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# Scale 2709: "Raga Kumud" ### 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).

## Common Names

Carnatic
Raga Kumud
Dozenal
Rasian
Unknown / Unsorted
Sankara
Shankara
Prabhati
Western Modern
Lydian Hexatonic
Zeitler
Thaptimic

## 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

{0,2,4,7,9,11}

#### Forte Number

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

6-32

#### 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.

[5.5]

#### 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.

1 (unhemitonic)

#### Cohemitonia

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

0 (ancohemitonic)

#### 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.

1

#### 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: 693

#### Generator

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

generator: 5
origin: 11

#### Deep Scale

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

yes

#### Interval Structure

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

[2, 2, 3, 2, 2, 1]

#### 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.

<1, 4, 3, 2, 5, 0>

#### Interval Spectrum

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

p5m2n3s4d

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

#### Spectra Variation

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

1.667

#### 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.

2.482

#### Polygon Perimeter

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

5.932

#### 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".

Proper

#### 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.

(0, 16, 51)

## Generator

This scale has a generator of 5, originating on 11.

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

G{7,11,2}131.5
am{9,0,4}131.5

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 2 no C, em G, am

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.

 Major: {7, 11, 2}Minor: {9, 0, 4}

## Modes

Modes are the rotational transformation of this scale. Scale 2709 can be rotated to make 5 other scales. The 1st mode is itself.

 2nd mode:Scale 1701 Dominant Seventh 3rd mode:Scale 1449 Raga Gopikavasantam 4th mode:Scale 693 Arezzo Major Diatonic Hexachord This is the prime mode 5th mode:Scale 1197 Minor Hexatonic 6th mode:Scale 1323 Ritsu

## Prime

The prime form of this scale is Scale 693

 Scale 693 Arezzo Major Diatonic Hexachord

## Complement

The hexatonic modal family [2709, 1701, 1449, 693, 1197, 1323] (Forte: 6-32) is the complement of the hexatonic modal family [693, 1197, 1323, 1449, 1701, 2709] (Forte: 6-32)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2709 is 1323

 Scale 1323 Ritsu

## 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> 2709       T0I <11,0> 1323
T1 <1,1> 1323      T1I <11,1> 2646
T2 <1,2> 2646      T2I <11,2> 1197
T3 <1,3> 1197      T3I <11,3> 2394
T4 <1,4> 2394      T4I <11,4> 693
T5 <1,5> 693      T5I <11,5> 1386
T6 <1,6> 1386      T6I <11,6> 2772
T7 <1,7> 2772      T7I <11,7> 1449
T8 <1,8> 1449      T8I <11,8> 2898
T9 <1,9> 2898      T9I <11,9> 1701
T10 <1,10> 1701      T10I <11,10> 3402
T11 <1,11> 3402      T11I <11,11> 2709
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3969      T0MI <7,0> 63
T1M <5,1> 3843      T1MI <7,1> 126
T2M <5,2> 3591      T2MI <7,2> 252
T3M <5,3> 3087      T3MI <7,3> 504
T4M <5,4> 2079      T4MI <7,4> 1008
T5M <5,5> 63      T5MI <7,5> 2016
T6M <5,6> 126      T6MI <7,6> 4032
T7M <5,7> 252      T7MI <7,7> 3969
T8M <5,8> 504      T8MI <7,8> 3843
T9M <5,9> 1008      T9MI <7,9> 3591
T10M <5,10> 2016      T10MI <7,10> 3087
T11M <5,11> 4032      T11MI <7,11> 2079

The transformations that map this set to itself are: T0, T11I

## 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 2711 Stolian Scale 2705 Raga Mamata Scale 2707 Banimic Scale 2713 Porimic Scale 2717 Epygian Scale 2693 Rajian Scale 2701 Hawaiian Scale 2725 Raga Nagagandhari Scale 2741 Major Scale 2773 Lydian Scale 2581 Raga Neroshta Scale 2645 Raga Mruganandana Scale 2837 Aelothimic Scale 2965 Darian Scale 2197 Raga Hamsadhvani Scale 2453 Raga Latika Scale 3221 Bycrimic Scale 3733 Gycrian Scale 661 Major Pentatonic Scale 1685 Zeracrimic

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.