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Scale 1083: "Goyian"

Scale 1083: Goyian, 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
Goyian

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

{0,1,3,4,5,10}

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-Z11

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: 2949

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.

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: 183

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, 5, 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, 3, 3, 2, 3, 1>

Interval Spectrum

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

p3m2n3s3d3t

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

Spectra Variation

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

3.667

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.

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.

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.

(27, 11, 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
Minor Triadsa♯m{10,1,5}110.5
Diminished Triadsa♯°{10,1,4}110.5

The following pitch classes are not present in any of the common triads: {0,3}

Parsimonious Voice Leading Between Common Triads of Scale 1083. Created by Ian Ring ©2019 a#° a#° a#m a#m a#°->a#m

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 1083 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2589
Scale 2589: Puvian, Ian Ring Music TheoryPuvian
3rd mode:
Scale 1671
Scale 1671: Kemian, Ian Ring Music TheoryKemian
4th mode:
Scale 2883
Scale 2883: Savian, Ian Ring Music TheorySavian
5th mode:
Scale 3489
Scale 3489: Vuvian, Ian Ring Music TheoryVuvian
6th mode:
Scale 237
Scale 237: Bijian, Ian Ring Music TheoryBijian

Prime

The prime form of this scale is Scale 183

Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian

Complement

The hexatonic modal family [1083, 2589, 1671, 2883, 3489, 237] (Forte: 6-Z11) is the complement of the hexatonic modal family [303, 753, 1929, 2199, 3147, 3621] (Forte: 6-Z40)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1083 is 2949

Scale 2949Scale 2949: Sikian, Ian Ring Music TheorySikian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1083 is chiral, and its enantiomorph is scale 2949

Scale 2949Scale 2949: Sikian, Ian Ring Music TheorySikian

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> 1083       T0I <11,0> 2949
T1 <1,1> 2166      T1I <11,1> 1803
T2 <1,2> 237      T2I <11,2> 3606
T3 <1,3> 474      T3I <11,3> 3117
T4 <1,4> 948      T4I <11,4> 2139
T5 <1,5> 1896      T5I <11,5> 183
T6 <1,6> 3792      T6I <11,6> 366
T7 <1,7> 3489      T7I <11,7> 732
T8 <1,8> 2883      T8I <11,8> 1464
T9 <1,9> 1671      T9I <11,9> 2928
T10 <1,10> 3342      T10I <11,10> 1761
T11 <1,11> 2589      T11I <11,11> 3522
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 303      T0MI <7,0> 3729
T1M <5,1> 606      T1MI <7,1> 3363
T2M <5,2> 1212      T2MI <7,2> 2631
T3M <5,3> 2424      T3MI <7,3> 1167
T4M <5,4> 753      T4MI <7,4> 2334
T5M <5,5> 1506      T5MI <7,5> 573
T6M <5,6> 3012      T6MI <7,6> 1146
T7M <5,7> 1929      T7MI <7,7> 2292
T8M <5,8> 3858      T8MI <7,8> 489
T9M <5,9> 3621      T9MI <7,9> 978
T10M <5,10> 3147      T10MI <7,10> 1956
T11M <5,11> 2199      T11MI <7,11> 3912

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 1081Scale 1081: Goxian, Ian Ring Music TheoryGoxian
Scale 1085Scale 1085: Gozian, Ian Ring Music TheoryGozian
Scale 1087Scale 1087: Gobian, Ian Ring Music TheoryGobian
Scale 1075Scale 1075: Gotian, Ian Ring Music TheoryGotian
Scale 1079Scale 1079: Gowian, Ian Ring Music TheoryGowian
Scale 1067Scale 1067: Gopian, Ian Ring Music TheoryGopian
Scale 1051Scale 1051: Gifian, Ian Ring Music TheoryGifian
Scale 1115Scale 1115: Superlocrian Hexamirror, Ian Ring Music TheorySuperlocrian Hexamirror
Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian
Scale 1211Scale 1211: Zadian, Ian Ring Music TheoryZadian
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 1595Scale 1595: Dacrian, Ian Ring Music TheoryDacrian
Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian
Scale 571Scale 571: Kynimic, Ian Ring Music TheoryKynimic
Scale 2107Scale 2107: Mutian, Ian Ring Music TheoryMutian
Scale 3131Scale 3131: Torian, Ian Ring Music TheoryTorian

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.