Major third: Difference between revisions
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== In just intonation == | == In just intonation == | ||
=== By prime limit === | === By prime limit === | ||
3-limit intervals in the range of major thirds include the | 3-limit intervals in the range of major thirds include the Pythagorean major third of [[81/64]], 407.8{{c}} in size, which corresponds to the mos-based interval category of the diatonic major third and is generated by [[stacking]] four just perfect fifths of [[3/2]], and the Pythagorean diminished fourth of [[8192/6561]], which is flat of 81/64 by one Pythagorean comma, and is about 384{{c}} in size. | ||
Much [[odd limit|simpler]] major thirds exist in higher [[prime limit|limits]], however, for example: | Much [[odd limit|simpler]] major thirds exist in higher [[prime limit|limits]], however, for example: | ||
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== In | == In edos == | ||
The following table lists the best tuning of 5/4 and 9/7, as well as other major thirds if present, in various significant [[ | The following table lists the best tuning of 5/4 and 9/7, as well as other major thirds if present, in various significant [[edo]]s. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! Edo | ||
! 5/4 | ! 5/4 | ||
! 9/7 | ! 9/7 | ||
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| {{nowrap|362{{c}} ≈ 21/17|408{{c}} ≈ 81/64|452{{c}} ≈ 13/10}} | | {{nowrap|362{{c}} ≈ 21/17|408{{c}} ≈ 81/64|452{{c}} ≈ 13/10}} | ||
|} | |} | ||
<nowiki />* These edos have an approximation to 9/7, but it | <nowiki/>* These edos have an approximation to 9/7, but it is sharper than 460{{c}}, not really a major third. | ||
<nowiki />** These edos have an approximation to 5/4, but it | <nowiki/>** These edos have an approximation to 5/4, but it is flatter than 360{{c}}, not really a major third. | ||
== In regular temperaments == | == In regular temperaments == | ||
The two simplest major | The two simplest major third ratios are 5/4 and 9/7. The following notable temperaments are generated by them: | ||
=== Temperaments that use 5/4 as a generator === | === Temperaments that use 5/4 as a generator === | ||
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* [[Father]], an exotemperament which equates [[4/3]] and 5/4 as a single "fourth-third" interval, from which it derives its name. | * [[Father]], an exotemperament which equates [[4/3]] and 5/4 as a single "fourth-third" interval, from which it derives its name. | ||
* [[Augmented (temperament)|Augmented]], which splits the octave into three equal parts, each representing [[5/4]]. | * [[Augmented (temperament)|Augmented]], which splits the octave into three equal parts, each representing [[5/4]]. | ||
** The 5-limit [[ | ** The 5-limit [[circular temperament]]s in general | ||
=== Temperaments that use 9/7 as a generator === | === Temperaments that use 9/7 as a generator === | ||
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* [[Squares]], generated by flat supermajor thirds representing [[9/7]] and [[14/11]], such that a stack of four gives [[8/3]]. | * [[Squares]], generated by flat supermajor thirds representing [[9/7]] and [[14/11]], such that a stack of four gives [[8/3]]. | ||
== In mos scales == | |||
Intervals between 360 and 480 cents generate the following [[mos]] scales: | |||
These tables start from the last monolarge mos generated by the interval range. | |||
Scales with more than 12 notes are not included. | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Range | ! Range | ||
! colspan="5" | | ! colspan="5" | Mos | ||
|- | |- | ||
| 360–400{{c}} | | 360–400{{c}} | ||