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Scale 847: "Ganian"

Scale 847: Ganian, 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

Zeitler
Ganian
Dozenal
Fekian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,3,6,8,9}

Forte Number

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

7-19

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: 3673

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.

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.

6

Prime Form

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

no
prime: 719

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, 1, 1, 3, 2, 1, 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.

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

Interval Spectrum

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

p4m3n4s3d4t3

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

Spectra Variation

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

2.286

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.

2.433

Polygon Perimeter

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

5.899

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.

(16, 29, 92)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsD{2,6,9}231.71
G♯{8,0,3}231.71
Minor Triadsf♯m{6,9,1}231.71
Diminished Triads{0,3,6}231.71
d♯°{3,6,9}231.71
f♯°{6,9,0}231.71
{9,0,3}231.71
Parsimonious Voice Leading Between Common Triads of Scale 847. Created by Ian Ring ©2019 d#° d#° c°->d#° G# G# c°->G# D D D->d#° f#m f#m D->f#m f#° f#° f#°->f#m f#°->a° G#->a°

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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 847 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2471
Scale 2471: Mela Ganamurti, Ian Ring Music TheoryMela Ganamurti
3rd mode:
Scale 3283
Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari
4th mode:
Scale 3689
Scale 3689: Katocrian, Ian Ring Music TheoryKatocrian
5th mode:
Scale 973
Scale 973: Mela Syamalangi, Ian Ring Music TheoryMela Syamalangi
6th mode:
Scale 1267
Scale 1267: Katynian, Ian Ring Music TheoryKatynian
7th mode:
Scale 2681
Scale 2681: Aerycrian, Ian Ring Music TheoryAerycrian

Prime

The prime form of this scale is Scale 719

Scale 719Scale 719: Kanian, Ian Ring Music TheoryKanian

Complement

The heptatonic modal family [847, 2471, 3283, 3689, 973, 1267, 2681] (Forte: 7-19) is the complement of the pentatonic modal family [203, 707, 1561, 2149, 2401] (Forte: 5-19)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 847 is 3673

Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 847 is chiral, and its enantiomorph is scale 3673

Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian

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> 847       T0I <11,0> 3673
T1 <1,1> 1694      T1I <11,1> 3251
T2 <1,2> 3388      T2I <11,2> 2407
T3 <1,3> 2681      T3I <11,3> 719
T4 <1,4> 1267      T4I <11,4> 1438
T5 <1,5> 2534      T5I <11,5> 2876
T6 <1,6> 973      T6I <11,6> 1657
T7 <1,7> 1946      T7I <11,7> 3314
T8 <1,8> 3892      T8I <11,8> 2533
T9 <1,9> 3689      T9I <11,9> 971
T10 <1,10> 3283      T10I <11,10> 1942
T11 <1,11> 2471      T11I <11,11> 3884
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1657      T0MI <7,0> 973
T1M <5,1> 3314      T1MI <7,1> 1946
T2M <5,2> 2533      T2MI <7,2> 3892
T3M <5,3> 971      T3MI <7,3> 3689
T4M <5,4> 1942      T4MI <7,4> 3283
T5M <5,5> 3884      T5MI <7,5> 2471
T6M <5,6> 3673      T6MI <7,6> 847
T7M <5,7> 3251      T7MI <7,7> 1694
T8M <5,8> 2407      T8MI <7,8> 3388
T9M <5,9> 719      T9MI <7,9> 2681
T10M <5,10> 1438      T10MI <7,10> 1267
T11M <5,11> 2876      T11MI <7,11> 2534

The transformations that map this set to itself are: T0, T6MI

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 845Scale 845: Raga Neelangi, Ian Ring Music TheoryRaga Neelangi
Scale 843Scale 843: Molimic, Ian Ring Music TheoryMolimic
Scale 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian
Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic
Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic
Scale 783Scale 783: Etuian, Ian Ring Music TheoryEtuian
Scale 815Scale 815: Bolian, Ian Ring Music TheoryBolian
Scale 911Scale 911: Radian, Ian Ring Music TheoryRadian
Scale 975Scale 975: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4
Scale 591Scale 591: Gaptimic, Ian Ring Music TheoryGaptimic
Scale 719Scale 719: Kanian, Ian Ring Music TheoryKanian
Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic
Scale 1359Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
Scale 1871Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic
Scale 2895Scale 2895: Aeoryllic, Ian Ring Music TheoryAeoryllic

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.