IFDO - Revision history

BudjarnLambeth: Link

2025-06-07T07:59:38Z

Link

← Older revision		Revision as of 07:59, 7 June 2025
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	== Relation to utonality and subharmonic series ==		== Relation to utonality and subharmonic series ==
	We can consider an IFDO system as a [[Otonality and utonality\|utonal system]]. ''Utonality'' is a term introduced by [[Harry Partch]] to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "[~~http://tonalsoft.com/enc/n/nexus.aspx Numerary~~ nexus]".		We can consider an IFDO system as a [[Otonality and utonality\|utonal system]]. ''Utonality'' is a term introduced by [[Harry Partch]] to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "[[numerary nexus]]".

	== Properties ==		== Properties ==

FloraC: /* Properties */

2025-01-08T12:39:06Z

Properties

← Older revision		Revision as of 12:39, 8 January 2025
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	== Properties ==		== Properties ==
	* ''n''-ifdo has [[maximum variety]] ''n''.		* ''n''-ifdo has [[maximum variety]] ''n''.
	* Except for 1ifdo and 2ifdo, ~~AFDOs~~ are [[chiral]]. The inverse of ''n''-ifdo is ''n''-afdo.		* Except for 1ifdo and 2ifdo, IFDOs are [[chiral]]. The inverse of ''n''-ifdo is ''n''-afdo.
	** 1ifdo is equivalent to 1afdo and 1edo;		** 1ifdo is equivalent to 1afdo and 1edo;
	** 2ifdo is equivalent to 2afdo.		** 2ifdo is equivalent to 2afdo.

	== Individual pages for IFDOs ==		== Individual pages for IFDOs ==

FloraC: +some properties

2025-01-08T12:32:17Z

+some properties

← Older revision		Revision as of 12:32, 8 January 2025
Line 45:		Line 45:
	== Relation to utonality and subharmonic series ==		== Relation to utonality and subharmonic series ==
	We can consider an IFDO system as a [[Otonality and utonality\|utonal system]]. ''Utonality'' is a term introduced by [[Harry Partch]] to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "[http://tonalsoft.com/enc/n/nexus.aspx Numerary nexus]".		We can consider an IFDO system as a [[Otonality and utonality\|utonal system]]. ''Utonality'' is a term introduced by [[Harry Partch]] to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "[http://tonalsoft.com/enc/n/nexus.aspx Numerary nexus]".

			== Properties ==
			* ''n''-ifdo has [[maximum variety]] ''n''.
			* Except for 1ifdo and 2ifdo, AFDOs are [[chiral]]. The inverse of ''n''-ifdo is ''n''-afdo.
			** 1ifdo is equivalent to 1afdo and 1edo;
			** 2ifdo is equivalent to 2afdo.

	== Individual pages for IFDOs ==		== Individual pages for IFDOs ==

	=== By size ===		=== By size ===
	{\| class="wikitable center-all"		{\| class="wikitable center-all"

FloraC: Note the potential difference from undertone scales

2024-10-15T12:54:11Z

Note the potential difference from undertone scales

← Older revision		Revision as of 12:54, 15 October 2024
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	For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All IFDOs are subsets of [[just intonation]], and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).		For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All IFDOs are subsets of [[just intonation]], and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).

	When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.		When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. However, an undertone scale often has an assumption of a tonic whereas an IFDO simply describes all the theoretically available pitch relations. Therefore, a passage built on 1/(12::22) could be said to be in Mode 12, but is technically covered by 11ifdo.

			An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

	== Formula ==		== Formula ==

BudjarnLambeth: /* By prime family */

2024-03-09T20:04:22Z

By prime family

← Older revision		Revision as of 20:04, 9 March 2024
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	'''~~Over~~-2''': {{IFDOs\|2, 4, 8, 16, 32, 64, 128, 256, 512}}		'''Under-2''': {{IFDOs\|2, 4, 8, 16, 32, 64, 128, 256, 512}}


	'''~~Over~~-2''': {{IFDOs\|3, 6, 12, 24, 48, 96, 192, 384}}		'''Under-3''': {{IFDOs\|3, 6, 12, 24, 48, 96, 192, 384}}


	'''~~Over~~-5''': {{IFDOs\|5, 10, 20, 40, 80, 160, 320}}		'''Under-5''': {{IFDOs\|5, 10, 20, 40, 80, 160, 320}}


	'''~~Over~~-7''': {{IFDOs\|7, 14, 28, 56, 112, 224, 448}}		'''Under-7''': {{IFDOs\|7, 14, 28, 56, 112, 224, 448}}


	'''~~Over~~-11''': {{IFDOs\|11, 22, 44, 88, 176, 352}}		'''Under-11''': {{IFDOs\|11, 22, 44, 88, 176, 352}}


	'''~~Over~~-13''': {{IFDOs\|13, 26, 52, 104, 208, 416}}		'''Under-13''': {{IFDOs\|13, 26, 52, 104, 208, 416}}


	'''~~Over~~-17''': {{IFDOs\|17, 34, 68, 136, 272, 544}}		'''Under-17''': {{IFDOs\|17, 34, 68, 136, 272, 544}}


	'''~~Over~~-19''': {{IFDOs\|19, 38, 76, 152, 304}}		'''Under-19''': {{IFDOs\|19, 38, 76, 152, 304}}


	'''~~Over~~-23''': {{IFDOs\|23, 46, 92, 184, 368}}		'''Under-23''': {{IFDOs\|23, 46, 92, 184, 368}}


	'''~~Over~~-29''': {{IFDOs\|29, 58, 116, 232, 464}}		'''Under-29''': {{IFDOs\|29, 58, 116, 232, 464}}


	'''~~Over~~-31''': {{IFDOs\|31, 62, 124, 248, 496}}		'''Under-31''': {{IFDOs\|31, 62, 124, 248, 496}}

	=== By other properties ===		=== By other properties ===

BudjarnLambeth: /* Individual pages for IFDOs */ Changed to match AFDO page

2024-03-09T20:03:27Z

Individual pages for IFDOs: Changed to match AFDO page

Show changes

FloraC: Move everything from the edl page with various improvements

2023-10-12T12:20:30Z

Move everything from the edl page with various improvements

@@ Line 11: / Line 11: @@
 <math>\displaystyle c = (2n)/(2n - k)</math>
 == Individual pages for IFDOs ==
@@ Line 24: / Line 56: @@
 ** [[AFDO]] – arithmetic frequency division of the octave
 ** [[EDO]] – equal division of the octave
 == Notes ==

FloraC: There's not a natural definition of non-integer ifdos

2023-10-12T11:46:01Z

There's not a natural definition of non-integer ifdos

← Older revision		Revision as of 11:46, 12 October 2023
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	The inverse-arithmetic progression is known in general mathematics as the {{W\|Harmonic progression (mathematics)\|harmonic progression}}, but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W\|arithmetic progression}}.		The inverse-arithmetic progression is known in general mathematics as the {{W\|Harmonic progression (mathematics)\|harmonic progression}}, but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W\|arithmetic progression}}.

	For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All ~~integer~~ IFDOs are subsets of [[just intonation]], and up to transposition, any ~~integer~~ IFDO is a superset of a smaller ~~integer~~ IFDO and a subset of a larger ~~integer~~ IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).		For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All IFDOs are subsets of [[just intonation]], and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).

	When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.		When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

FloraC: Oops, wiki markup error

2023-10-08T16:45:22Z

Oops, wiki markup error

← Older revision		Revision as of 16:45, 8 October 2023
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	An '''IFDO''' ('''inverse-arithmetic frequency division of the octave'''), or '''UDO''' ('''utonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic progression of frequency.		An '''IFDO''' ('''inverse-arithmetic frequency division of the octave'''), or '''UDO''' ('''utonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic progression of frequency.

	The inverse-arithmetic progression is known in general mathematics as the {{W\|Harmonic progression (mathematics)\|harmonic progression]], but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W\|arithmetic progression}}.		The inverse-arithmetic progression is known in general mathematics as the {{W\|Harmonic progression (mathematics)\|harmonic progression}}, but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W\|arithmetic progression}}.

	For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All integer IFDOs are subsets of [[just intonation]], and up to transposition, any integer IFDO is a superset of a smaller integer IFDO and a subset of a larger integer IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).		For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All integer IFDOs are subsets of [[just intonation]], and up to transposition, any integer IFDO is a superset of a smaller integer IFDO and a subset of a larger integer IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).

FloraC: Mean -> progression and other changes according to the AFDO article

2023-10-08T16:39:56Z

Mean -> progression and other changes according to the AFDO article

← Older revision		Revision as of 16:39, 8 October 2023
Line 1:		Line 1:
	An '''IFDO''' ('''inverse-arithmetic frequency division of the octave'''), or '''UDO''' ('''utonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic ~~mean~~ of frequency.		An '''IFDO''' ('''inverse-arithmetic frequency division of the octave'''), or '''UDO''' ('''utonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic progression of frequency.

	The inverse-arithmetic ~~mean~~ is known in general mathematics as the ~~[[Wikipedia:~~ Harmonic ~~mean~~\|harmonic ~~mean~~]], but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic ~~mean~~" was coined to avoid this conflict, as well as to point to its relationship with the ~~[[Wikipedia: Arithmetic mean~~\|arithmetic ~~mean]] (see [[Pythagorean means]])~~.		The inverse-arithmetic progression is known in general mathematics as the {{W\|Harmonic progression (mathematics)\|harmonic progression]], but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W\|arithmetic progression}}.

	When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An ''n''-~~IFDO~~ includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts~~, and thus may also be called an ''n''-ELDO ([[equal length division]] of the octave)~~; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of [[just intonation]]. As divisions of the octave, which is a rational interval, all IFDOs are subsets of JI, and thus the more precise and appropriate equivalence of an ''n''-IFDO is to an ''n''-UDO ([[utonal division]] of the octave).		For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All integer IFDOs are subsets of [[just intonation]], and up to transposition, any integer IFDO is a superset of a smaller integer IFDO and a subset of a larger integer IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).

			When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

	== Formula ==		== Formula ==

← Older revision		Revision as of 20:04, 9 March 2024
Line 275:		Line 275:


	'''~~Over~~-2''': {{IFDOs\|2, 4, 8, 16, 32, 64, 128, 256, 512}}		'''Under-2''': {{IFDOs\|2, 4, 8, 16, 32, 64, 128, 256, 512}}


	'''~~Over~~-2''': {{IFDOs\|3, 6, 12, 24, 48, 96, 192, 384}}		'''Under-3''': {{IFDOs\|3, 6, 12, 24, 48, 96, 192, 384}}


	'''~~Over~~-5''': {{IFDOs\|5, 10, 20, 40, 80, 160, 320}}		'''Under-5''': {{IFDOs\|5, 10, 20, 40, 80, 160, 320}}


	'''~~Over~~-7''': {{IFDOs\|7, 14, 28, 56, 112, 224, 448}}		'''Under-7''': {{IFDOs\|7, 14, 28, 56, 112, 224, 448}}


	'''~~Over~~-11''': {{IFDOs\|11, 22, 44, 88, 176, 352}}		'''Under-11''': {{IFDOs\|11, 22, 44, 88, 176, 352}}


	'''~~Over~~-13''': {{IFDOs\|13, 26, 52, 104, 208, 416}}		'''Under-13''': {{IFDOs\|13, 26, 52, 104, 208, 416}}


	'''~~Over~~-17''': {{IFDOs\|17, 34, 68, 136, 272, 544}}		'''Under-17''': {{IFDOs\|17, 34, 68, 136, 272, 544}}


	'''~~Over~~-19''': {{IFDOs\|19, 38, 76, 152, 304}}		'''Under-19''': {{IFDOs\|19, 38, 76, 152, 304}}


	'''~~Over~~-23''': {{IFDOs\|23, 46, 92, 184, 368}}		'''Under-23''': {{IFDOs\|23, 46, 92, 184, 368}}


	'''~~Over~~-29''': {{IFDOs\|29, 58, 116, 232, 464}}		'''Under-29''': {{IFDOs\|29, 58, 116, 232, 464}}


	'''~~Over~~-31''': {{IFDOs\|31, 62, 124, 248, 496}}		'''Under-31''': {{IFDOs\|31, 62, 124, 248, 496}}

	=== By other properties ===		=== By other properties ===