Omnitetrachordality - Revision history

VectorGraphics at 00:06, 28 June 2025

2025-06-28T00:06:25Z

← Older revision		Revision as of 00:06, 28 June 2025
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	All omnitetrachordal scales are of the form ABA, where A is any word spanning an approximation of 4/3 and B is any word spanning an approximation of 9/8.		All omnitetrachordal scales are of the form ABA, where A is any word spanning an approximation of 4/3 and B is any word spanning an approximation of 9/8.

	{{Theorem\|contents=All strongly omnitetrachordal scales are of the form ABABA, where A is any [[word]] spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.}}		{{Theorem\|contents=All strongly omnitetrachordal scales are of the form ABABA, where A is any [[word]] (an abstract scale pattern, without regard to relative step sizes) spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.}}
	{{Proof\|contents=An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.		{{Proof\|contents=An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.

FloraC: Clarify that this doesn't require four-note segments

2025-01-29T11:52:21Z

Clarify that this doesn't require four-note segments

← Older revision		Revision as of 11:52, 29 January 2025
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	A [[scale]] is '''weakly omnitetrachordal''' if any [[mode]] of the scale (that is, any particular octave span of the ~~infinite~~ scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[4/3]], plus a [[9/8]] that may or may not be divided into smaller steps (a "disjunction"). The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.		An [[period\|octave-repeating]] [[scale]] is '''weakly omnitetrachordal''' if any [[mode]] of the scale (that is, any particular octave span of the scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[4/3]], plus a [[9/8]] that may or may not be divided into smaller steps (a "disjunction"). Although the property is called ''omnitetrachordality'', it does not require exactly four notes (three steps) in the sequence. The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.

	If, for each mode of the scale, the entire pattern of scale steps within the disjunction also occurs within the tetrachord, the scale is '''strongly omnitetrachordal''' ([[Paul Erlich]]'s original definition of omnitetrachordal). For example, the scale "abc abc ac" is weakly omnitetrachordal but not strongly omnitetrachordal, because the disjunction "ac" does not occur in the tetrachords "abc".		If, for each mode of the scale, the entire pattern of scale steps within the disjunction also occurs within the tetrachord, the scale is '''strongly omnitetrachordal''' ([[Paul Erlich]]'s original definition of omnitetrachordal). For example, the scale "abc abc ac" is weakly omnitetrachordal but not strongly omnitetrachordal, because the disjunction "ac" does not occur in the tetrachords "abc".

FloraC: Style and formatting improvements

2025-01-29T11:42:54Z

Style and formatting improvements

← Older revision		Revision as of 11:42, 29 January 2025
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	A scale is '''weakly omnitetrachordal''' if any mode of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[~~4/3\|~~4/3]], plus a [[~~9/8\|~~9/8]] that may or may not be divided into smaller steps (a "disjunction"). The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.		A [[scale]] is '''weakly omnitetrachordal''' if any [[mode]] of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[4/3]], plus a [[9/8]] that may or may not be divided into smaller steps (a "disjunction"). The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.

	If, for each mode of the scale, the entire pattern of scale steps within the disjunction also occurs within the tetrachord, the scale is '''strongly omnitetrachordal''' (Paul Erlich's original definition of omnitetrachordal). For example, the scale "abc abc ac" is weakly omnitetrachordal but not strongly omnitetrachordal, because the disjunction "ac" does not occur in the tetrachords "abc."		If, for each mode of the scale, the entire pattern of scale steps within the disjunction also occurs within the tetrachord, the scale is '''strongly omnitetrachordal''' ([[Paul Erlich]]'s original definition of omnitetrachordal). For example, the scale "abc abc ac" is weakly omnitetrachordal but not strongly omnitetrachordal, because the disjunction "ac" does not occur in the tetrachords "abc".

	There are other related definitions in between the above two, that all differ by the restrictions placed on the disjunction, for example:		There are other related definitions in between the above two, that all differ by the restrictions placed on the disjunction, for example:
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	All omnitetrachordal scales are of the form ABA, where A is any word spanning an approximation of 4/3 and B is any word spanning an approximation of 9/8.		All omnitetrachordal scales are of the form ABA, where A is any word spanning an approximation of 4/3 and B is any word spanning an approximation of 9/8.

	~~'''~~Theorem~~:'''~~ All strongly omnitetrachordal scales are of the form ABABA, where A is any word spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.		{{Theorem\|contents=All strongly omnitetrachordal scales are of the form ABABA, where A is any [[word]] spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.}}
			{{Proof\|contents=An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.
	~~'''~~Proof~~:'''~~ An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.

	Any word may be written as a product of three words: ABC (where any of A, B, or C may be empty).		Any word may be written as a product of three words: ABC (where any of A, B, or C may be empty).
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	Clearly ADADA and CDCDC are equivalent in form to ABABA and therefore all strongly OTC scales have the form ABABA. For ABABA to be tetrachordal, it is also required for A to span an approximation of 9/8 and ABA to span an approximation of 4/3. Therefore B must span an approximation of 32/27.		Clearly ADADA and CDCDC are equivalent in form to ABABA and therefore all strongly OTC scales have the form ABABA. For ABABA to be tetrachordal, it is also required for A to span an approximation of 9/8 and ABA to span an approximation of 4/3. Therefore B must span an approximation of 32/27.
			}}

	Any [[~~MOS Cradle Scales\|MOS Cradle Scale~~]] with parent scale ababa, or any [[SN scale]] generated from ababa, is strongly omnitetrachordal.		Any [[mos cradle]] scale with parent scale ababa, or any [[SN scale]] generated from ababa, is strongly omnitetrachordal.

	Below we will look at strongly omnitetrachordal scales.		Below we will look at strongly omnitetrachordal scales.

	== Strongly ~~Omnitetrachordal Examples~~ ==		== Strongly omnitetrachordal examples ==
			This definition could be difficult to understand, so take the [[5L 2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.
	This definition could be difficult to understand, so take the [[~~5L_2s\|~~5L+2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.

	C D E F G A B C = (C D E F) + 9/8 + (G A B C) (both tetrachords are LLs)		C D E F G A B C = (C D E F) + 9/8 + (G A B C) (both tetrachords are LLs)
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	Since each mode can be expressed as two tetrachords each spanning 4/3 and a leftover 9/8 (some in more than one way), the diatonic scale is omnitetrachordal.		Since each mode can be expressed as two tetrachords each spanning 4/3 and a leftover 9/8 (some in more than one way), the diatonic scale is omnitetrachordal.

	If you understand [[~~MOSScales\|MOS scales~~]] well it should be clear that any ~~MOS~~ of a temperament in which the period represents 2/1 and the generator represents 4/3 (including [[~~Meantone\|~~meantone]], [[~~Mavila\|~~mavila]], [[~~Superpyth\|~~superpyth]], [[~~Schismatic_family\|~~schismatic]], etc.) will be omnitetrachordal. However, these are not the only possible omnitetrachordal scales. For an example of a different kind of omnitetrachordal scale, take the [[~~MODMOS\|MODMOS~~]] of the [[~~2L_8s\|~~2L+8s]] scale (in [[~~pajara\|~~pajara]] ~~for example)~~ with the pattern LsssLsssss.		If you understand [[mos scale]]s well it should be clear that any mos of a temperament in which the period represents 2/1 and the generator represents 4/3 (including [[meantone]], [[mavila]], [[superpyth]], [[schismatic]], etc.) will be omnitetrachordal. However, these are not the only possible omnitetrachordal scales. For an example of a different kind of omnitetrachordal scale, take the [[modmos]] of the [[2L 8s]] scale in [[pajara]] with the pattern LsssLsssss.

	(Lsss)(Lsss)(ss)		(Lsss)(Lsss)(ss)
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	(sLss)(sLss)(ss)		(sLss)(sLss)(ss)

	In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific ~~MODMOS~~ of pajara was named the "pentachordal decatonic scale" by ~~[[Paul_Erlich\|~~Paul Erlich]] (who is believed to have originated the concept of omnitetrachordality, circa 1993). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).		In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific modmos of pajara was named the "pentachordal decatonic scale" by Paul Erlich (who is believed to have originated the concept of omnitetrachordality, circa 1993). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).

	See also [[~~Gallery_of_omnitetrachordal_scales\|~~Gallery of omnitetrachordal scales]].		== See also ==
			* [[Gallery of omnitetrachordal scales]]

	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]

Inthar at 22:31, 9 February 2024

2024-02-09T22:31:05Z

← Older revision		Revision as of 22:31, 9 February 2024
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	[[Category:Scale]]		[[Category:Scale]]
	[[Category:~~Articles~~ with proofs]]		[[Category:Pages with proofs]]

BudjarnLambeth: Category:Articles with proofs

2024-01-29T01:15:41Z

Category:Articles with proofs

← Older revision		Revision as of 01:15, 29 January 2024
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	[[Category:Scale]]		[[Category:Scale]]
			[[Category:Articles with proofs]]

Fredg999: Categories

2022-03-25T15:04:30Z

Lhearne: Fixed up my theorems and proofs by changin 'any word' to 'any word spanning an approximation of...

2022-02-06T12:45:10Z

Fixed up my theorems and proofs by changin 'any word' to 'any word spanning an approximation of...

← Older revision		Revision as of 12:45, 6 February 2022
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	* That the entire disjunction must simply occur elsewhere in the scale, not within an individual tetrachord (so "aa aba aba" is allowed, since the "aa" occurs between the two tetrachords, but "abc abc ac" not allowed)		* That the entire disjunction must simply occur elsewhere in the scale, not within an individual tetrachord (so "aa aba aba" is allowed, since the "aa" occurs between the two tetrachords, but "abc abc ac" not allowed)
	* That only each individual step size in the disjunction must occur elsewhere in the scale (so "abc abc ac" is now allowed, but not "ab ab ac"		* That only each individual step size in the disjunction must occur elsewhere in the scale (so "abc abc ac" is now allowed, but not "ab ab ac"
	All omnitetrachordal scales are of the form ABA, where A and B ~~are~~ any ~~words~~.		All omnitetrachordal scales are of the form ABA, where A is any word spanning an approximation of 4/3 and B is any word spanning an approximation of 9/8.

	'''Theorem:''' All strongly omnitetrachordal scales are of the form ABABA, where A and B ~~are~~ any ~~words:~~		'''Theorem:''' All strongly omnitetrachordal scales are of the form ABABA, where A is any word spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.

	'''Proof:''' An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.		'''Proof:''' An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.
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	If the disjunction is C, then the scale is ABCABCC in one of its modes. Writing AB as D, we have DCDCC, or in another mode, CDCDC.		If the disjunction is C, then the scale is ABCABCC in one of its modes. Writing AB as D, we have DCDCC, or in another mode, CDCDC.

	Clearly ADADA and CDCDC are equivalent in form to ABABA, and therefore all strongly OTC scales have the form ABABA, ~~where~~ A and B ~~are any words~~.		Clearly ADADA and CDCDC are equivalent in form to ABABA and therefore all strongly OTC scales have the form ABABA. For ABABA to be tetrachordal, it is also required for A to span an approximation of 9/8 and ABA to span an approximation of 4/3. Therefore B must span an approximation of 32/27.

	Any [[MOS Cradle Scales\|MOS Cradle Scale]] with parent scale ababa, or any [[SN scale]] generated from ababa, is strongly omnitetrachordal.		Any [[MOS Cradle Scales\|MOS Cradle Scale]] with parent scale ababa, or any [[SN scale]] generated from ababa, is strongly omnitetrachordal.

Lhearne: /* Strongly Omnitetrachordal Examples */ changed year of origin of the concept from 2002 and 1993 after advice from Paul

2021-05-05T04:58:41Z

Strongly Omnitetrachordal Examples: changed year of origin of the concept from 2002 and 1993 after advice from Paul

← Older revision		Revision as of 04:58, 5 May 2021
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	(sLss)(sLss)(ss)		(sLss)(sLss)(ss)

	In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific MODMOS of pajara was named the "pentachordal decatonic scale" by [[Paul_Erlich\|Paul Erlich]] (who is believed to have originated the concept of omnitetrachordality, circa ~~2002~~). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).		In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific MODMOS of pajara was named the "pentachordal decatonic scale" by [[Paul_Erlich\|Paul Erlich]] (who is believed to have originated the concept of omnitetrachordality, circa 1993). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).

	See also [[Gallery_of_omnitetrachordal_scales\|Gallery of omnitetrachordal scales]].		See also [[Gallery_of_omnitetrachordal_scales\|Gallery of omnitetrachordal scales]].

Lhearne: Added proof of strong omnitetrachordality

2021-05-03T13:25:03Z

Added proof of strong omnitetrachordality

@@ Line 6: / Line 6: @@
 * That the entire disjunction must simply occur elsewhere in the scale, not within an individual tetrachord (so "aa aba aba" is allowed, since the "aa" occurs between the two tetrachords, but "abc abc ac" not allowed)
-* That only each individual step size in the disjunction must occur elsewhere in the scale (so "abc abc ac" is now allowed, but not "ab ab ac")
+* That only each individual step size in the disjunction must occur elsewhere in the scale (so "abc abc ac" is now allowed, but not "ab ab ac"
 Below we will look at strongly omnitetrachordal scales.

Mike Battaglia: new definitions

2019-02-28T19:31:47Z

new definitions

← Older revision		Revision as of 19:31, 28 February 2019
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	A scale is '''omnitetrachordal''' if any mode of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[4/3\|4/3]], plus a [[9/8\|9/8]] that may or may not be divided into smaller steps. The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.		A scale is '''weakly omnitetrachordal''' if any mode of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[4/3\|4/3]], plus a [[9/8\|9/8]] that may or may not be divided into smaller steps (a "disjunction"). The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.

			If, for each mode of the scale, the entire pattern of scale steps within the disjunction also occurs within the tetrachord, the scale is '''strongly omnitetrachordal''' (Paul Erlich's original definition of omnitetrachordal). For example, the scale "abc abc ac" is weakly omnitetrachordal but not strongly omnitetrachordal, because the disjunction "ac" does not occur in the tetrachords "abc."

			There are other related definitions in between the above two, that all differ by the restrictions placed on the disjunction, for example:

			* That the entire disjunction must simply occur elsewhere in the scale, not within an individual tetrachord (so "aa aba aba" is allowed, since the "aa" occurs between the two tetrachords, but "abc abc ac" not allowed)
			* That only each individual step size in the disjunction must occur elsewhere in the scale (so "abc abc ac" is now allowed, but not "ab ab ac")

			Below we will look at strongly omnitetrachordal scales.

			== Strongly Omnitetrachordal Examples ==

	This definition could be difficult to understand, so take the [[5L_2s\|5L+2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.		This definition could be difficult to understand, so take the [[5L_2s\|5L+2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.

← Older revision		Revision as of 15:04, 25 March 2022
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	See also [[Gallery_of_omnitetrachordal_scales\|Gallery of omnitetrachordal scales]].		See also [[Gallery_of_omnitetrachordal_scales\|Gallery of omnitetrachordal scales]].

			[[Category:Scale]]