Editing Meantone temperament (section)

==The tone as a mean==
The name "meantone temperament" derives from the fact that in all such temperaments the size of the whole tone, within the diatonic scale, is somewhere between the [[Major second#Major and minor tones|major and minor tones]] (9:8 and 10:9 respectively) of [[just intonation]], which differ from each other by a [[syntonic comma]]. In any regular system <ref name ="Barbour"/> the whole tone (as {{sc|'''C D'''}}) is reached after two fifths (as {{sc|'''C G D'''}}) (lowered by an octave), while the major third is reached after four fifths {{nobr|({{sc|'''C G D A E'''}})}} (lowered by two octaves). It follows that in {{sfrac| 1 | 4 }} comma meantone the whole tone is exactly half of the just major third (in cents) or, equivalently, the square root of the frequency ratio of {{sfrac| 5 | 4 }}.

Thus, one sense in which the tone is a mean is that, as a frequency ratio,  it is the [[geometric mean]] of the major tone  and the minor tone: 
<math>\ \sqrt{ \tfrac{ 10 }{\ 9\ } \cdot \tfrac{\ 9\ }{ 8 }\ } = \sqrt{ \tfrac{\ 5\ }{ 4 }\ } = 1.1180340 </math>, 
equivalent to 193.157&nbsp;[[musical cent|cents]]: the quarter-comma whole-tone size. However, ''any'' intermediate tone qualifies as a "mean" in the sense of being intermediate, and hence as a valid choice for some meantone system.

In the case of quarter-comma meantone,  where the major third is made narrower by a syntonic comma, the whole tone is made half a comma narrower than the major tone of just intonation (9:8), or half a comma wider than the minor tone (10:9). This is the sense in which quarter-tone temperament is often considered "the" exemplary meantone temperament since, in it, the whole tone lies midway (in [[musical cents|cents]]) between its possible extremes.<ref name ="Barbour"/>