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Scale 2945: "Sihian"

Scale 2945: Sihian, 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
Sihian

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,7,8,9,11}

Forte Number

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

5-3

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.

none

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.

yes
enantiomorph: 59

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.

1 (uncohemitonic)

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.

4

Prime Form

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

no
prime: 55

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.

[7, 1, 1, 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.

<3, 2, 2, 2, 1, 0>

Interval Spectrum

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

pm2n2s2d3

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

Spectra Variation

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

4.8

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.

0.933

Polygon Perimeter

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

4.485

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.

none

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.

(15, 2, 30)

Common Triads

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 2945 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 55
Scale 55: Aspian, Ian Ring Music TheoryAspianThis is the prime mode
3rd mode:
Scale 2075
Scale 2075: Mozian, Ian Ring Music TheoryMozian
4th mode:
Scale 3085
Scale 3085: Tepian, Ian Ring Music TheoryTepian
5th mode:
Scale 1795
Scale 1795: Lakian, Ian Ring Music TheoryLakian

Prime

The prime form of this scale is Scale 55

Scale 55Scale 55: Aspian, Ian Ring Music TheoryAspian

Complement

The pentatonic modal family [2945, 55, 2075, 3085, 1795] (Forte: 5-3) is the complement of the heptatonic modal family [319, 1009, 2207, 3151, 3623, 3859, 3977] (Forte: 7-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2945 is 59

Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2945 is chiral, and its enantiomorph is scale 59

Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian

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> 2945       T0I <11,0> 59
T1 <1,1> 1795      T1I <11,1> 118
T2 <1,2> 3590      T2I <11,2> 236
T3 <1,3> 3085      T3I <11,3> 472
T4 <1,4> 2075      T4I <11,4> 944
T5 <1,5> 55      T5I <11,5> 1888
T6 <1,6> 110      T6I <11,6> 3776
T7 <1,7> 220      T7I <11,7> 3457
T8 <1,8> 440      T8I <11,8> 2819
T9 <1,9> 880      T9I <11,9> 1543
T10 <1,10> 1760      T10I <11,10> 3086
T11 <1,11> 3520      T11I <11,11> 2077
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2705      T0MI <7,0> 299
T1M <5,1> 1315      T1MI <7,1> 598
T2M <5,2> 2630      T2MI <7,2> 1196
T3M <5,3> 1165      T3MI <7,3> 2392
T4M <5,4> 2330      T4MI <7,4> 689
T5M <5,5> 565      T5MI <7,5> 1378
T6M <5,6> 1130      T6MI <7,6> 2756
T7M <5,7> 2260      T7MI <7,7> 1417
T8M <5,8> 425      T8MI <7,8> 2834
T9M <5,9> 850      T9MI <7,9> 1573
T10M <5,10> 1700      T10MI <7,10> 3146
T11M <5,11> 3400      T11MI <7,11> 2197

The transformations that map this set to itself are: T0

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 2947Scale 2947: Sijian, Ian Ring Music TheorySijian
Scale 2949Scale 2949: Sikian, Ian Ring Music TheorySikian
Scale 2953Scale 2953: Ionylimic, Ian Ring Music TheoryIonylimic
Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic
Scale 2977Scale 2977: Sobian, Ian Ring Music TheorySobian
Scale 3009Scale 3009: Suvian, Ian Ring Music TheorySuvian
Scale 2817Scale 2817, Ian Ring Music Theory
Scale 2881Scale 2881: Satian, Ian Ring Music TheorySatian
Scale 2689Scale 2689: Ragian, Ian Ring Music TheoryRagian
Scale 2433Scale 2433: Pacian, Ian Ring Music TheoryPacian
Scale 3457Scale 3457: Vobian, Ian Ring Music TheoryVobian
Scale 3969Scale 3969: Hexatonic Chromatic Descending, Ian Ring Music TheoryHexatonic Chromatic Descending
Scale 897Scale 897: Fopian, Ian Ring Music TheoryFopian
Scale 1921Scale 1921: Lukian, Ian Ring Music TheoryLukian

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.