Diaschismic: Difference between revisions

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== Chords and harmony ==

Diaschismic finds the 5-limit triads very simply, with [[4:5:6|1–5/4–3/2]] (≈0–392–704{{c}}) and [[10:12:15|1–6/5–3/2]] (≈0–312–704{{c}}) each occuring six times in the 12-note [[10L 2s]] mos scale. In optimal tunings, the [[3/2]] perfect fifth and [[5/4]] major third are both tuned slightly sharp, giving major triads an overall bright sound. Unlike in [[meantone]], the syntonic comma [[81/80]] is not tempered out, meaning the [[9/8]] and [[10/9]] whole tones are distinguished. Due to the sharp fifth, the [[Pythagorean tuning|Pythagorean]] major and minor triads are [[neogothic major and minor|neogothic]] in quality, representing [[22:28:33|1–14/11–3/2]] (≈0–416–704{{c}}) and [[22:26:33|1–13/11–3/2]] (≈0–288–704{{c}}) respectively. It can thus be seen as an expansion on the harmony of [[12edo]], which adds new flavors and distinctions.

Prime 17 also plays a role: For example, the interval [[17/8]], which is one octave above [[17/16]], can be used as a minor ninth in chords such as 1–5/4–3/2–17/8 (≈0–392–704{{c}}). Additionally, the semioctave represents [[17/12]]~[[24/17]], and can act as a [[tritone]] in [[diminished triad]]s such as [[85:102:120|1–6/5–24/17]] (≈0–312–600{{c}}) and its melodic inverse [[17:20:24|1–20/17–24/17]] (≈0–288–600{{c}}). The sharp ~9/8 whole tone is equated with a flat ~[[17/15]], and [[20/17]] is equated with [[32/27]]~[[13/11]]. Additionally, the [[dominant seventh chord]] can be seen as representing [[68:85:102:120|1–5/4–3/2–30/17]] (≈0–392–704–992{{c}}), thus becoming [[dyadic chord|dyadically consonant]] in the [[17-odd-limit]], meaning any interval between two notes has an interpretation with an [[odd limit]] of 17 or less. Its {{W|negative harmony}} version is [[10:12:15:17|1–6/5–3/2–17/10]] (≈0–312–704–912{{c}}), a {{W|minor sixth chord}} with a relatively simple otonal signature of 10:12:15:17. The [[2.3.5.17 subgroup|2.3.5.17-subgroup]] is optimized with a somewhat sharper fifth, around 705 cents, close to [[34edo]] or [[80edo]].

== Tunings ==

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 == Chords and harmony ==
 {{See also| Chords of diaschismic }}
+Diaschismic finds the 5-limit triads very simply, with [[4:5:6|1–5/4–3/2]] (≈0–392–704{{c}}) and [[10:12:15|1–6/5–3/2]] (≈0–312–704{{c}}) each occuring six times in the 12-note [[10L 2s]] mos scale. In optimal tunings, the [[3/2]] perfect fifth and [[5/4]] major third are both tuned slightly sharp, giving major triads an overall bright sound. Unlike in [[meantone]], the syntonic comma [[81/80]] is not tempered out, meaning the [[9/8]] and [[10/9]] whole tones are distinguished. Due to the sharp fifth, the [[Pythagorean tuning|Pythagorean]] major and minor triads are [[neogothic major and minor|neogothic]] in quality, representing [[22:28:33|1–14/11–3/2]] (≈0–416–704{{c}}) and [[22:26:33|1–13/11–3/2]] (≈0–288–704{{c}}) respectively. It can thus be seen as an expansion on the harmony of [[12edo]], which adds new flavors and distinctions.
+Prime 17 also plays a role: For example, the interval [[17/8]], which is one octave above [[17/16]], can be used as a minor ninth in chords such as 1–5/4–3/2–17/8 (≈0–392–704{{c}}). Additionally, the semioctave represents [[17/12]]~[[24/17]], and can act as a [[tritone]] in [[diminished triad]]s such as [[85:102:120|1–6/5–24/17]] (≈0–312–600{{c}}) and its melodic inverse [[17:20:24|1–20/17–24/17]] (≈0–288–600{{c}}). The sharp ~9/8 whole tone is equated with a flat ~[[17/15]], and [[20/17]] is equated with [[32/27]]~[[13/11]]. Additionally, the [[dominant seventh chord]] can be seen as representing [[68:85:102:120|1–5/4–3/2–30/17]] (≈0–392–704–992{{c}}), thus becoming [[dyadic chord|dyadically consonant]] in the [[17-odd-limit]], meaning any interval between two notes has an interpretation with an [[odd limit]] of 17 or less. Its {{W|negative harmony}} version is [[10:12:15:17|1–6/5–3/2–17/10]] (≈0–312–704–912{{c}}), a {{W|minor sixth chord}} with a relatively simple otonal signature of 10:12:15:17. The [[2.3.5.17 subgroup|2.3.5.17-subgroup]] is optimized with a somewhat sharper fifth, around 705 cents, close to [[34edo]] or [[80edo]].
 == Tunings ==