Pythagorean tuning: Difference between revisions

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~~The~~ '''Pythagorean tuning''' is the 3-limit ~~version of~~ [[just intonation]].

'''Pythagorean tuning''' is a system where all intervals are determined by perfect fifths tuned to [[3/2]] and [[2/1|octaves]]. As such, Pythagorean tuning contains the same intervals as [[3-limit]] [[just intonation]],

~~See~~ [[~~3-limit~~]] ~~for more information~~.

When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].

== History ==

Pythagorean tuning ~~is a~~ [[~~Tuning system~~|~~system of musical tuning~~]] ~~based on the mathematical ratios of pitches~~. ~~It is named after the ancient Greek philosopher {{w~~|~~Pythagoras}}~~, ~~who, according to legend, discovered the foundational principles of this tuning system through an experiment with hammers of different weights. Pythagoras' fascination with numerical ratios~~ and ~~their relation to~~ the ~~cosmos, particularly his concept of the 'music of the spheres', significantly influenced this tuning method~~.

Pythagorean tuning was not actually invented by [[Pythagoras of Samos|Pythagoras]]. The earliest records are from [[Mesopotamian music|Ancient Mesopotamia]], and it was later inherited by the [[Ancient Greek]]s.

The ~~Greeks used two systems~~ of ~~tuning based on ideal integer ratios:~~ Pythagorean ~~and Ptolemaic. The major difference is, Ptolemaic~~ tuning ~~uses simpler ratios, where as Pythagorean tuning uses a~~ [[~~chain of fifths~~|~~chain of fifths and fourths~~]]~~. For example~~, ~~a major third in Pythagorean would be [[81/64]]~~ where ~~as in Ptolemaic~~ it ~~is [[5/4]]~~. ~~Later music theorists, such as~~ {{~~w|Gioseffo Zarlino~~}}<ref>Chisholm, Hugh (1911). ''The Encyclopædia Britannica'', Vol. 28, p. 961. The Encyclopædia Britannica Company.</ref>during the Renaissance, would prefer the Ptolemaic tuning. Tuning systems based on those ratios are called [[just intonation]].

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''' (十二律).{{clear}}

Pythagorean tuning ~~was developed using method called~~ the ~~'chain of fifths'~~, where ~~you multiply~~ the ~~pitch/frequency~~ by ~~a fifth (~~3/2) ~~until you pass an octave~~. ~~When you pass an octave, you take that same note, and move it down an octave by multiplying it by another ratio. Every ratio~~ can be ~~generated by~~ a ~~combination~~ of 3/~~2 and~~ 4/3. ~~One old account of this method is ascribed to an anonymous source in~~ a ~~book by Iacobus de Ispania~~ in the ~~13th~~ century)<ref>~~Schulter~~, ~~Margo “~~[~~https~~:~~//web~~.~~archive~~.~~org/web/20120215000445/http://www~~.~~medieval~~.~~org:80/emfaq/harmony/pyth4~~.~~html Pythagorean Tuning and Medieval Polyphony~~]"</~~ref>~~

== Relation to temperaments ==

Pythagorean tuning can be considered a [[trivial temperament|trivial]] [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the [[3-limit|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 [[extension]]s of Pythagorean tuning.

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. C–F♭) approximates [[5/4]], since the [[schisma]] is so small. Mark Lindley argues such a system was used in Europe during the 15th century<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref>, with keyboards tuned to nearly pure fifths as

: {{dash| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.

This makes triads such as {{dash|D, G♭, A}} ({{dash|D, vF#, A}} more intuitively) very close to [[4:5:6]] in this tuning.

It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] [[Schismatic family#Tridecaschismic (2.3.5.13)|temperament]], as the triple-augmented fourth C– F♯♯♯ is incredibly close to [[13/8]], differing by the [[tridecapyth comma]] which is even smaller than the schisma.

== Scales ==

* [[Pythagorean5]] - proper [[2L 3s]]~~. Also~~ known as pythagorean pentatonic scale

Pythagorean tuning generates the following [[mos|MOS]] scales:

* [[Pythagorean7]] - improper [[5L 2s]]~~. Also~~ known as pythagorean diatonic scale

* [[Pythagorean5]] – proper [[2L 3s]], also known as pentic, the ''pythagorean pentatonic scale''.

* [[Pythagorean12]] - proper [[5L 7s]]~~. Also~~ known as pythagorean chromatic scale

* [[Pythagorean7]] – improper [[5L 2s]], also known as diatonic,the ''pythagorean diatonic scale''.

* [[Pythagorean17]] - improper [[12L 5s]]~~. Also~~ known as pythagorean enharmonic scale

* [[Pythagorean12]] – proper [[5L 7s]], also known as p-chromatic, the ''pythagorean chromatic scale''.

* [[Pythagorean29]] - improper [[12L 17s]]

* [[Pythagorean17]] – improper [[12L 5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''.

* [[Pythagorean41]] - proper [[12L 29s]]

* [[Pythagorean29]] – improper [[12L 17s]], sometimes known as ''pythagotonic''.

* [[Pythagorean53]] - proper [[41L 12s]]

* [[Pythagorean41]] – proper [[12L 29s]], sometimes known as ''pythamystonic.''

* [[Pythagorean53]] – proper [[41L 12s]], sometimes known as ''pythomerc''.

The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, 3.846 for enharmonic, 2.8459 for pythagotonic, 1.8459 for pythamystonic, and 1.1822 for pythomerc. Pythamystonic, pentic, p-chromatic and pythomerc generate the softest MOS scales generated by a just fifth as their equalized tunings represent edos with convergent fifths.

== Approaches ==

There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.

[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice music of the West. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53.

[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.

== Music ==

See [[Music ~~in just intonation~~]].

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 ==

[[Category:3-limit| ]]

[[Category:Rank-2 temperaments]]

[[Category:Historical]]

[[Category:Tuning]]

[[Category:Listen]]

@@ Line 6: / Line 6: @@
 }}
 {{Wikipedia|Pythagorean tuning}}
-The '''Pythagorean tuning''' is the 3-limit version of [[just intonation]].
+'''Pythagorean tuning''' is a system where all intervals are determined by perfect fifths tuned to [[3/2]] and [[2/1|octaves]]. As such, Pythagorean tuning contains the same intervals as [[3-limit]] [[just intonation]],
-See [[3-limit]] for more information.
+When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].
 == History ==
-Pythagorean tuning is a [[Tuning system|system of musical tuning]] based on the mathematical ratios of pitches. It is named after the ancient Greek philosopher {{w|Pythagoras}}, who, according to legend, discovered the foundational principles of this tuning system through an experiment with hammers of different weights. Pythagoras' fascination with numerical ratios and their relation to the cosmos, particularly his concept of the 'music of the spheres', significantly influenced this tuning method.
+{{wikipedia|Music of Mesopotamia#Music theory}}
+{{wikipedia|Shi'er lü}}
+Pythagorean tuning was not actually invented by [[Pythagoras of Samos|Pythagoras]]. The earliest records are from [[Mesopotamian music|Ancient Mesopotamia]], and it was later inherited by the [[Ancient Greek]]s.
-The Greeks used two systems of tuning based on ideal integer ratios: Pythagorean and Ptolemaic. The major difference is, Ptolemaic tuning uses simpler ratios, where as Pythagorean tuning uses a [[chain of fifths|chain of fifths and fourths]]. For example, a major third in Pythagorean would be [[81/64]] where as in Ptolemaic it is [[5/4]]. Later music theorists, such as {{w|Gioseffo Zarlino}}<ref>Chisholm, Hugh (1911). ''The Encyclopædia Britannica'', Vol. 28, p. 961. The Encyclopædia Britannica Company.</ref>during the Renaissance, would prefer the Ptolemaic tuning. Tuning systems based on those ratios are called [[just intonation]].
+The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music|Ancient China]] between 600&nbsp;BCE and 240&nbsp;CE, where it was called '''shi'er lü''' (十二律).{{clear}}
-Pythagorean tuning was developed using method called the 'chain of fifths', where you multiply the pitch/frequency by a fifth (3/2) until you pass an octave. When you pass an octave, you take that same note, and move it down an octave by multiplying it by another ratio. Every ratio can be generated by a combination of 3/2 and 4/3. One old account of this method is ascribed to an anonymous source in a book by Iacobus de Ispania in the 13th century)<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>
+== Relation to temperaments ==
+Pythagorean tuning can be considered a [[trivial temperament|trivial]] [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the [[3-limit|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 [[extension]]s of Pythagorean tuning.
+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. C–F♭) approximates [[5/4]], since the [[schisma]] is so small. Mark Lindley argues such a system was used in Europe during the 15th century<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref>, with keyboards tuned to nearly pure fifths as
+: {{dash| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.
+This makes triads such as {{dash|D, G♭, A}} ({{dash|D, vF#, A}} more intuitively) very close to [[4:5:6]] in this tuning.
+It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] [[Schismatic family#Tridecaschismic (2.3.5.13)|temperament]], as the triple-augmented fourth C–&nbsp;F♯♯♯ is incredibly close to [[13/8]], differing by the [[tridecapyth comma]] which is even smaller than the schisma.
 == Scales ==
-* [[Pythagorean5]] - proper [[2L 3s]]. Also known as pythagorean pentatonic scale
+Pythagorean tuning generates the following [[mos|MOS]] scales:
-* [[Pythagorean7]] - improper [[5L 2s]]. Also known as pythagorean diatonic scale
+* [[Pythagorean5]] – proper [[2L&nbsp;3s]], also known as pentic, the ''pythagorean pentatonic scale''.
-* [[Pythagorean12]] - proper [[5L 7s]]. Also known as pythagorean chromatic scale
+* [[Pythagorean7]] – improper [[5L&nbsp;2s]], also known as diatonic,the ''pythagorean diatonic scale''.
-* [[Pythagorean17]] - improper [[12L 5s]]. Also known as pythagorean enharmonic scale
+* [[Pythagorean12]] – proper [[5L&nbsp;7s]], also known as p-chromatic, the ''pythagorean chromatic scale''.
-* [[Pythagorean29]] - improper [[12L 17s]]
+* [[Pythagorean17]] – improper [[12L&nbsp;5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''.
-* [[Pythagorean41]] - proper [[12L 29s]]
+* [[Pythagorean29]] – improper [[12L&nbsp;17s]], sometimes known as ''pythagotonic''.
-* [[Pythagorean53]] - proper [[41L 12s]]
+* [[Pythagorean41]] – proper [[12L&nbsp;29s]], sometimes known as ''pythamystonic.''
+* [[Pythagorean53]] – proper [[41L&nbsp;12s]], sometimes known as ''pythomerc''.
+The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, 3.846 for enharmonic, 2.8459 for pythagotonic, 1.8459 for pythamystonic, and 1.1822 for pythomerc. Pythamystonic, pentic, p-chromatic and pythomerc generate the softest MOS scales generated by a just fifth as their equalized tunings represent edos with convergent fifths.
+== Approaches ==
+There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.
+[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice music of the West. This gives MOS sizes of 2, 3, 5 ([[2L&nbsp;3s]] pentic), 7 ([[5L&nbsp;2s]] diatonic), 12 ([[5L&nbsp;7s]] chromatic), 17 ([[12L&nbsp;5s]] enharmonic), 29, 41, and 53.
+[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L&nbsp;5s), 11 (8L&nbsp;3s), 19 (8L&nbsp;11s), 27 (19L&nbsp;8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.
 == Music ==
-See [[Music in just intonation]].
+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 ==
+<references/>
 [[Category:3-limit| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
 [[Category:Historical]]
 [[Category:Tuning]]
 [[Category:Listen]]