Diaschismic: Difference between revisions
→As a detemperament of 12et: add more info |
start scales section |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 12: | Line 12: | ||
| Odd limit 2 = 17-limit 21 | Mistuning 2 = ??? | Complexity 2 = 46 | | Odd limit 2 = 17-limit 21 | Mistuning 2 = ??? | Complexity 2 = 46 | ||
}} | }} | ||
'''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]], ideally tuned slightly sharp, or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out|tempered out]], and a half-octave | '''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]], ideally tuned slightly sharp, or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]] = 9/8 ÷ (16/15)<sup>2</sup>, is [[tempering out|tempered out]], and a half-octave represents [[45/32]], and subtracting a whole tone from that gives [[5/4]]. Since the whole tone has been split in two equal halves of [[~]]16/15, it makes sense to equate that semitone with [[17/16]] and [[18/17]], by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]], thus giving the [[2.3.5.17 subgroup|2.3.5.17-subgroup]] version of diaschismic, sometimes known as ''srutal archagall''. | ||
The canonical [[extension]] to the [[7-limit]] reaches [[7/4]] at -8 fifths (a diminished fourth) plus a semioctave, tempering out the starling comma, [[126/125]], as well as the aberschisma, [[5120/5103]]. This equates the [[64/63|septimal comma]] with the [[81/80|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. | The canonical [[extension]] to the [[7-limit]] reaches [[7/4]] at -8 fifths (a diminished fourth) [[octave-reduced]] plus a semioctave, tempering out the starling comma, [[126/125]], as well as the aberschisma, [[5120/5103]]. This equates the [[64/63|septimal comma]] with the [[81/80|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. This extension works best between [[46edo]] and [[58edo]]. | ||
In this tuning range, a stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), or two comma steps, is close in size to a [[quartertone]], and that interval plus a half-octave can be used to represent [[16/11]]. Three more fifths on top of 16/11 gives [[16/13]] when octave-reduced, and thus [[13/11]] is equated with the Pythagorean minor third [[32/27]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp. Adding the mapping of 17/16 to the semitone, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. While this provides much more harmonic resources, the 2.3.5.17-subgroup version is simpler and has a more flexible tuning range. | |||
An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is again mapped to the semitone. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler (and arguably more elegant) representation of the 7-limit. | |||
An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped | |||
See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic|#Septimal diaschismic]] for technical data. | See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic|#Septimal diaschismic]] for technical data. | ||
| Line 162: | Line 160: | ||
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]]. | 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]]. | ||
== Scales == | |||
{{Todo|complete section}} | |||
=== 10-note (proper) === | |||
{{Main| 2L 8s }} | |||
=== 12-note (proper) === | |||
{{Main| 10L 2s }} | |||
== Tunings == | == Tunings == | ||