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# Scale 633: "Kydimic" ### Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

### Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Zeitler
Kydimic
Dozenal
Durian

## Analysis

#### Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

#### Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,3,4,5,6,9}

#### Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z42

#### Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

#### Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

[4.5]

#### Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

#### Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

no

#### Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

#### Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

2 (dicohemitonic)

#### Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

4

#### Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

5

#### Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 591

#### Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

#### Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

#### Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[3, 1, 1, 1, 3, 3]

#### Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<3, 2, 4, 2, 2, 2>

#### Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p2m2n4s2d3t2

#### Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,3}
<2> = {2,4,6}
<3> = {3,5,7,9}
<4> = {6,8,10}
<5> = {9,11}

#### Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

3

#### Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

#### Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

#### Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.25

#### Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.796

#### Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

#### Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

#### Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.



#### Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Improper

#### Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(28, 10, 55)

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

d♯°{3,6,9}231.5
f♯°{6,9,0}231.5
{9,0,3}231.5

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter 3 3 yes

## Modes

Modes are the rotational transformation of this scale. Scale 633 can be rotated to make 5 other scales. The 1st mode is itself.

 2nd mode:Scale 591 Gaptimic This is the prime mode 3rd mode:Scale 2343 Tharimic 4th mode:Scale 3219 Ionaphimic 5th mode:Scale 3657 Epynimic 6th mode:Scale 969 Ionogimic

## Prime

The prime form of this scale is Scale 591

 Scale 591 Gaptimic

## Complement

The hexatonic modal family [633, 591, 2343, 3219, 3657, 969] (Forte: 6-Z42) is the complement of the hexatonic modal family [219, 1563, 1731, 2157, 2829, 2913] (Forte: 6-Z13)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 633 is 969

 Scale 969 Ionogimic

## Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 633       T0I <11,0> 969
T1 <1,1> 1266      T1I <11,1> 1938
T2 <1,2> 2532      T2I <11,2> 3876
T3 <1,3> 969      T3I <11,3> 3657
T4 <1,4> 1938      T4I <11,4> 3219
T5 <1,5> 3876      T5I <11,5> 2343
T6 <1,6> 3657      T6I <11,6> 591
T7 <1,7> 3219      T7I <11,7> 1182
T8 <1,8> 2343      T8I <11,8> 2364
T9 <1,9> 591      T9I <11,9> 633
T10 <1,10> 1182      T10I <11,10> 1266
T11 <1,11> 2364      T11I <11,11> 2532
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 843      T0MI <7,0> 2649
T1M <5,1> 1686      T1MI <7,1> 1203
T2M <5,2> 3372      T2MI <7,2> 2406
T3M <5,3> 2649      T3MI <7,3> 717
T4M <5,4> 1203      T4MI <7,4> 1434
T5M <5,5> 2406      T5MI <7,5> 2868
T6M <5,6> 717      T6MI <7,6> 1641
T7M <5,7> 1434      T7MI <7,7> 3282
T8M <5,8> 2868      T8MI <7,8> 2469
T9M <5,9> 1641      T9MI <7,9> 843
T10M <5,10> 3282      T10MI <7,10> 1686
T11M <5,11> 2469      T11MI <7,11> 3372

The transformations that map this set to itself are: T0, T9I

## Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

 Scale 635 Epolian Scale 637 Debussy's Heptatonic Scale 625 Ionyptitonic Scale 629 Aeronimic Scale 617 Katycritonic Scale 601 Bycritonic Scale 569 Mothitonic Scale 697 Lagimic Scale 761 Ponian Scale 889 Borian Scale 121 Asoian Scale 377 Kathimic Scale 1145 Zygimic Scale 1657 Ionothian Scale 2681 Aerycrian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.