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== Definition ==
The '''odd limit''' is a metric that limits the [[complexity]] of the [[ratio]]s used in a piece of music. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. [[Integer limit]] and [[prime limit]] are related concepts, albeit prime-limit is unbound.  
{{odd-limit navigation}}
 
'''Odd limit''' has two meanings. In the original sense of the term, discussed first, an odd limit is a set of [[Ratio|ratios]]. 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.  
== As a set of ratios ==
The ''q''-odd limit is the set of irreducible ratios between 1 and 2 whose numerator and denominator, once all factors of two are removed, are both less than or equal to ''q''.  


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.
Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond]]s''. 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. Examples: some ratios in the 9-odd-limit are: 3/2, 5/4, 7/6, 10/7, 12/7, 9/8 and 14/9; but not 11/9 nor 13/8 nor 16/15 (these have are odd terms greater than 9, thus not in the set).


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).
To find the set of all ratios of ''q''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''q'' in every combination. Discard any ratios that can be simplified (e.g. 5/5 or 15/3). Transposing any of these ratios by an octave does not change the odd limit. Hence an odd limit set is theoretically infinite. For convenience, the odd limit set is usually written out in [[octave-reduced]] form.


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).
For example, to find the 5-odd-limit set of ratios, pair off 1, 3 and 5: {1/1, 3/1, 5/1, 1/3, (3/3), 5/3, 1/5, 3/5, (5/5)}. Discard the two redundant ratios in parentheses. Octave-reduced and in ascending order, the 5-odd-limit set is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}.


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|'''prime limit''']] are related concepts.
The 3-odd-limit set of ratios is {1/1, 4/3, 3/2}. The 3-odd-limit set is contained in the 5-odd-limit set, both sets are contained in the 7-odd-limit set, and so on.


== Mathematical Definition ==
Note that the 5-odd-limit set contains no seconds or sevenths. It is rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the second and/or the seventh of the scale, but a piece of music being in a certain odd limit usually means merely that at any given moment, no interval in the piece exceeds the odd limit. In other words, all vertical intervals within a chord, or between a melody note and a chord note, are within the odd limit. Even this looser definition excludes many songs. It perhaps includes {{w|Kumbaya|"Kum Ba Yah"}}, depending on the exact chords used. But even the simplest pentatonic songs usually have a melody note that is a major second from some chord note. The major second ratio is usually 10/9 or 9/8, making the piece 9-odd-limit. An even looser definition ignores the melody notes and requires only 5-limit ''chords''. This definition includes any song that uses only major and minor triads.


The '''q''' '''odd limit''', where ''q'' is an odd positive integer, consists of everything of the form <code>2^i*u/v</code>, or <math>2^\mathbb Z\frac u v</math>, where ''u'' and ''v'' are odd positive integers less than or equal to q. It may be identified with the [[Tonality diamond|q-limit 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).
== As a property of a ratio ==
Given a ratio of positive integers ''n''/''d'', its odd limit is found by removing all factors of two and all other common factors from ''n''/''d'', producing a ratio ''a''/''b'' of relatively prime odd numbers. Thus the odd-limit of ''n''/''d'' is the maximum of ''a'' and ''b'', max(''a'', ''b'').


== Odd limit of a ratio ==
The odd limit is also called the [[Kees semi-height]] of the interval, named after [[Kees van Prooijen]] who showed what this metric looks like geometrically on the lattice. To find the odd limit of a ratio: if either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two odd numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.


From the definition above, we can see that an interval like 3/2 is not only part of the 3-odd-limit, but also the 5-odd-limit, the 7-odd-limit, and so on. However, it is also useful to refer to the *smallest* such odd limit that some interval fits into. This is often simply just called the "odd limit" of the ratio.
== Individual pages for odd limits ==
{| class="wikitable center-all"
|-
| [[1-odd-limit]] || [[3-odd-limit]] || [[5-odd-limit]] || [[7-odd-limit]] || [[9-odd-limit]] || [[11-odd-limit]]
|-
| [[13-odd-limit]] || [[15-odd-limit]] || [[17-odd-limit]] || [[19-odd-limit]] || [[21-odd-limit]] || [[23-odd-limit]]
|-
| [[25-odd-limit]] || [[27-odd-limit]] || [[29-odd-limit]] || [[31-odd-limit]] || [[33-odd-limit]] || [[35-odd-limit]]
|-
| [[37-odd-limit]] || [[39-odd-limit]] || [[41-odd-limit]] || [[43-odd-limit]] || [[45-odd-limit]] || [[47-odd-limit]]
|}


To find the odd limit of a ratio: If either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 > 3, thus the odd limit is 7.
== Integer limit ==
[[File:WilsonHeightIntegerLimit.png|200px|thumb|right|Diagram by Lériendil showing the integer limit (horizontal axis) versus [[Wilson height]] (vertical axis) of simple intervals.]]The '''integer limit''' of a ratio is simply the larger of the ratio's numerator and denominator. For example, the integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio, and is the same as the [[Weil height]]. The set of all ratios with an integer limit up to ''n'' is the same as the {{w|Farey sequence}} of order ''n''.


This is also called the [[Kees expressibility]] of the interval, named after Kees van Prooijen who showed what this metric looks like geometrically on the lattice.
The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd-limit can be thought of as the best-case integer limit, when assuming [[octave equivalence]].
 
== Relationship to other limits ==
 
The '''integer limit''' of a ratio is simply the larger of the ratio's two numbers, which is always the numerator. The integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio; it is equal to the exponentiation base two of the [[Weil height]]. But the odd limit is far more common, because the integer limit depends on the voicing of the interval, and the odd limit does not. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd limit can be thought of as the best-case-scenario integer limit. The odd limit reflects the complexity of the ratio in a context in which [[octave equivalence]] is assumed.


== Generalizations ==
=== Odd limit of a chord ===
Odd limit can be generalized to apply to chords in a number of ways. The '''intervallic limit''' looks at each interval of the chord, and the odd limit of that interval. The chord's odd limit is the largest of these odd limits. For example, both 4:5:6 and 10:12:15 have component intervals 3/2, 5/4, and 6/5. The intervals' odd limits are 3, 5, and 5. Thus both chords' intervallic limits are 5.
Odd limit can be generalized to apply to chords in a number of ways. The '''intervallic limit''' looks at each interval of the chord, and the odd limit of that interval. The chord's odd limit is the largest of these odd limits. For example, both 4:5:6 and 10:12:15 have component intervals 3/2, 5/4, and 6/5. The intervals' odd limits are 3, 5, and 5. Thus both chords' intervallic limits are 5.


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For example, 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, and thus the chord's otonal limit is 15. By contrast, 4:5:6's otonal limit is 5. 10:12:15 is sometimes considered more complex than 4:5:6, and the otonal limit is the measure that reflects that. However, 10:12:15 can be written as 1/(6:5:4), so the chord's utonal limit is 5, same as 4:5:6's otonal limit. Thus the ambitonal limits of both chords are 5, bringing them back to the same complexity level by recognizing each chord's more prominent otonal or utonal identity.  
For example, 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, and thus the chord's otonal limit is 15. By contrast, 4:5:6's otonal limit is 5. 10:12:15 is sometimes considered more complex than 4:5:6, and the otonal limit is the measure that reflects that. However, 10:12:15 can be written as 1/(6:5:4), so the chord's utonal limit is 5, same as 4:5:6's otonal limit. Thus the ambitonal limits of both chords are 5, bringing them back to the same complexity level by recognizing each chord's more prominent otonal or utonal identity.  


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 Extensions ==
=== Proposed extensions ===
[[KiteGiedraitis|Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit.
[[Kite Giedraitis]] has proposed several extensions to the concepts of odd limit and integer limit.


The '''double odd limit''' or '''DOL''' of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL.
The '''double odd limit''' or '''DOL''' of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL.


The '''double integer limit''' or '''DIL''' of a ratio a/b is (b, a). For any interval, the voicing which has the smallest DIL is the '''all-odd voicing''' or '''AOV''', in which both the numerator and the denominator are odd. The AOV of a ratio is found by taking the odd limit of each number in the ratio, and combining them into a new ratio. For 12/7, the AOV is 7/3. For 3/2, the AOV is 3/1.
The '''double integer limit''' or '''DIL''' of a ratio ''a''/''b'' is (''b'', ''a''). For any interval, the voicing which has the smallest DIL is the '''all-odd voicing''' or '''AOV''', in which both the numerator and the denominator are odd. The AOV of a ratio is found by taking the odd limit of each number in the ratio (i.e. factoring out all the twos), and combining them into a new ratio. For 12/7, the AOV is 7/3. For 3/2, the AOV is 3/1.


The concept of integer limit can be generalized to apply to a chord either intervallicly or otonally. Either way, the integer limit is the highest (final) number of the extended ratio.
The concept of integer limit can be generalized to apply to a chord either intervallicly or otonally. Either way, the integer limit is the highest (final) number of the extended ratio.


The '''multiple integer limit''' or '''MIL''' of a chord is simply the numbers of the extended ratio, listed highest to lowest. For any chord, the voicing which has the smallest MIL is the AOV, in which every number of the extended ratio is odd. The AOV of a chord is found by taking the odd limit of each number in the extended ratio, sorting them by size, and assembling them into a new extended ratio. For 4:5:6, the AOV is 1:3:5. For 10:12:15, the AOV is 3:5:15.
The '''multiple integer limit''' or '''MIL''' of a chord is simply the numbers of the extended ratio, listed highest to lowest. For any chord, the voicing which has the smallest MIL is the AOV, in which every number of the extended ratio is odd. The AOV of a chord is found by taking the odd limit of each number in the extended ratio, sorting them by size, and assembling them into a new extended ratio. For 4:5:6, the AOV is 1:3:5. For 10:12:15, the AOV is 3:5:15.
A chord in AOV is often impractically wide. The '''condensed all-odd voicing''' or '''CAOV''' octave-reduces every interval between adjacent notes. For example, 1:3:5 has a large gap between the two lowest voices, and 2:3:5 is more practical. To find the CAOV, begin with the AOV. Starting at the top, when you come to an interval wider than an octave, double all the numbers below it. Keep going until you reach the bottom. For example, the AOV of 10:12:15 is 3:5:15, and the CAOV is 6:10:15.


Kite has conjectured that the all-odd voicing of a just intonation ratio or chord is in general the most consonant voicing, with several caveats. Timbre matters. Register matters. Musical context matters. This conjecture may fail for ratios and chords with a high odd limit. For example, narrow all-odd ratios like 65/63 = 54¢ are better voiced widened by an octave. Also, the best voicing of 301/200 is not 301/25 but 301/100, because 301/200 is very close to a ratio with a much smaller odd limit, 3/2. Finally, it's difficult to judge the consonance of extremely wide intervals such as 11/1.
Kite has conjectured that the all-odd voicing of a just intonation ratio or chord is in general the most consonant voicing, with several caveats. Timbre matters. Register matters. Musical context matters. This conjecture may fail for ratios and chords with a high odd limit. For example, narrow all-odd ratios like 65/63 = 54¢ are better voiced widened by an octave. Also, the best voicing of 301/200 is not 301/25 but 301/100, because 301/200 is very close to a ratio with a much smaller odd limit, 3/2. Finally, it's difficult to judge the consonance of extremely wide intervals such as 11/1.


This conjecture has two implications. First, a given JI chord has an ideal voicing. This voicing may be rather far-flung, and a more compact voicing may be almost as consonant. For example, 1:3:5:7 has a large gap between the two lowest voices, and 2:3:5:7 is more practical. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.
This conjecture has two implications. First, a given JI chord has an ideal voicing. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.
 
==== Non-octave settings ====
{{Main| Equave limit }}


The concept of odd limit can be generalized to prime three in a [[Nonoctave|non-octave]] ("no-twos") tritave-equivalent context such as [[Bohlen-Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words '''threeven''' and '''throdd''' for divisibility by three. The '''throdd limit''' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The '''double throdd limit''' of 15/7 is (7,5). Its '''all-throdd voicing''' is 7/5. The 1/1 - 9/7 - 9/5 - 3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35.
The concept of odd limit can be generalized to [[nonoctave|non-octave]] contexts such as 3/1-equivalent [[Bohlen–Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words ''threeven'' and ''throdd'' for divisibility by three. The ''throdd limit'' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The ''double throdd limit'' of 15/7 is (7, 5). Its ''all-throdd voicing'' is 7/5. The 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its ''intervallic throdd limit'' is 7, and its ''otonal throdd limit'' is 35.


== See also ==
== See also ==
* [[p-limit]] - or prime [[harmonic limit]]
* [[Harmonic limit|''p''-limit]] – or prime [[harmonic limit]]
* [[Cubic and octahedral limits]]
* [[Shadow]] (a psychoacoustic effect based on integer limit)
* [[List of 47-odd-limit intervals]]


== External links ==
== External links ==
* [http://tonalsoft.com/enc/l/limit.aspx Limit - Tonalsoft Encyclopedia of Microtonal Music Theory]
* [http://tonalsoft.com/enc/l/limit.aspx Limit – Tonalsoft Encyclopedia of Microtonal Music Theory]


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Latest revision as of 01:18, 19 November 2025

English Wikipedia has an article on:

The odd limit is a metric that limits the complexity of the ratios used in a piece of music. Odd limit can refer to the set of all ratios that are within this limit, or it can refer to the metric itself, applied to individual ratios. Integer limit and prime limit are related concepts, albeit prime-limit is unbound.

As a set of ratios

The q-odd limit is the set of irreducible ratios between 1 and 2 whose numerator and denominator, once all factors of two are removed, are both less than or equal to q.

Odd limits are more or less equivalent to what Harry Partch calls Tonality diamonds. 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. Examples: some ratios in the 9-odd-limit are: 3/2, 5/4, 7/6, 10/7, 12/7, 9/8 and 14/9; but not 11/9 nor 13/8 nor 16/15 (these have are odd terms greater than 9, thus not in the set).

To find the set of all ratios of q-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to q in every combination. Discard any ratios that can be simplified (e.g. 5/5 or 15/3). Transposing any of these ratios by an octave does not change the odd limit. Hence an odd limit set is theoretically infinite. For convenience, the odd limit set is usually written out in octave-reduced form.

For example, to find the 5-odd-limit set of ratios, pair off 1, 3 and 5: {1/1, 3/1, 5/1, 1/3, (3/3), 5/3, 1/5, 3/5, (5/5)}. Discard the two redundant ratios in parentheses. Octave-reduced and in ascending order, the 5-odd-limit set is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}.

The 3-odd-limit set of ratios is {1/1, 4/3, 3/2}. The 3-odd-limit set is contained in the 5-odd-limit set, both sets are contained in the 7-odd-limit set, and so on.

Note that the 5-odd-limit set contains no seconds or sevenths. It is rare for an entire piece of music to strictly fall within the 5-odd-limit, because melodies generally do use the second and/or the seventh of the scale, but a piece of music being in a certain odd limit usually means merely that at any given moment, no interval in the piece exceeds the odd limit. In other words, all vertical intervals within a chord, or between a melody note and a chord note, are within the odd limit. Even this looser definition excludes many songs. It perhaps includes "Kum Ba Yah", depending on the exact chords used. But even the simplest pentatonic songs usually have a melody note that is a major second from some chord note. The major second ratio is usually 10/9 or 9/8, making the piece 9-odd-limit. An even looser definition ignores the melody notes and requires only 5-limit chords. This definition includes any song that uses only major and minor triads.

As a property of a ratio

Given a ratio of positive integers n/d, its odd limit is found by removing all factors of two and all other common factors from n/d, producing a ratio a/b of relatively prime odd numbers. Thus the odd-limit of n/d is the maximum of a and b, max(a, b).

The odd limit is also called the Kees semi-height of the interval, named after Kees van Prooijen who showed what this metric looks like geometrically on the lattice. To find the odd limit of a ratio: if either the numerator or the denominator is even, divide it by two until it is odd. The larger of the two odd numbers is the odd limit. Example: 12/7 becomes 3/7, and 7 is greater than 3, thus the odd limit is 7.

Individual pages for odd limits

1-odd-limit 3-odd-limit 5-odd-limit 7-odd-limit 9-odd-limit 11-odd-limit
13-odd-limit 15-odd-limit 17-odd-limit 19-odd-limit 21-odd-limit 23-odd-limit
25-odd-limit 27-odd-limit 29-odd-limit 31-odd-limit 33-odd-limit 35-odd-limit
37-odd-limit 39-odd-limit 41-odd-limit 43-odd-limit 45-odd-limit 47-odd-limit

Integer limit

Diagram by Lériendil showing the integer limit (horizontal axis) versus Wilson height (vertical axis) of simple intervals.

The integer limit of a ratio is simply the larger of the ratio's numerator and denominator. For example, the integer limit of 12/7 is 12. The integer limit more directly reflects the complexity of the ratio, and is the same as the Weil height. The set of all ratios with an integer limit up to n is the same as the Farey sequence of order n.

The odd limit is more common, because it does not depend on the voicing of the interval, while the integer limit does. For example, 12/7 voiced an octave wider is 24/7, integer limit 24. Consider all possible voicings of an interval, and the integer limit of each one. The smallest of all these integer limits is the odd limit. For 12/7, voicings 7/6 and 7/3 both have integer limit 7. Thus the odd-limit can be thought of as the best-case integer limit, when assuming octave equivalence.

Generalizations

Odd limit of a chord

Odd limit can be generalized to apply to chords in a number of ways. The intervallic limit looks at each interval of the chord, and the odd limit of that interval. The chord's odd limit is the largest of these odd limits. For example, both 4:5:6 and 10:12:15 have component intervals 3/2, 5/4, and 6/5. The intervals' odd limits are 3, 5, and 5. Thus both chords' intervallic limits are 5.

The otonal limit of a chord looks at each number in the extended ratio a:b:c:…, and the odd limit of that number. The odd limit of a number is defined as the number itself if odd, and if even, the number divided by two until it is odd. The chord's otonal limit is the largest of these odd limits. The utonal limit is defined analogously. Combining the otonal and utonal limits, we can define the ambitonal limit, which is the smaller value of the otonal and utonal limits of a chord.

For example, 10:12:15 has numbers 10, 12 and 15, the odd limits of which are 5, 3 and 15, and thus the chord's otonal limit is 15. By contrast, 4:5:6's otonal limit is 5. 10:12:15 is sometimes considered more complex than 4:5:6, and the otonal limit is the measure that reflects that. However, 10:12:15 can be written as 1/(6:5:4), so the chord's utonal limit is 5, same as 4:5:6's otonal limit. Thus the ambitonal limits of both chords are 5, bringing them back to the same complexity level by recognizing each chord's more prominent otonal or utonal identity.

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 extensions

Kite Giedraitis has proposed several extensions to the concepts of odd limit and integer limit.

The double odd limit or DOL of a ratio is simply the odd limit of each number in the ratio, with the higher one listed first. DOL (12/7) = (7, 3). The DOL is useful as a tiebreaker when comparing the complexity of two ratios with the same odd limit. For example, 50/49 and 49/48 are both odd limit 49. But DOL (50/49) = (49, 25) and DOL (49/48) = (49, 3). Since 3 < 25, 49/48 has a lower DOL.

The double integer limit or DIL of a ratio a/b is (b, a). For any interval, the voicing which has the smallest DIL is the all-odd voicing or AOV, in which both the numerator and the denominator are odd. The AOV of a ratio is found by taking the odd limit of each number in the ratio (i.e. factoring out all the twos), and combining them into a new ratio. For 12/7, the AOV is 7/3. For 3/2, the AOV is 3/1.

The concept of integer limit can be generalized to apply to a chord either intervallicly or otonally. Either way, the integer limit is the highest (final) number of the extended ratio.

The multiple integer limit or MIL of a chord is simply the numbers of the extended ratio, listed highest to lowest. For any chord, the voicing which has the smallest MIL is the AOV, in which every number of the extended ratio is odd. The AOV of a chord is found by taking the odd limit of each number in the extended ratio, sorting them by size, and assembling them into a new extended ratio. For 4:5:6, the AOV is 1:3:5. For 10:12:15, the AOV is 3:5:15.

A chord in AOV is often impractically wide. The condensed all-odd voicing or CAOV octave-reduces every interval between adjacent notes. For example, 1:3:5 has a large gap between the two lowest voices, and 2:3:5 is more practical. To find the CAOV, begin with the AOV. Starting at the top, when you come to an interval wider than an octave, double all the numbers below it. Keep going until you reach the bottom. For example, the AOV of 10:12:15 is 3:5:15, and the CAOV is 6:10:15.

Kite has conjectured that the all-odd voicing of a just intonation ratio or chord is in general the most consonant voicing, with several caveats. Timbre matters. Register matters. Musical context matters. This conjecture may fail for ratios and chords with a high odd limit. For example, narrow all-odd ratios like 65/63 = 54¢ are better voiced widened by an octave. Also, the best voicing of 301/200 is not 301/25 but 301/100, because 301/200 is very close to a ratio with a much smaller odd limit, 3/2. Finally, it's difficult to judge the consonance of extremely wide intervals such as 11/1.

This conjecture has two implications. First, a given JI chord has an ideal voicing. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.

Non-octave settings

Main article: Equave limit

The concept of odd limit can be generalized to non-octave contexts such as 3/1-equivalent Bohlen–Pierce. Just as the words even and odd refer to divisibility by two, mathematicians use the words threeven and throdd for divisibility by three. The throdd limit of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The double throdd limit of 15/7 is (7, 5). Its all-throdd voicing is 7/5. The 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its intervallic throdd limit is 7, and its otonal throdd limit is 35.

See also

External links

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