Otonality and utonality: Difference between revisions

(4 intermediate revisions by 3 users not shown)

Line 19:

If a chord can be voiced as a "palindrome", it inverts to itself, and is ambitonal. Such a voicing makes the lowest interval the same as the highest, the next lowest the same as the next highest, etc. For example, the min7 chord can be voiced as 1-m3-P5-m7 = min 3rd, maj 3rd, min 3rd, therefore it must be ambitonal. Note that some ambitonal chords, such as the maj7no5, cannot be voiced as a palindrome.

By this definition all ~~monads, dyads~~ and ~~intervals~~ are ambitonal. (Dyads and intervals are not the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)

=== Dyads vs. intervals ===

By this definition all [[monad]]s and [[dyad]]s are ambitonal. (Dyads and intervals are not the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)

Therefore take note that while [[43/32]] may be the "prime harmonic fourth" (in that it is rooted/of the form ''k'' / 2''n''), it is only because we are seeing it as an ''interval'' that it is so, because seeing it as a ''dyad'' would mean seeing it as 32:43:64 so that it isn't clear whether it is otonal or utonal as [[64/43]] is the "prime subharmonic fifth", so interpreting it as a dyad means that whether it is harmonic or subharmonic (or neither) depends on the voicing and/or inversion used.

Note that a dyad (consisting of ''two'' [[pitch class]]es) thus has ''two'' possible ''inversions'' (which is a distinct concept to [[octave complement]]s!). For further clarity, see the section directly below.

=== Telling inversion of an ''n''-ad ===

To determine the inversion of an (''n''+''d'')-note chord consisting of ''n'' pitches up to [[octave equivalence]] (that is, given an ''n''-ad), go through all the pitches from lowest to highest until every pitch class is accounted for; that representation will then tell you which inversion the ''n''-ad has.

Example: going through the pitches of the 5-note chord 5:8:10:16:20 lowest to highest, we find that 5:8 accounts for all higher pitches (in that all higher pitches are a whole number of octaves above one of those harmonics); therefore this chord is a ''dyad'' (''n''=2); in this case, as one of the integers in the ''interval'' is a power of 2, we can classify this inversion of the dyad as ''subharmonic''.

== Properties of types of chords ==

Line 26:

Line 36:

* If we represent an otonal chord as a set of integers in the form A1:A2: ... :An, we may add any additional integers without affecting the chord's otonality.

* All chords with [[~~Linear chord|isoratios~~]] that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal.

* All chords with [[delta signature]]s that can be reduced (scaled by a positive real number) to +1+1, +1+1+1, +1+1+1+1, etc., are otonal.

=== Utonal ===

* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]

* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]{{dead link}}

=== Ambitonal ===

Line 46:

Line 56:

For N = 15, the factors are 1, 3, 5 and 15, and the ambitonal chords are {1, 3, 5, 15}, {1, 3, 15} and {1, 5, 15}. These [[octave-reduce]] to {1/1, 3/2, 5/4, 15/8} = maj7 chord, {1/1, 3/2, 15/8} = maj7no3 chord, and {1/1, 5/4, 15/8} = maj7no5 chord.

For N = 45, the factors are 1, 3, 5, 9, 15 and 45. One ambitonal chord is {1, 3, 5, 9, 15, 45}, which octave-reduces to {1/1, 5/4, 3/2, 15/8, 9/4, 45/16} = 16:20:24:30:36:45 = maj9(#11) chord. Any note or notes can be dropped except the root and the 11th, and the chord will still be ambitonal. The only other chord is {3, 5, 9, 15} = {1/1, 5/4, 3/2, 5/3} = maj6 chord, or its homonym the min7 chord. {3, 9, 15} is not ambitonal because the GCD isn't 1. {3, 5, 15} is not ambitonal because the LCM isn't 45.

For N = 45, the factors are 1, 3, 5, 9, 15 and 45. One ambitonal chord is {1, 3, 5, 9, 15, 45}, which octave-reduces to {1/1, 5/4, 3/2, 15/8, 9/4, 45/16} = 16:20:24:30:36:45 = maj9(#11) chord. Any note or notes can be dropped except the root and the 11th, and the chord will still be ambitonal. The only other chord is {3, 5, 9, 15} = {1/1, 5/4, 3/2, 5/3} = maj6 chord, or its [[Chord homonym|homonym]] the min7 chord. {3, 9, 15} is not ambitonal because the GCD isn't 1. {3, 5, 15} is not ambitonal because the LCM isn't 45.

== Scales ==

@@ Line 19: / Line 19: @@
 If a chord can be voiced as a "palindrome", it inverts to itself, and is ambitonal. Such a voicing makes the lowest interval the same as the highest, the next lowest the same as the next highest, etc. For example, the min7 chord can be voiced as 1-m3-P5-m7 = min 3rd, maj 3rd, min 3rd, therefore it must be ambitonal. Note that some ambitonal chords, such as the maj7no5, cannot be voiced as a palindrome.
-By this definition all monads, dyads and intervals are ambitonal. (Dyads and intervals are <u>not</u> the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)
+=== Dyads vs. intervals ===
+By this definition all [[monad]]s and [[dyad]]s are ambitonal. (Dyads and intervals are <u>not</u> the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)
+Therefore take note that while [[43/32]] may be the "prime harmonic fourth" (in  that it is rooted/of the form ''k'' / 2<sup>''n''</sup>), it is only because we are seeing it as an ''interval'' that it is so, because seeing it as a ''dyad'' would mean seeing it as 32:43:64 so that it isn't clear whether it is otonal or utonal as [[64/43]] is the "prime subharmonic fifth", so interpreting it as a dyad means that whether it is harmonic or subharmonic (or neither) depends on the voicing and/or inversion used.
+Note that a dyad (consisting of ''two'' [[pitch class]]es) thus has ''two'' possible ''inversions'' (which is a distinct concept to [[octave complement]]s!). For further clarity, see the section directly below.
+=== Telling inversion of an ''n''-ad ===
+To determine the inversion of an (''n''+''d'')-note chord consisting of ''n'' pitches up to [[octave equivalence]] (that is, given an ''n''-ad), go through all the pitches from lowest to highest until every pitch class is accounted for; that representation will then tell you which inversion the ''n''-ad has.
+Example: going through the pitches of the 5-note chord 5:8:10:16:20 lowest to highest, we find that 5:8 accounts for all higher pitches (in that all higher pitches are a whole number of octaves above one of those harmonics); therefore this chord is a ''dyad'' (''n''=2); in this case, as one of the integers in the ''interval'' is a power of 2, we can classify this inversion of the dyad as ''subharmonic''.
 == Properties of types of chords ==
@@ Line 26: / Line 36: @@
 * If we represent an otonal chord as a set of integers in the form A<sub>1</sub>:A<sub>2</sub>: ... :A<sub>n</sub>, we may add any additional integers without affecting the chord's otonality.
-* All chords with [[Linear chord|isoratios]] that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal.
+* All chords with [[delta signature]]s that can be reduced (scaled by a positive real number) to +1+1, +1+1+1, +1+1+1+1, etc., are otonal.
 === Utonal ===
-* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]
+* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]{{dead link}}
 === Ambitonal ===
@@ Line 46: / Line 56: @@
 For N = 15, the factors are 1, 3, 5 and 15, and the ambitonal chords are {1, 3, 5, 15}, {1, 3, 15} and {1, 5, 15}. These [[octave-reduce]] to {1/1, 3/2, 5/4, 15/8} = maj7 chord, {1/1, 3/2, 15/8} = maj7no3 chord, and {1/1, 5/4, 15/8} = maj7no5 chord.
-For N = 45, the factors are 1, 3, 5, 9, 15 and 45. One ambitonal chord is {1, 3, 5, 9, 15, 45}, which octave-reduces to {1/1, 5/4, 3/2, 15/8, 9/4, 45/16} = 16:20:24:30:36:45 = maj9(#11) chord. Any note or notes can be dropped except the root and the 11th, and the chord will still be ambitonal. The only other chord is {3, 5, 9, 15} = {1/1, 5/4, 3/2, 5/3} = maj6 chord, or its homonym the min7 chord. {3, 9, 15} is not ambitonal because the GCD isn't 1. {3, 5, 15} is not ambitonal because the LCM isn't 45.
+For N = 45, the factors are 1, 3, 5, 9, 15 and 45. One ambitonal chord is {1, 3, 5, 9, 15, 45}, which octave-reduces to {1/1, 5/4, 3/2, 15/8, 9/4, 45/16} = 16:20:24:30:36:45 = maj9(#11) chord. Any note or notes can be dropped except the root and the 11th, and the chord will still be ambitonal. The only other chord is {3, 5, 9, 15} = {1/1, 5/4, 3/2, 5/3} = maj6 chord, or its [[Chord homonym|homonym]] the min7 chord. {3, 9, 15} is not ambitonal because the GCD isn't 1. {3, 5, 15} is not ambitonal because the LCM isn't 45.
 == Scales ==