Odd limit: Difference between revisions
→Relationship to other limits: integer limit = exponentiation of the weil height. +utonal limit, ambitonal limit, and their musical utility |
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'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[ | '''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[ratio]]s. In the newer sense, discussed [[Odd limit#Odd limit of a ratio or chord|below]], the odd limit ''of a ratio'' is a specific number. | ||
An odd limit is the set of all ratios for which neither the numerator nor denominator exceed some maximum value, once all powers of 2 are removed. Typically, the maximum value is some small odd number, such as 3, 5, 7, 9, 11, etc. Each odd number gives rise to a different odd limit, so that there is a 3-odd-limit, a 5-odd-limit, and so on. This forms an increasing sequence of odd-limits, so that each numbered odd limit in this sequence is a subset of the next, so that the 3-odd-limit is a subset of the 5-odd-limit, which is in turn a subset of the 7-odd-limit, and so on. | An odd limit is the set of all ratios for which neither the numerator nor denominator exceed some maximum value, once all powers of 2 are removed. Typically, the maximum value is some small odd number, such as 3, 5, 7, 9, 11, etc. Each odd number gives rise to a different odd limit, so that there is a 3-odd-limit, a 5-odd-limit, and so on. This forms an increasing sequence of odd-limits, so that each numbered odd limit in this sequence is a subset of the next, so that the 3-odd-limit is a subset of the 5-odd-limit, which is in turn a subset of the 7-odd-limit, and so on. | ||
Odd-limits are more or less equivalent to what Harry Partch calls | Odd-limits are more or less equivalent to what Harry Partch calls ''[[Tonality diamond|Tonality Diamonds]]'', in his theory. More precisely, a Tonality Diamond can be viewed as a particular geometric representation of a certain odd-limit, and the two terms are often used together (e.g., the 11-odd-limit Tonality Diamond). The sequence of increasing odd limits can be visualized as as a smaller tonality diamond being embedded in a set of progressively larger ones. | ||
The purpose of an odd-limit or tonality diamond is to provide a "simple" subset of JI intervals to play, given one particularly natural definition of "simple." The removal of powers of 2 makes it so that for any interval that is viewed as "simple enough," the set of all its octave transpositions is also included in the set. Increasing the cutoff number increases the set of ratios viewed as being "simple enough" to be in the set. These are musically useful because such intervals will often tend to be play nicely with one another when forming chords (or at least, more so than some random JI intervals). | The purpose of an odd-limit or tonality diamond is to provide a "simple" subset of JI intervals to play, given one particularly natural definition of "simple." The removal of powers of 2 makes it so that for any interval that is viewed as "simple enough," the set of all its octave transpositions is also included in the set. Increasing the cutoff number increases the set of ratios viewed as being "simple enough" to be in the set. These are musically useful because such intervals will often tend to be play nicely with one another when forming chords (or at least, more so than some random JI intervals). | ||
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As an example, the 5-odd-limit is the set of intervals {1/1, 3/1, 1/3, 5/1, 1/5, 5/3, 3/5}, as well as every octave transpositions of the above (e.g. 2/1, 4/1, 3/2, 6/1, 5/4 and so on). | As an example, the 5-odd-limit is the set of intervals {1/1, 3/1, 1/3, 5/1, 1/5, 5/3, 3/5}, as well as every octave transpositions of the above (e.g. 2/1, 4/1, 3/2, 6/1, 5/4 and so on). | ||
As a result, the '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the harmonies used in a piece of music, and hence of the music itself. Integer limit and | As a result, the '''odd limit''' is a metric that places an upper bound on (i.e. limits) the complexity of the harmonies used in a piece of music, and hence of the music itself. Integer limit and prime limit are related concepts. | ||
== Mathematical | == Mathematical definition == | ||
The ' | The '''''q''-odd-limit''', where ''q'' is an odd positive integer, consists of every number of the form <math>2^i \cdot u/v</math> where ''i'' is an integer and where ''u'' and ''v'' are odd positive integers less than or equal to ''q''. It may be identified with the [[Tonality diamond|''q''-odd-limit tonality diamond]]. Examples: some ratios in the 9-limit are: 3/2, 5/4, 7/6, 10/7, 12/7, 9/8 and 14/9. But not 11/9 (11 is a prime greater than 9) nor 15/7 (since 15 is 3 × 5, both less then 9, but with product greater than 9). | ||
== Odd limit of a ratio == | == Odd limit of a ratio == | ||
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Note that the ambitonal limit is often equal to the intervallic limit, but not always, e.g. the chord 1-6/5-10/7-8/5 (left to readers as an exercise). | Note that the ambitonal limit is often equal to the intervallic limit, but not always, e.g. the chord 1-6/5-10/7-8/5 (left to readers as an exercise). | ||
== Proposed | == Proposed extensions == | ||
[[KiteGiedraitis|Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit. | [[KiteGiedraitis|Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit. | ||