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Scale 1075: "Gotian"

Scale 1075: Gotian, 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
Gotian

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

{0,1,4,5,10}

Forte Number

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

5-Z18

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: 2437

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.

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.

4

Prime Form

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

no
prime: 179

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, 3, 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.

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

Interval Spectrum

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

p2m2n2sd2t

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

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.

1.683

Polygon Perimeter

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

5.381

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.

(9, 5, 36)

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}

Parsimonious Voice Leading Between Common Triads of Scale 1075. 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 1075 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2585
Scale 2585: Otlian, Ian Ring Music TheoryOtlian
3rd mode:
Scale 835
Scale 835: Fecian, Ian Ring Music TheoryFecian
4th mode:
Scale 2465
Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani
5th mode:
Scale 205
Scale 205: Bepian, Ian Ring Music TheoryBepian

Prime

The prime form of this scale is Scale 179

Scale 179Scale 179: Beyian, Ian Ring Music TheoryBeyian

Complement

The pentatonic modal family [1075, 2585, 835, 2465, 205] (Forte: 5-Z18) is the complement of the heptatonic modal family [755, 815, 1945, 2425, 2455, 3275, 3685] (Forte: 7-Z18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1075 is 2437

Scale 2437Scale 2437: Pafian, Ian Ring Music TheoryPafian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1075 is chiral, and its enantiomorph is scale 2437

Scale 2437Scale 2437: Pafian, Ian Ring Music TheoryPafian

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> 1075       T0I <11,0> 2437
T1 <1,1> 2150      T1I <11,1> 779
T2 <1,2> 205      T2I <11,2> 1558
T3 <1,3> 410      T3I <11,3> 3116
T4 <1,4> 820      T4I <11,4> 2137
T5 <1,5> 1640      T5I <11,5> 179
T6 <1,6> 3280      T6I <11,6> 358
T7 <1,7> 2465      T7I <11,7> 716
T8 <1,8> 835      T8I <11,8> 1432
T9 <1,9> 1670      T9I <11,9> 2864
T10 <1,10> 3340      T10I <11,10> 1633
T11 <1,11> 2585      T11I <11,11> 3266
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 295      T0MI <7,0> 3217
T1M <5,1> 590      T1MI <7,1> 2339
T2M <5,2> 1180      T2MI <7,2> 583
T3M <5,3> 2360      T3MI <7,3> 1166
T4M <5,4> 625      T4MI <7,4> 2332
T5M <5,5> 1250      T5MI <7,5> 569
T6M <5,6> 2500      T6MI <7,6> 1138
T7M <5,7> 905      T7MI <7,7> 2276
T8M <5,8> 1810      T8MI <7,8> 457
T9M <5,9> 3620      T9MI <7,9> 914
T10M <5,10> 3145      T10MI <7,10> 1828
T11M <5,11> 2195      T11MI <7,11> 3656

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 1073Scale 1073: Gosian, Ian Ring Music TheoryGosian
Scale 1077Scale 1077: Govian, Ian Ring Music TheoryGovian
Scale 1079Scale 1079: Gowian, Ian Ring Music TheoryGowian
Scale 1083Scale 1083: Goyian, Ian Ring Music TheoryGoyian
Scale 1059Scale 1059: Gikian, Ian Ring Music TheoryGikian
Scale 1067Scale 1067: Gopian, Ian Ring Music TheoryGopian
Scale 1043Scale 1043: Gizian, Ian Ring Music TheoryGizian
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
Scale 1139Scale 1139: Aerygimic, Ian Ring Music TheoryAerygimic
Scale 1203Scale 1203: Pagimic, Ian Ring Music TheoryPagimic
Scale 1331Scale 1331: Raga Vasantabhairavi, Ian Ring Music TheoryRaga Vasantabhairavi
Scale 1587Scale 1587: Raga Rudra Pancama, Ian Ring Music TheoryRaga Rudra Pancama
Scale 51Scale 51: Arfian, Ian Ring Music TheoryArfian
Scale 563Scale 563: Thacritonic, Ian Ring Music TheoryThacritonic
Scale 2099Scale 2099: Raga Megharanji, Ian Ring Music TheoryRaga Megharanji
Scale 3123Scale 3123: Tomian, Ian Ring Music TheoryTomian

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.