Diaschismic - Revision history

Overthink: start scales section

2026-06-21T03:02:23Z

start scales section

← Older revision		Revision as of 03:02, 21 June 2026
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	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 ==

Overthink: mention 17 earlier

2026-06-21T02:54:20Z

mention 17 earlier

← Older revision		Revision as of 02:54, 21 June 2026
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	\| 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 ~~minus~~ a whole tone ~~represents~~ [[5/4]].		'''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]].

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), ~~in this tuning range~~, is close in size to a quartertone, and that plus a ~~period~~ can be used to represent [[16/11]]. Three more fifths on top of 16/11 ~~give~~ [[16/13]]. 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.		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.

	Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S\|16}}), [[289/288]] ({{S\|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup\|2.3.5.17-subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.		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 ~~as in srutal archagall~~. 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 representation of the 7-limit.

	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.

Overthink: link this way

2026-06-21T02:42:20Z

link this way

← Older revision		Revision as of 02:42, 21 June 2026
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	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.

	Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S\|16}}), [[289/288]] ({{S\|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup\|2.3.5.17-~~]][[~~subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.		Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S\|16}}), [[289/288]] ({{S\|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup\|2.3.5.17-subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.

	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 as in srutal archagall. 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 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 as in srutal archagall. 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 representation of the 7-limit.

Overthink: /* As a detemperament of 12et */ add more info

2026-06-21T02:41:30Z

As a detemperament of 12et: add more info

← Older revision		Revision as of 02:41, 21 June 2026
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	[[File: Diaschismic 12et Detempering.png\|thumb\|Diaschismic as a 58-tone 12et detempering]]		[[File: Diaschismic 12et Detempering.png\|thumb\|Diaschismic as a 58-tone 12et detempering]]

	Diaschismic is naturally considered as a [[detemperament]] of the [[12edo\|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.		Diaschismic is naturally considered as a [[detemperament]] of the [[12edo\|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14 steps. 58 is the largest number of tones for a [[mos]] where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region. In general, ratios closer to intervals of 12edo, such as [[17/16]], are simpler in terms of generator steps, while ratios further from 12edo intervals, such as [[11/9]], are more complex. The 13th harmonic is just beyond the specified generator range, so it does not appear in the diagram.

	The 13th harmonic is just beyond the specified generator range, so the diagram does not show it. It is found as a comma step below [[28/17]], or alternatively [[16/13]] is found as the [[fifth complement]] of [[11/9]].		Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise, though it requires using the sharp, second-best approximation of [[5/4]] (aka the 104c [[val]]).

	Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise.
	{{Clear}}		{{Clear}}

Overthink: /* Chords and harmony */ expand

2026-06-21T02:35:45Z

Chords and harmony: expand

← Older revision		Revision as of 02:35, 21 June 2026
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	== Chords and harmony ==		== Chords and harmony ==
	{{See also\| Chords of diaschismic }}		{{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 ==		== Tunings ==

Overthink: /* As a detemperament of 12et */ add info

2026-06-21T01:44:01Z

As a detemperament of 12et: add info

← Older revision		Revision as of 01:44, 21 June 2026
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	[[File: Diaschismic 12et Detempering.png\|thumb\|Diaschismic as a 58-tone 12et detempering]]		[[File: Diaschismic 12et Detempering.png\|thumb\|Diaschismic as a 58-tone 12et detempering]]

	Diaschismic is naturally considered as a [[detemperament]] of the [[12edo\|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 ~~generator steps~~, which ~~represent~~ the generic comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.		Diaschismic is naturally considered as a [[detemperament]] of the [[12edo\|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.

	The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.		The 13th harmonic is just beyond the specified generator range, so the diagram does not show it. It is found as a comma step below [[28/17]], or alternatively [[16/13]] is found as the [[fifth complement]] of [[11/9]].

	Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.		Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise.
	{{Clear}}		{{Clear}}

Overthink: explain this differently; - duplicate info

2026-06-21T01:43:04Z

explain this differently; - duplicate info

← Older revision		Revision as of 01:43, 21 June 2026
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	'''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 minus a whole tone represents [[5/4]].		'''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 minus a whole tone represents [[5/4]].

	The canonical [[extension]] to the [[7-limit]] ~~lies roughly between~~ [[~~46edo]] and [[58edo]], such that stacking eight of them [[octave reduced~~]] (~~an augmented fifth~~) ~~and down~~ a semioctave ~~approximates [[8/7]]~~, 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) 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.

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.

Overthink: explain some stuff differently

2026-06-21T01:35:18Z

explain some stuff differently

← Older revision		Revision as of 01:35, 21 June 2026
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	\| 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]] 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 ~~we also have~~ a whole tone ~~plus a period represent~~ [[8/5]].		'''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 minus a whole tone represents [[5/4]].

	The canonical [[extension]] to the [[7-limit]] lies ~~where the fifth is tuned a little sharp~~ such that eight of them octave reduced (an augmented fifth) ~~minus~~ a ~~period approximate~~ [[8/7]], tempering out the starling comma, [[126/125]], as well as the ~~argent comma~~, [[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]] lies roughly between [[46edo]] and [[58edo]], such that stacking eight of them [[octave reduced]] (an augmented fifth) and down a semioctave approximates [[8/7]], 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.

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.

FloraC: Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

2026-06-06T12:39:48Z

Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

← Older revision		Revision as of 12:39, 6 June 2026
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	[[Category:Diaschismic family]]		[[Category:Diaschismic family]]
	[[Category:Starling temperaments]]		[[Category:Starling temperaments]]
	[[Category:~~Hemifamity~~ temperaments]]		[[Category:Aberschismic temperaments]]

FloraC at 10:18, 9 February 2026

2026-02-09T10:18:51Z

← Older revision		Revision as of 10:18, 9 February 2026
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	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Diaschismic		\| Title = Diaschismic
	\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17		\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
	\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)		\| Comma basis = [[2048/2025]] (5-limit); <br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)
	\| Edo join 1 = 46 \| Edo join 2 = 58		\| Edo join 1 = 46 \| Edo join 2 = 58
			\| Mapping = 2; 1 -2 -8 -12 -15 1
	\| Generators = 3/2 \| Generators tuning = 703.9 \| Optimization method = CWE		\| Generators = 3/2 \| Generators tuning = 703.9 \| Optimization method = CWE
	\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]		\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
	~~\| Mapping = 2; 1 -2 -8 -12 -15 1~~
	\| Pergen = (P8/2, P5)		\| Pergen = (P8/2, P5)
	\| Color name = Saguguti		\| Color name = Saguguti

← Older revision		Revision as of 02:54, 21 June 2026
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	\| 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 ~~minus~~ a whole tone ~~represents~~ [[5/4]].		'''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]].

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), ~~in this tuning range~~, is close in size to a quartertone, and that plus a ~~period~~ can be used to represent [[16/11]]. Three more fifths on top of 16/11 ~~give~~ [[16/13]]. 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.		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.

	Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S\|16}}), [[289/288]] ({{S\|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup\|2.3.5.17-subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.		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 ~~as in srutal archagall~~. 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 representation of the 7-limit.

	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.

← Older revision		Revision as of 01:43, 21 June 2026
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	'''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 minus a whole tone represents [[5/4]].		'''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 minus a whole tone represents [[5/4]].

	The canonical [[extension]] to the [[7-limit]] ~~lies roughly between~~ [[~~46edo]] and [[58edo]], such that stacking eight of them [[octave reduced~~]] (~~an augmented fifth~~) ~~and down~~ a semioctave ~~approximates [[8/7]]~~, 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) 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.

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.

← Older revision		Revision as of 01:35, 21 June 2026
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	\| 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]] 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 ~~we also have~~ a whole tone ~~plus a period represent~~ [[8/5]].		'''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 minus a whole tone represents [[5/4]].

	The canonical [[extension]] to the [[7-limit]] lies ~~where the fifth is tuned a little sharp~~ such that eight of them octave reduced (an augmented fifth) ~~minus~~ a ~~period approximate~~ [[8/7]], tempering out the starling comma, [[126/125]], as well as the ~~argent comma~~, [[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]] lies roughly between [[46edo]] and [[58edo]], such that stacking eight of them [[octave reduced]] (an augmented fifth) and down a semioctave approximates [[8/7]], 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.

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. 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.