Superpyth: Difference between revisions

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<nowiki/>* In 7-limit CWE tuning, octave reduced

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== Notation ==

Because superpyth is generated by an octave and a fifth, [[chain-of-fifths notation]] can be used. However, if one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth does not temper out [[81/80]]. For example, the [[4:5:6|classical major triad]] on C is written as C–D♯–G rather than C–E–G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C–vE–G and the minor triad as C–^E♭–G. 81/80 is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the equivalences {{nowrap| ^C {{=}} D♭ }}, {{nowrap| E {{=}} vF }}, etc. The limma (C–D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C–C♯) as in meantone, which is a submajor second in size in superpyth.

== Chords and harmony ==

Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the ~~[[4:5:6|~~1–5/4–3/2]] and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads is not as ~~expressive~~ as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone.

Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the 1–5/4–3/2 and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads is not as effective as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone.

If one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth does not temper out [[81/80]]. For example, the major triad on C is written as C–D♯–G rather than C–E–G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C–vE–G and the minor triad as C–^E♭–G. 81/80 is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the enharmonic equivalences {{nowrap| ^C {{=}} D♭ }}, {{nowrap| E {{=}} vF }}, etc. The limma (C–D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C–C♯) as in meantone, which is a submajor second in size in superpyth.

Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. [[Chromatic]] or [[enharmonic]] alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2.

@@ Line 121: / Line 121: @@
 <nowiki/>* In 7-limit CWE tuning, octave reduced
 </div></div>
+== Notation ==
+Because superpyth is generated by an octave and a fifth, [[chain-of-fifths notation]] can be used. However, if one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth does not temper out [[81/80]]. For example, the [[4:5:6|classical major triad]] on C is written as C–D♯–G rather than C–E–G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C–vE–G and the minor triad as C–^E♭–G. 81/80 is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the equivalences {{nowrap| ^C {{=}} D♭ }}, {{nowrap| E {{=}} vF }}, etc. The limma (C–D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C–C♯) as in meantone, which is a submajor second in size in superpyth.
 == Chords and harmony ==
 {{See also| Chords of superpyth }}
-Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the [[4:5:6|1–5/4–3/2]] and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads is not as expressive as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone.
+Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the 1–5/4–3/2 and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads is not as effective as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone.
-If one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth does not temper out [[81/80]]. For example, the major triad on C is written as C–D♯–G rather than C–E–G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C–vE–G and the minor triad as C–^E♭–G. 81/80 is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the enharmonic equivalences {{nowrap| ^C {{=}} D♭ }}, {{nowrap| E {{=}} vF }}, etc. The limma (C–D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C–C♯) as in meantone, which is a submajor second in size in superpyth.
 Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. [[Chromatic]] or [[enharmonic]] alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2.