Odd harmonic: Difference between revisions

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An '''odd harmonic''' is a [[harmonic]] where the [[frequency ratio]] is an odd number. Odd harmonics are significant in that they create distinct [[pitch class]]es, since any even harmonic is a whole number of [[~~2/1|~~octave]]s above an odd harmonic.

An '''odd harmonic''' is a [[harmonic]] where the [[frequency ratio]] is an odd number. The first few odd harmonics are [[1/1]], [[3/1]], [[5/1]], [[7/1]], [[9/1]], [[11/1]], etc. Odd harmonics are significant in that they create distinct [[pitch class]]es, since any even harmonic is a whole number of [[octave]]s above an odd harmonic.

An [[odd limit]] is the set of all [[just interval]]s where the largest odd factor in the numerator and denominator both do not exceed a specified bound. For example, the [[5-odd-limit]] consists of all ratios where the only allowable odd factors are 1, 3, and 5; those being [[1/1]], [[6/5]], [[5/4]], [[4/3]], [[3/2]], [[8/5]], [[5/3]], and any whole number of octaves above those intervals.

We can also restrict the set of usable intervals to only allow odd harmonics. Such intervals include [[3/1]], [[5/3]], [[9/7]], [[27/25]], etc. Here, using [[3/1]] as the [[interval of equivalence]] (a "tritave") is natural, since the 3rd harmonic is the smallest odd harmonic after the fundamental. [[Edt]]s divide the tritave into a whole number of equal steps, analogous to how [[edo]]s divide the octave.

[[Category:Terms]]

[[Category:Odd limit]]

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-An '''odd harmonic''' is a [[harmonic]] where the [[frequency ratio]] is an odd number. Odd harmonics are significant in that they create distinct [[pitch class]]es, since any even harmonic is a whole number of [[2/1|octave]]s above an odd harmonic.
+An '''odd harmonic''' is a [[harmonic]] where the [[frequency ratio]] is an odd number. The first few odd harmonics are [[1/1]], [[3/1]], [[5/1]], [[7/1]], [[9/1]], [[11/1]], etc. Odd harmonics are significant in that they create distinct [[pitch class]]es, since any even harmonic is a whole number of [[octave]]s above an odd harmonic.
 An [[odd limit]] is the set of all [[just interval]]s where the largest odd factor in the numerator and denominator both do not exceed a specified bound. For example, the [[5-odd-limit]] consists of all ratios where the only allowable odd factors are 1, 3, and 5; those being [[1/1]], [[6/5]], [[5/4]], [[4/3]], [[3/2]], [[8/5]], [[5/3]], and any whole number of octaves above those intervals.
+We can also restrict the set of usable intervals to only allow odd harmonics. Such intervals include [[3/1]], [[5/3]], [[9/7]], [[27/25]], etc. Here, using [[3/1]] as the [[interval of equivalence]] (a "tritave") is natural, since the 3rd harmonic is the smallest odd harmonic after the fundamental. [[Edt]]s divide the tritave into a whole number of equal steps, analogous to how [[edo]]s divide the octave.
 [[Category:Terms]]
 [[Category:Odd limit]]
-{{Stub}}