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# Scale 2107: "Mutian" ### 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).

Dozenal
Mutian

## 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

{0,1,3,4,5,11}

#### 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-Z4

#### 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.



#### 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.

4 (multihemitonic)

#### 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: 119

#### 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.

[1, 2, 1, 1, 6, 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.

<4, 3, 2, 3, 2, 1>

#### Interval Spectrum

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

p2m3n2s3d4t

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

#### Spectra Variation

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

4

#### 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.

1.433

#### Polygon Perimeter

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

5.071

#### 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.

(34, 5, 51)

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

## Modes

Modes are the rotational transformation of this scale. Scale 2107 can be rotated to make 5 other scales. The 1st mode is itself.

 2nd mode:Scale 3101 Tiyian 3rd mode:Scale 1799 Lamian 4th mode:Scale 2947 Sijian 5th mode:Scale 3521 Wanian 6th mode:Scale 119 Smoian This is the prime mode

## Prime

The prime form of this scale is Scale 119

 Scale 119 Smoian

## Complement

The hexatonic modal family [2107, 3101, 1799, 2947, 3521, 119] (Forte: 6-Z4) is the complement of the hexatonic modal family [287, 497, 2191, 3143, 3619, 3857] (Forte: 6-Z37)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2107 is 2947

 Scale 2947 Sijian

## 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> 2107       T0I <11,0> 2947
T1 <1,1> 119      T1I <11,1> 1799
T2 <1,2> 238      T2I <11,2> 3598
T3 <1,3> 476      T3I <11,3> 3101
T4 <1,4> 952      T4I <11,4> 2107
T5 <1,5> 1904      T5I <11,5> 119
T6 <1,6> 3808      T6I <11,6> 238
T7 <1,7> 3521      T7I <11,7> 476
T8 <1,8> 2947      T8I <11,8> 952
T9 <1,9> 1799      T9I <11,9> 1904
T10 <1,10> 3598      T10I <11,10> 3808
T11 <1,11> 3101      T11I <11,11> 3521
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 427      T0MI <7,0> 2737
T1M <5,1> 854      T1MI <7,1> 1379
T2M <5,2> 1708      T2MI <7,2> 2758
T3M <5,3> 3416      T3MI <7,3> 1421
T4M <5,4> 2737      T4MI <7,4> 2842
T5M <5,5> 1379      T5MI <7,5> 1589
T6M <5,6> 2758      T6MI <7,6> 3178
T7M <5,7> 1421      T7MI <7,7> 2261
T8M <5,8> 2842      T8MI <7,8> 427
T9M <5,9> 1589      T9MI <7,9> 854
T10M <5,10> 3178      T10MI <7,10> 1708
T11M <5,11> 2261      T11MI <7,11> 3416

The transformations that map this set to itself are: T0, T4I

## 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 2105 Rigian Scale 2109 Muvian Scale 2111 Heptatonic Chromatic 2 Scale 2099 Raga Megharanji Scale 2103 Murian Scale 2091 Mukian Scale 2075 Mozian Scale 2139 Namian Scale 2171 Negian Scale 2235 Bathian Scale 2363 Kataptian Scale 2619 Ionyrian Scale 3131 Torian Scale 59 Ahuian Scale 1083 Goyian

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.