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Scale 1815: "Godian"

Scale 1815: Godian, 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
Godian
Dozenal
Lewian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,4,8,9,10}

Forte Number

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

7-13

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: 3357

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

6

Prime Form

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

no
prime: 375

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, 1, 2, 4, 1, 1, 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.

<4, 4, 3, 5, 3, 2>

Interval Spectrum

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

p3m5n3s4d4t2

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

Spectra Variation

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

2.857

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.

2.299

Polygon Perimeter

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

5.803

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.

(38, 26, 90)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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
Major TriadsA{9,1,4}321
Minor Triadsc♯m{1,4,8}221.2
am{9,0,4}221.2
Augmented TriadsC+{0,4,8}231.4
Diminished Triadsa♯°{10,1,4}131.6

The following pitch classes are not present in any of the common triads: {2}

Parsimonious Voice Leading Between Common Triads of Scale 1815. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m am am C+->am A A c#m->A am->A a#° a#° A->a#°

view full size

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesc♯m, am, A
Peripheral VerticesC+, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 1815 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2955
Scale 2955: Thorian, Ian Ring Music TheoryThorian
3rd mode:
Scale 3525
Scale 3525: Zocrian, Ian Ring Music TheoryZocrian
4th mode:
Scale 1905
Scale 1905: Katacrian, Ian Ring Music TheoryKatacrian
5th mode:
Scale 375
Scale 375: Sodian, Ian Ring Music TheorySodianThis is the prime mode
6th mode:
Scale 2235
Scale 2235: Bathian, Ian Ring Music TheoryBathian
7th mode:
Scale 3165
Scale 3165: Mylian, Ian Ring Music TheoryMylian

Prime

The prime form of this scale is Scale 375

Scale 375Scale 375: Sodian, Ian Ring Music TheorySodian

Complement

The heptatonic modal family [1815, 2955, 3525, 1905, 375, 2235, 3165] (Forte: 7-13) is the complement of the pentatonic modal family [279, 369, 1809, 2187, 3141] (Forte: 5-13)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1815 is 3357

Scale 3357Scale 3357: Phrodian, Ian Ring Music TheoryPhrodian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1815 is chiral, and its enantiomorph is scale 3357

Scale 3357Scale 3357: Phrodian, Ian Ring Music TheoryPhrodian

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> 1815       T0I <11,0> 3357
T1 <1,1> 3630      T1I <11,1> 2619
T2 <1,2> 3165      T2I <11,2> 1143
T3 <1,3> 2235      T3I <11,3> 2286
T4 <1,4> 375      T4I <11,4> 477
T5 <1,5> 750      T5I <11,5> 954
T6 <1,6> 1500      T6I <11,6> 1908
T7 <1,7> 3000      T7I <11,7> 3816
T8 <1,8> 1905      T8I <11,8> 3537
T9 <1,9> 3810      T9I <11,9> 2979
T10 <1,10> 3525      T10I <11,10> 1863
T11 <1,11> 2955      T11I <11,11> 3726
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1845      T0MI <7,0> 1437
T1M <5,1> 3690      T1MI <7,1> 2874
T2M <5,2> 3285      T2MI <7,2> 1653
T3M <5,3> 2475      T3MI <7,3> 3306
T4M <5,4> 855      T4MI <7,4> 2517
T5M <5,5> 1710      T5MI <7,5> 939
T6M <5,6> 3420      T6MI <7,6> 1878
T7M <5,7> 2745      T7MI <7,7> 3756
T8M <5,8> 1395      T8MI <7,8> 3417
T9M <5,9> 2790      T9MI <7,9> 2739
T10M <5,10> 1485      T10MI <7,10> 1383
T11M <5,11> 2970      T11MI <7,11> 2766

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 1813Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
Scale 1811Scale 1811: Kyptimic, Ian Ring Music TheoryKyptimic
Scale 1819Scale 1819: Pydian, Ian Ring Music TheoryPydian
Scale 1823Scale 1823: Phralyllic, Ian Ring Music TheoryPhralyllic
Scale 1799Scale 1799: Lamian, Ian Ring Music TheoryLamian
Scale 1807Scale 1807: Larian, Ian Ring Music TheoryLarian
Scale 1831Scale 1831: Pothian, Ian Ring Music TheoryPothian
Scale 1847Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic
Scale 1879Scale 1879: Mixoryllic, Ian Ring Music TheoryMixoryllic
Scale 1943Scale 1943: Luxian, Ian Ring Music TheoryLuxian
Scale 1559Scale 1559: Jowian, Ian Ring Music TheoryJowian
Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian
Scale 1303Scale 1303: Epolimic, Ian Ring Music TheoryEpolimic
Scale 791Scale 791: Aeoloptimic, Ian Ring Music TheoryAeoloptimic
Scale 2839Scale 2839: Lyptian, Ian Ring Music TheoryLyptian
Scale 3863Scale 3863: Eparyllic, Ian Ring Music TheoryEparyllic

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.