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Scale 305: "Gonic"

Scale 305: Gonic, 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
Gonic
Dozenal
Buyian

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

{0,4,5,8}

Forte Number

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

4-19

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: 401

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

1 (unhemitonic)

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.

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.

3

Prime Form

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

no
prime: 275

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.

[4, 1, 3, 4]

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.

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

Interval Spectrum

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

pm3nd

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

Spectra Variation

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

2.5

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.

5.396

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".

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, 16)

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 Triadsfm{5,8,0}110.5
Augmented TriadsC+{0,4,8}110.5
Parsimonious Voice Leading Between Common Triads of Scale 305. Created by Ian Ring ©2019 C+ C+ fm fm C+->fm

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

2nd mode:
Scale 275
Scale 275: Dalic, Ian Ring Music TheoryDalicThis is the prime mode
3rd mode:
Scale 2185
Scale 2185: Dygic, Ian Ring Music TheoryDygic
4th mode:
Scale 785
Scale 785: Aeoloric, Ian Ring Music TheoryAeoloric

Prime

The prime form of this scale is Scale 275

Scale 275Scale 275: Dalic, Ian Ring Music TheoryDalic

Complement

The tetratonic modal family [305, 275, 2185, 785] (Forte: 4-19) is the complement of the octatonic modal family [887, 1847, 1907, 2491, 2971, 3001, 3293, 3533] (Forte: 8-19)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 305 is 401

Scale 401Scale 401: Epogic, Ian Ring Music TheoryEpogic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 305 is chiral, and its enantiomorph is scale 401

Scale 401Scale 401: Epogic, Ian Ring Music TheoryEpogic

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> 305       T0I <11,0> 401
T1 <1,1> 610      T1I <11,1> 802
T2 <1,2> 1220      T2I <11,2> 1604
T3 <1,3> 2440      T3I <11,3> 3208
T4 <1,4> 785      T4I <11,4> 2321
T5 <1,5> 1570      T5I <11,5> 547
T6 <1,6> 3140      T6I <11,6> 1094
T7 <1,7> 2185      T7I <11,7> 2188
T8 <1,8> 275      T8I <11,8> 281
T9 <1,9> 550      T9I <11,9> 562
T10 <1,10> 1100      T10I <11,10> 1124
T11 <1,11> 2200      T11I <11,11> 2248
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 275      T0MI <7,0> 2321
T1M <5,1> 550      T1MI <7,1> 547
T2M <5,2> 1100      T2MI <7,2> 1094
T3M <5,3> 2200      T3MI <7,3> 2188
T4M <5,4> 305       T4MI <7,4> 281
T5M <5,5> 610      T5MI <7,5> 562
T6M <5,6> 1220      T6MI <7,6> 1124
T7M <5,7> 2440      T7MI <7,7> 2248
T8M <5,8> 785      T8MI <7,8> 401
T9M <5,9> 1570      T9MI <7,9> 802
T10M <5,10> 3140      T10MI <7,10> 1604
T11M <5,11> 2185      T11MI <7,11> 3208

The transformations that map this set to itself are: T0, T4M

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 307Scale 307: Raga Megharanjani, Ian Ring Music TheoryRaga Megharanjani
Scale 309Scale 309: Palitonic, Ian Ring Music TheoryPalitonic
Scale 313Scale 313: Goritonic, Ian Ring Music TheoryGoritonic
Scale 289Scale 289: Valian, Ian Ring Music TheoryValian
Scale 297Scale 297: Mynic, Ian Ring Music TheoryMynic
Scale 273Scale 273: Augmented Triad, Ian Ring Music TheoryAugmented Triad
Scale 337Scale 337: Koptic, Ian Ring Music TheoryKoptic
Scale 369Scale 369: Laditonic, Ian Ring Music TheoryLaditonic
Scale 433Scale 433: Raga Zilaf, Ian Ring Music TheoryRaga Zilaf
Scale 49Scale 49: Aguian, Ian Ring Music TheoryAguian
Scale 177Scale 177: Bexian, Ian Ring Music TheoryBexian
Scale 561Scale 561: Phratic, Ian Ring Music TheoryPhratic
Scale 817Scale 817: Zothitonic, Ian Ring Music TheoryZothitonic
Scale 1329Scale 1329: Epygitonic, Ian Ring Music TheoryEpygitonic
Scale 2353Scale 2353: Raga Girija, Ian Ring Music TheoryRaga Girija

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.