Odd limit: Difference between revisions

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{{interwiki

| en = Odd limit

| de = Odd-Limit

~~| en = Odd limit~~

| ja = アドリミット

}}

~~{{Odd-limit navigation}}~~

The '''odd limit''' is a metric that ~~places an upper bound on (i.e.~~ 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.

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.

== 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''.

~~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~~.

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).

To find the set of all ratios of ''N''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''N'' 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 ~~doesn't~~ 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.

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 ~~2nds~~ or ~~7ths~~. It's rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the ~~2nd~~ and/or the ~~7th~~ 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 ~~most~~ 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 ~~2nd~~ from some chord note. The major ~~2nd~~ 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.

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.

Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond|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.

== ~~Mathematical definitions~~ ==

== 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 ~~as a set~~ of ~~ratios ===~~

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.

~~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 as a property of a ratio~~ ===

== Individual pages for odd limits ==

{| class="wikitable center-all"

~~Given a ratio of positive integers ''p''/''q'', its~~ odd limit ~~is found by first removing factors of two and all common factors from ''p''/''q'', producing a ratio ''a''/''b'' of relatively prime~~ odd ~~numbers. The~~ odd limit ~~equals max(''a'', ''b''). It's also called the~~ [[~~Kees expressibility~~]] ~~of the interval, named after~~ [[~~Kees van Prooijen~~]] ~~who showed what this metric looks like geometrically on the lattice.~~

|-

| [[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 ==

[[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''.

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]].

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 ==

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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 '''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~~. 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 ====

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 {{interwiki
+| en = Odd limit
 | de = Odd-Limit
-| en = Odd limit
 | ja = アドリミット
 }}
-{{Odd-limit navigation}}
 {{Wikipedia|Limit (music)}}
-The '''odd limit''' is a metric that places an upper bound on (i.e. 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.
+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.
+== 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''.
-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.
+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).
-To find the set of all ratios of ''N''-odd-limit, construct a set of ratios by pairing off all the odd numbers less than or equal to ''N'' 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 doesn't 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.
+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 2nds or 7ths. It's rare for an entire piece of music to ''strictly'' fall within the 5-odd-limit, because melodies generally do use the 2nd and/or the 7th 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 most 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 2nd from some chord note. The major 2nd 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.
+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.
-Odd limits are more or less equivalent to what [[Harry Partch]] calls ''[[Tonality diamond|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.
-== Mathematical definitions ==
+== 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 as a set of ratios ===
+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.
-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 as a property of a ratio ===
+== Individual pages for odd limits ==
+{| class="wikitable center-all"
-Given a ratio of positive integers ''p''/''q'', its odd limit is found by first removing factors of two and all common factors from ''p''/''q'', producing a ratio ''a''/''b'' of relatively prime odd numbers. The odd limit equals max(''a'', ''b''). It's also called the [[Kees expressibility]] of the interval, named after [[Kees van Prooijen]] who showed what this metric looks like geometrically on the lattice.
+|-
+| [[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 ==
 [[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''.
-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]].
+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 ==
@@ Line 50: / Line 58: @@
 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 '''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. 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 ====