Meantone temperament
Meantone temperament issystemmusical tuning. In it,major thirdtuned toparticular ratio,then dividedhalfmake two whole tonesequal size. Since two fifths upan octave down make upwhole tone, four fifths uptwo octaves down makemajor thirdmeantone temperament,hence four fifthsmeantone temperament make an interval ofseventeenth, whichtwo octaves abovemajor third,so hasratio at or about 1:5. Meantone tuning involves flatteningfifth so asbringseventeenth more nearly, or exactly, equalthis ratio.The most common formmeantone temperament tunes allmajor thirds tojust ratio4:5 (so,instance, if Atuned440 Hz, C#'tuned550 Hz). Thisachieved by tuningperfect fifthquarter ofsyntonic comma flatter thanjust ratio2:3. Itthis that givessystem its namequarter comma meantone or 1/4-comma meantone.
This system gives whole tones inratio 2:sqrt(5), diatonic semitones inratio ratio 5(5/4):8,perfect fifths inratio1:5(1/4), which1.495349.., compared withjustly tuned fifth2:3, which1.5. One offifths will bewolf interval, which meanswill be so sharpwill not sound at allsame asperfect fifth,will not normally be usedcommon practice music. Thisbecause twelve perfect fifths, each flattened byquarter ofsyntonic comma, do not add upan exact numberoctaves.
The term meantone temperamentsometimes usedrefer specifically1/4-comma meantone. However, systems which flattenfifth by differing amounts but which still equatemajor whole tone, whichjust intonation9/8, withminor whole tone, tuned justly10/9,also called meantone systems. Since (9/8)/(10/9) = 81/80,syntonic comma,fundamental character ofmeantone tuningthat all intervalsgenerated from fifths, andsyntonic commatempered tounison.
Meantones can be specifiedvarious ways. We can, as above, specify what fraction (logarithmically) ofsyntonic commafifthbeing flattened by, what equal temperament hasmeantone fifthquestion, or whatratio ofwhole tone todiatonic semitone is. This ratio was termed "R" by American composer, pianisttheoretician Easley Blackwood, buteffect has beenusemuch longer than that. Ituseful becausegives us an idea ofmelodic qualities oftuning,because if R isrational number, so(3R+1)/(5R+2), which issizefifthtermslogarithms base 2,which immediately tells us what division ofoctave we will have. If we multiply by 1200, we havesizefifthcents.
In these terms, some historically important meantone tuningslisted below. The relationship betweenfirst two columnsexact, while that between them andthirdclosely approximate.
| R | Equal temperament | Fraction ofcomma |
|---|---|---|
| 2 | 7/12 | 1/11 |
| 9/5 | 32/55 | 1/6 |
| 7/4 | 25/43 | 1/5 |
| 5/3 | 18/31 | 7/29 |
| 33/20 | 119/205 | 1/4 |
| 8/5 | 29/50 | 2/7 |
| 3/2 | 11/19 | 1/3 |
Because ofwolf interval which arises when twelve notes tooctavetuned tomeantonefifths significantly flatter than1/11-commaequal temperament, well temperamentseventually equal temperament became more popular.
