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Scale 1067: "Gopian"

Scale 1067: Gopian, 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
Gopian

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

{0,1,3,5,10}

Forte Number

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

5-23

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.

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: 2693

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.

2

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: 173

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, 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.

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

Interval Spectrum

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

p3mn2s3d

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

Spectra Variation

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

3.2

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.

1.799

Polygon Perimeter

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

5.449

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.

(10, 4, 30)

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}000

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

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 1067 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2581
Scale 2581: Raga Neroshta, Ian Ring Music TheoryRaga Neroshta
3rd mode:
Scale 1669
Scale 1669: Raga Matha Kokila, Ian Ring Music TheoryRaga Matha Kokila
4th mode:
Scale 1441
Scale 1441: Jabian, Ian Ring Music TheoryJabian
5th mode:
Scale 173
Scale 173: Raga Purnalalita, Ian Ring Music TheoryRaga PurnalalitaThis is the prime mode

Prime

The prime form of this scale is Scale 173

Scale 173Scale 173: Raga Purnalalita, Ian Ring Music TheoryRaga Purnalalita

Complement

The pentatonic modal family [1067, 2581, 1669, 1441, 173] (Forte: 5-23) is the complement of the heptatonic modal family [701, 1199, 1513, 1957, 2647, 3371, 3733] (Forte: 7-23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1067 is 2693

Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1067 is chiral, and its enantiomorph is scale 2693

Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian

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> 1067       T0I <11,0> 2693
T1 <1,1> 2134      T1I <11,1> 1291
T2 <1,2> 173      T2I <11,2> 2582
T3 <1,3> 346      T3I <11,3> 1069
T4 <1,4> 692      T4I <11,4> 2138
T5 <1,5> 1384      T5I <11,5> 181
T6 <1,6> 2768      T6I <11,6> 362
T7 <1,7> 1441      T7I <11,7> 724
T8 <1,8> 2882      T8I <11,8> 1448
T9 <1,9> 1669      T9I <11,9> 2896
T10 <1,10> 3338      T10I <11,10> 1697
T11 <1,11> 2581      T11I <11,11> 3394
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 47      T0MI <7,0> 3713
T1M <5,1> 94      T1MI <7,1> 3331
T2M <5,2> 188      T2MI <7,2> 2567
T3M <5,3> 376      T3MI <7,3> 1039
T4M <5,4> 752      T4MI <7,4> 2078
T5M <5,5> 1504      T5MI <7,5> 61
T6M <5,6> 3008      T6MI <7,6> 122
T7M <5,7> 1921      T7MI <7,7> 244
T8M <5,8> 3842      T8MI <7,8> 488
T9M <5,9> 3589      T9MI <7,9> 976
T10M <5,10> 3083      T10MI <7,10> 1952
T11M <5,11> 2071      T11MI <7,11> 3904

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 1065Scale 1065: Gonian, Ian Ring Music TheoryGonian
Scale 1069Scale 1069: Goqian, Ian Ring Music TheoryGoqian
Scale 1071Scale 1071: Gorian, Ian Ring Music TheoryGorian
Scale 1059Scale 1059: Gikian, Ian Ring Music TheoryGikian
Scale 1063Scale 1063: Gomian, Ian Ring Music TheoryGomian
Scale 1075Scale 1075: Gotian, Ian Ring Music TheoryGotian
Scale 1083Scale 1083: Goyian, Ian Ring Music TheoryGoyian
Scale 1035Scale 1035: Givian, Ian Ring Music TheoryGivian
Scale 1051Scale 1051: Gifian, Ian Ring Music TheoryGifian
Scale 1099Scale 1099: Dyritonic, Ian Ring Music TheoryDyritonic
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1195Scale 1195: Raga Gandharavam, Ian Ring Music TheoryRaga Gandharavam
Scale 1323Scale 1323: Ritsu, Ian Ring Music TheoryRitsu
Scale 1579Scale 1579: Sagimic, Ian Ring Music TheorySagimic
Scale 43Scale 43: Alfian, Ian Ring Music TheoryAlfian
Scale 555Scale 555: Aeolycritonic, Ian Ring Music TheoryAeolycritonic
Scale 2091Scale 2091: Mukian, Ian Ring Music TheoryMukian
Scale 3115Scale 3115: Tihian, Ian Ring Music TheoryTihian

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.