IFDO: Difference between revisions
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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 == | ||
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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 "[ | 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 == | |||
* ''n''-ifdo has [[maximum variety]] ''n''. | |||
* Except for 1ifdo and 2ifdo, IFDOs 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" | ||
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''' | '''Under-2''': {{IFDOs|2, 4, 8, 16, 32, 64, 128, 256, 512}} | ||
''' | '''Under-3''': {{IFDOs|3, 6, 12, 24, 48, 96, 192, 384}} | ||
''' | '''Under-5''': {{IFDOs|5, 10, 20, 40, 80, 160, 320}} | ||
''' | '''Under-7''': {{IFDOs|7, 14, 28, 56, 112, 224, 448}} | ||
''' | '''Under-11''': {{IFDOs|11, 22, 44, 88, 176, 352}} | ||
''' | '''Under-13''': {{IFDOs|13, 26, 52, 104, 208, 416}} | ||
''' | '''Under-17''': {{IFDOs|17, 34, 68, 136, 272, 544}} | ||
''' | '''Under-19''': {{IFDOs|19, 38, 76, 152, 304}} | ||
''' | '''Under-23''': {{IFDOs|23, 46, 92, 184, 368}} | ||
''' | '''Under-29''': {{IFDOs|29, 58, 116, 232, 464}} | ||
''' | '''Under-31''': {{IFDOs|31, 62, 124, 248, 496}} | ||
=== By other properties === | === By other properties === | ||