98
| 13
| 43.8311
| [[40/39]]
|-
|-
| [[49/24]]
| [[576/289]]
| 1235.70
| 17
| 6.0008
| [[289/288]]
|-
|-
| [[72/35]]
| [[255/128]]
| 1248.77
| 17
| 6.7759
| [[256/255]]
|}
|}


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