Pythagorean tuning: Difference between revisions
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{{Wikipedia|Pythagorean tuning}} | {{Wikipedia|Pythagorean tuning}} | ||
'''Pythagorean tuning''' is a system where all intervals are determined by [[3/2|pure fifths]] and [[2/1|octaves]]. This makes it essentially the same as [[3-limit]] [[just intonation]]. | |||
When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]]. | |||
== History == | == History == | ||
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The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music|Ancient China]] between 600 BCE and 240 CE, where it was called '''shi'er lü'''. | The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music|Ancient China]] between 600 BCE and 240 CE, where it was called '''shi'er lü'''. | ||
== Relation to temperaments == | |||
Pythagorean tuning can be considered a [[trivial temperament|trivial]] rank-2 temperament in the 2.3 subgroup, where it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher (e.g. meantone) can be seen as extensions of Pythagorean temperament. | |||
Because the [[schisma]] is so small, a series of just fifths can also be considered a reasonable tuning of the [[schismatic]] temperament, where the [[Pythagorean diminished fourth|diminished fourth]] (e.g. {{dash|C, F♭}}) approximates [[5/4]]. | |||
Mark Lindley<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref> argues such a system was used in Europe during the 15th century, with keyboards tuned to nearly pure fifths as | |||
:{{dash|G♭, C♭, A♭, E♭, B, F, C, G, D, A, E, B}}. | |||
When respelled enharmonically, triads such as {{dash|D, F♯, A}} are close to 4:5:6 in this tuning. | |||
== Scales == | == Scales == | ||
Pythagorean tuning generates the following [[MOS]] scales: | |||
* [[Pythagorean5]] – proper [[2L 3s]]. Also known as pythagorean pentic scale | * [[Pythagorean5]] – proper [[2L 3s]]. Also known as pythagorean pentic scale | ||
*[[Pythagorean7]] – improper [[5L 2s]]. Also known as pythagorean diatonic scale | * [[Pythagorean7]] – improper [[5L 2s]]. Also known as pythagorean diatonic scale | ||
*[[Pythagorean12]] – proper [[5L 7s]]. Also known as pythagorean chromatic scale | * [[Pythagorean12]] – proper [[5L 7s]]. Also known as pythagorean chromatic scale | ||
*[[Pythagorean17]] – improper [[12L 5s]]. Also known as pythagorean enharmonic scale | * [[Pythagorean17]] – improper [[12L 5s]]. Also known as pythagorean enharmonic scale | ||
*[[Pythagorean29]] – improper [[12L 17s]] | * [[Pythagorean29]] – improper [[12L 17s]] | ||
*[[Pythagorean41]] – proper [[12L 29s]] | * [[Pythagorean41]] – proper [[12L 29s]] | ||
*[[Pythagorean53]] – proper [[41L 12s]] | * [[Pythagorean53]] – proper [[41L 12s]] | ||
The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, and 3.846 for enharmonic. | The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, and 3.846 for enharmonic. | ||
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== See also == | == See also == | ||
* [[3-limit]], the JI subgroup | * [[3-limit]], the corresponding JI subgroup. | ||
* [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning | * [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning. | ||
== References == | |||
<references /> | |||
[[Category:3-limit| ]] <!-- main article --> | [[Category:3-limit| ]] <!-- main article --> | ||
Revision as of 11:43, 21 April 2025
Pythagorean tuning is a system where all intervals are determined by pure fifths and octaves. This makes it essentially the same as 3-limit just intonation.
When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the chain of fifths.
History
Pythagorean tuning was not actually invented by Pythagoras. The earliest records are from Ancient Mesopotamia, and it was later inherited by the Ancient Greeks.
The 12-tone form of Pythagorean tuning was (probably independently) invented in Ancient China between 600 BCE and 240 CE, where it was called shi'er lü.
Relation to temperaments
Pythagorean tuning can be considered a trivial rank-2 temperament in the 2.3 subgroup, where it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher (e.g. meantone) can be seen as extensions of Pythagorean temperament.
Because the schisma is so small, a series of just fifths can also be considered a reasonable tuning of the schismatic temperament, where the diminished fourth (e.g. C – F♭) approximates 5/4. Mark Lindley[1] argues such a system was used in Europe during the 15th century, with keyboards tuned to nearly pure fifths as
- G♭ – C♭ – A♭ – E♭ – B – F – C – G – D – A – E – B.
When respelled enharmonically, triads such as D – F♯ – A are close to 4:5:6 in this tuning.
Scales
Pythagorean tuning generates the following MOS scales:
- Pythagorean5 – proper 2L 3s. Also known as pythagorean pentic scale
- Pythagorean7 – improper 5L 2s. Also known as pythagorean diatonic scale
- Pythagorean12 – proper 5L 7s. Also known as pythagorean chromatic scale
- Pythagorean17 – improper 12L 5s. Also known as pythagorean enharmonic scale
- Pythagorean29 – improper 12L 17s
- Pythagorean41 – proper 12L 29s
- Pythagorean53 – proper 41L 12s
The hardnesses of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, and 3.846 for enharmonic.
Music
See 3-limit #Music.
See also
- 3-limit, the corresponding JI subgroup.
- Chain of fifths, a harmonic structure based on the concepts of Pythagorean tuning.
References
- ↑ Mark Lindley, Pythagorean Intonation and the Rise of the Triad, Royal Musical Association Research Chronicle, 1980