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Scale 1953: "Macian"

Scale 1953: Macian, Ian Ring Music Theory

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

Dozenal
Macian

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

{0,5,7,8,9,10}

Forte Number

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

6-8

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.

[2.5]

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.

3

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: 189

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.

[5, 2, 1, 1, 1, 2]

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, 4, 3, 2, 3, 0>

Interval Spectrum

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

p3m2n3s4d3

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

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.

1.866

Polygon Perimeter

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

5.485

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.

[5]

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.

(30, 16, 59)

Common Triads

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

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsF{5,9,0}110.5
Minor Triadsfm{5,8,0}110.5

The following pitch classes are not present in any of the common triads: {7,10}

Parsimonious Voice Leading Between Common Triads of Scale 1953. Created by Ian Ring ©2019 fm fm F F fm->F

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1953 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 189
Scale 189: Befian, Ian Ring Music TheoryBefianThis is the prime mode
3rd mode:
Scale 1071
Scale 1071: Gorian, Ian Ring Music TheoryGorian
4th mode:
Scale 2583
Scale 2583: Purian, Ian Ring Music TheoryPurian
5th mode:
Scale 3339
Scale 3339: Smuian, Ian Ring Music TheorySmuian
6th mode:
Scale 3717
Scale 3717: Xidian, Ian Ring Music TheoryXidian

Prime

The prime form of this scale is Scale 189

Scale 189Scale 189: Befian, Ian Ring Music TheoryBefian

Complement

The hexatonic modal family [1953, 189, 1071, 2583, 3339, 3717] (Forte: 6-8) is the complement of the hexatonic modal family [189, 1071, 1953, 2583, 3339, 3717] (Forte: 6-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1953 is 189

Scale 189Scale 189: Befian, Ian Ring Music TheoryBefian

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> 1953       T0I <11,0> 189
T1 <1,1> 3906      T1I <11,1> 378
T2 <1,2> 3717      T2I <11,2> 756
T3 <1,3> 3339      T3I <11,3> 1512
T4 <1,4> 2583      T4I <11,4> 3024
T5 <1,5> 1071      T5I <11,5> 1953
T6 <1,6> 2142      T6I <11,6> 3906
T7 <1,7> 189      T7I <11,7> 3717
T8 <1,8> 378      T8I <11,8> 3339
T9 <1,9> 756      T9I <11,9> 2583
T10 <1,10> 1512      T10I <11,10> 1071
T11 <1,11> 3024      T11I <11,11> 2142
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2583      T0MI <7,0> 3339
T1M <5,1> 1071      T1MI <7,1> 2583
T2M <5,2> 2142      T2MI <7,2> 1071
T3M <5,3> 189      T3MI <7,3> 2142
T4M <5,4> 378      T4MI <7,4> 189
T5M <5,5> 756      T5MI <7,5> 378
T6M <5,6> 1512      T6MI <7,6> 756
T7M <5,7> 3024      T7MI <7,7> 1512
T8M <5,8> 1953       T8MI <7,8> 3024
T9M <5,9> 3906      T9MI <7,9> 1953
T10M <5,10> 3717      T10MI <7,10> 3906
T11M <5,11> 3339      T11MI <7,11> 3717

The transformations that map this set to itself are: T0, T5I, T8M, T9MI

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 1955Scale 1955: Sonian, Ian Ring Music TheorySonian
Scale 1957Scale 1957: Pyrian, Ian Ring Music TheoryPyrian
Scale 1961Scale 1961: Soptian, Ian Ring Music TheorySoptian
Scale 1969Scale 1969: Stylian, Ian Ring Music TheoryStylian
Scale 1921Scale 1921: Lukian, Ian Ring Music TheoryLukian
Scale 1937Scale 1937: Galimic, Ian Ring Music TheoryGalimic
Scale 1985Scale 1985: Mewian, Ian Ring Music TheoryMewian
Scale 2017Scale 2017: Meqian, Ian Ring Music TheoryMeqian
Scale 1825Scale 1825: Lecian, Ian Ring Music TheoryLecian
Scale 1889Scale 1889: Loqian, Ian Ring Music TheoryLoqian
Scale 1697Scale 1697: Raga Kuntvarali, Ian Ring Music TheoryRaga Kuntvarali
Scale 1441Scale 1441: Jabian, Ian Ring Music TheoryJabian
Scale 929Scale 929: Fujian, Ian Ring Music TheoryFujian
Scale 2977Scale 2977: Sobian, Ian Ring Music TheorySobian
Scale 4001Scale 4001: Ziyian, Ian Ring Music TheoryZiyian

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.