Pythagorean tuning: Difference between revisions

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== 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~~.

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. ~~{{dash|C, F♭}}~~) approximates [[5/4]], since 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. 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

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♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.

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

When respelled enharmonically, triads such as ~~{{dash|D, F♯, A}}~~ are very close to [[4:5:6]] in this tuning.

When respelled enharmonically, triads such as D–F♯–A are 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~~]] temperament, as the triple-augmented fourth ~~{{dash|C, F♯♯♯}}~~ is incredibly close to [[13/8]], differing by the [[~~Tridecapyth~~ comma]] which is even smaller than the schisma.

It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] 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 ==

@@ Line 18: / Line 18: @@
 == 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.
+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. {{dash|C, F♭}}) approximates [[5/4]], since 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. 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
-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♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.
+: {{dash| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.
-When respelled enharmonically, triads such as {{dash|D, F♯, A}} are very close to [[4:5:6]] in this tuning.
+When respelled enharmonically, triads such as D–F♯–A are 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]] temperament, as the triple-augmented fourth {{dash|C, F♯♯♯}} is incredibly close to [[13/8]], differing by the [[Tridecapyth comma]] which is even smaller than the schisma.
+It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] 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 ==