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Scale 1051: "Gifian"

Scale 1051: Gifian, 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
Gifian

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

{0,1,3,4,10}

Forte Number

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

5-10

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: 2821

Hemitonia

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

2 (dihemitonic)

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.

4

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: 91

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, 6, 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.

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

Interval Spectrum

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

pmn3s2d2t

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> = {1,2,6}
<2> = {3,7,8}
<3> = {4,5,9}
<4> = {6,10,11}

Spectra Variation

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

4

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.

1.366

Polygon Perimeter

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

5.035

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.

(14, 1, 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
Diminished Triadsa♯°{10,1,4}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 1051 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2573
Scale 2573: Pulian, Ian Ring Music TheoryPulian
3rd mode:
Scale 1667
Scale 1667: Kekian, Ian Ring Music TheoryKekian
4th mode:
Scale 2881
Scale 2881: Satian, Ian Ring Music TheorySatian
5th mode:
Scale 109
Scale 109: Amsian, Ian Ring Music TheoryAmsian

Prime

The prime form of this scale is Scale 91

Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian

Complement

The pentatonic modal family [1051, 2573, 1667, 2881, 109] (Forte: 5-10) is the complement of the heptatonic modal family [607, 761, 1993, 2351, 3223, 3659, 3877] (Forte: 7-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1051 is 2821

Scale 2821Scale 2821: Rukian, Ian Ring Music TheoryRukian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1051 is chiral, and its enantiomorph is scale 2821

Scale 2821Scale 2821: Rukian, Ian Ring Music TheoryRukian

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> 1051       T0I <11,0> 2821
T1 <1,1> 2102      T1I <11,1> 1547
T2 <1,2> 109      T2I <11,2> 3094
T3 <1,3> 218      T3I <11,3> 2093
T4 <1,4> 436      T4I <11,4> 91
T5 <1,5> 872      T5I <11,5> 182
T6 <1,6> 1744      T6I <11,6> 364
T7 <1,7> 3488      T7I <11,7> 728
T8 <1,8> 2881      T8I <11,8> 1456
T9 <1,9> 1667      T9I <11,9> 2912
T10 <1,10> 3334      T10I <11,10> 1729
T11 <1,11> 2573      T11I <11,11> 3458
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 301      T0MI <7,0> 1681
T1M <5,1> 602      T1MI <7,1> 3362
T2M <5,2> 1204      T2MI <7,2> 2629
T3M <5,3> 2408      T3MI <7,3> 1163
T4M <5,4> 721      T4MI <7,4> 2326
T5M <5,5> 1442      T5MI <7,5> 557
T6M <5,6> 2884      T6MI <7,6> 1114
T7M <5,7> 1673      T7MI <7,7> 2228
T8M <5,8> 3346      T8MI <7,8> 361
T9M <5,9> 2597      T9MI <7,9> 722
T10M <5,10> 1099      T10MI <7,10> 1444
T11M <5,11> 2198      T11MI <7,11> 2888

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 1049Scale 1049: Gidian, Ian Ring Music TheoryGidian
Scale 1053Scale 1053: Gigian, Ian Ring Music TheoryGigian
Scale 1055Scale 1055: Gihian, Ian Ring Music TheoryGihian
Scale 1043Scale 1043: Gizian, Ian Ring Music TheoryGizian
Scale 1047Scale 1047: Gician, Ian Ring Music TheoryGician
Scale 1035Scale 1035: Givian, Ian Ring Music TheoryGivian
Scale 1067Scale 1067: Gopian, Ian Ring Music TheoryGopian
Scale 1083Scale 1083: Goyian, Ian Ring Music TheoryGoyian
Scale 1115Scale 1115: Superlocrian Hexamirror, Ian Ring Music TheorySuperlocrian Hexamirror
Scale 1179Scale 1179: Sonimic, Ian Ring Music TheorySonimic
Scale 1307Scale 1307: Katorimic, Ian Ring Music TheoryKatorimic
Scale 1563Scale 1563: Joyian, Ian Ring Music TheoryJoyian
Scale 27Scale 27: Adoian, Ian Ring Music TheoryAdoian
Scale 539Scale 539: Delian, Ian Ring Music TheoryDelian
Scale 2075Scale 2075: Mozian, Ian Ring Music TheoryMozian
Scale 3099Scale 3099: Tixian, Ian Ring Music TheoryTixian

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.