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Scale 1065: "Gonian"

Scale 1065: Gonian, 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
Gonian

Analysis

Cardinality

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

4 (tetratonic)

Pitch Class Set

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

{0,3,5,10}

Forte Number

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

4-23

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.

[1.5]

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.

0 (anhemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

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.

1

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.

3

Prime Form

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

no
prime: 165

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 5
origin: 0

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.

[3, 2, 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.

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

Interval Spectrum

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

p3ns2

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> = {2,3,5}
<2> = {5,7}
<3> = {7,9,10}

Spectra Variation

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

2

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.

1.616

Polygon Perimeter

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

5.346

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.

[3]

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".

Proper

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.

(0, 4, 14)

Generator

This scale has a generator of 5, originating on 0.

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 1065 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 645
Scale 645: Duyian, Ian Ring Music TheoryDuyian
3rd mode:
Scale 1185
Scale 1185: Genus Primum Inverse, Ian Ring Music TheoryGenus Primum Inverse
4th mode:
Scale 165
Scale 165: Genus Primum, Ian Ring Music TheoryGenus PrimumThis is the prime mode

Prime

The prime form of this scale is Scale 165

Scale 165Scale 165: Genus Primum, Ian Ring Music TheoryGenus Primum

Complement

The tetratonic modal family [1065, 645, 1185, 165] (Forte: 4-23) is the complement of the octatonic modal family [1455, 1515, 1725, 1965, 2775, 2805, 3435, 3765] (Forte: 8-23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1065 is 645

Scale 645Scale 645: Duyian, Ian Ring Music TheoryDuyian

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> 1065       T0I <11,0> 645
T1 <1,1> 2130      T1I <11,1> 1290
T2 <1,2> 165      T2I <11,2> 2580
T3 <1,3> 330      T3I <11,3> 1065
T4 <1,4> 660      T4I <11,4> 2130
T5 <1,5> 1320      T5I <11,5> 165
T6 <1,6> 2640      T6I <11,6> 330
T7 <1,7> 1185      T7I <11,7> 660
T8 <1,8> 2370      T8I <11,8> 1320
T9 <1,9> 645      T9I <11,9> 2640
T10 <1,10> 1290      T10I <11,10> 1185
T11 <1,11> 2580      T11I <11,11> 2370
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 15      T0MI <7,0> 3585
T1M <5,1> 30      T1MI <7,1> 3075
T2M <5,2> 60      T2MI <7,2> 2055
T3M <5,3> 120      T3MI <7,3> 15
T4M <5,4> 240      T4MI <7,4> 30
T5M <5,5> 480      T5MI <7,5> 60
T6M <5,6> 960      T6MI <7,6> 120
T7M <5,7> 1920      T7MI <7,7> 240
T8M <5,8> 3840      T8MI <7,8> 480
T9M <5,9> 3585      T9MI <7,9> 960
T10M <5,10> 3075      T10MI <7,10> 1920
T11M <5,11> 2055      T11MI <7,11> 3840

The transformations that map this set to itself are: T0, T3I

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 1067Scale 1067: Gopian, Ian Ring Music TheoryGopian
Scale 1069Scale 1069: Goqian, Ian Ring Music TheoryGoqian
Scale 1057Scale 1057: Sansagari, Ian Ring Music TheorySansagari
Scale 1061Scale 1061: Gilian, Ian Ring Music TheoryGilian
Scale 1073Scale 1073: Gosian, Ian Ring Music TheoryGosian
Scale 1081Scale 1081: Goxian, Ian Ring Music TheoryGoxian
Scale 1033Scale 1033: Allian, Ian Ring Music TheoryAllian
Scale 1049Scale 1049: Gidian, Ian Ring Music TheoryGidian
Scale 1097Scale 1097: Aeraphic, Ian Ring Music TheoryAeraphic
Scale 1129Scale 1129: Raga Jayakauns, Ian Ring Music TheoryRaga Jayakauns
Scale 1193Scale 1193: Minor Pentatonic, Ian Ring Music TheoryMinor Pentatonic
Scale 1321Scale 1321: Blues Minor, Ian Ring Music TheoryBlues Minor
Scale 1577Scale 1577: Raga Chandrakauns (Kafi), Ian Ring Music TheoryRaga Chandrakauns (Kafi)
Scale 41Scale 41: Vietnamese Tritonic, Ian Ring Music TheoryVietnamese Tritonic
Scale 553Scale 553: Rothic 2, Ian Ring Music TheoryRothic 2
Scale 2089Scale 2089: Mujian, Ian Ring Music TheoryMujian
Scale 3113Scale 3113: Tigian, Ian Ring Music TheoryTigian

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.