AFDO: Difference between revisions
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For example, in [[12afdo]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an AFDO system, the ''difference'' between interval ratios is equal (they form an arithmetic progression), rather than their ''ratios'' between interval ratios being equal as in [[EDO]] systems (a geometric progression). All integer AFDOs are subsets of [[just intonation]], and up to transposition, any integer AFDO is a superset of a smaller integer AFDO and a subset of a larger integer AFDO (i.e. ''n''-afdo is a superset of (''n'' - 1)-afdo and a subset of (''n'' + 1)-afdo for any integer ''n'' > 1). | For example, in [[12afdo]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an AFDO system, the ''difference'' between interval ratios is equal (they form an arithmetic progression), rather than their ''ratios'' between interval ratios being equal as in [[EDO]] systems (a geometric progression). All integer AFDOs are subsets of [[just intonation]], and up to transposition, any integer AFDO is a superset of a smaller integer AFDO and a subset of a larger integer AFDO (i.e. ''n''-afdo is a superset of (''n'' - 1)-afdo and a subset of (''n'' + 1)-afdo for any integer ''n'' > 1). | ||
When treated as a scale, the AFDO is equivalent to the [[overtone scale]]. An AFDO is equivalent to an ODO ([[otonal division]] of the octave). It may also be called an EFDO ([[equal frequency division]] of the octave), however, this more general acronym is typically reserved for divisions of irrational intervals | When treated as a scale, the AFDO is equivalent to the [[overtone scale]]. An AFDO is equivalent to an ODO ([[otonal division]] of the octave). It may also be called an EFDO ([[equal frequency 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. | ||
== Formula == | == Formula == | ||
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== Relation to string lengths == | == Relation to string lengths == | ||
If the first division has ratio of ''r''<sub>1</sub> and length of ''l''<sub>1</sub> and the last, ''r''<sub>''n''</sub> and | If the first division has ratio of ''r''<sub>1</sub> and length of ''l''<sub>1</sub> and the last, ''r''<sub>''n''</sub> and ''l''<sub>''n''</sub> , we have: ''l''<sub>''n''</sub> = 1/''r''<sub>''n''</sub> and if ''r''<sub>''n''</sub> > … > ''r''<sub>3</sub> > ''r''<sub>2</sub> > ''r''<sub>1</sub>, then ''l''<sub>1</sub> > ''l''<sub>2</sub> > ''l''<sub>3</sub> > … > ''l''<sub>''n''</sub> | ||
[[File:ADO-4.jpg|350px|center]] | [[File:ADO-4.jpg|350px|center]] | ||
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* [[AFS]] | * [[AFS]] | ||
* [[Frequency temperament]] | * [[Frequency temperament]] | ||
== Notes == | == Notes == | ||