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Scale 3417: "Golian"

Scale 3417: Golian, 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
Golian
Dozenal
Vician

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

{0,3,4,6,8,10,11}

Forte Number

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

7-30

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: 855

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

6

Prime Form

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

no
prime: 855

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.

[3, 1, 2, 2, 2, 1, 1]

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.

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

Interval Spectrum

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

p4m5n3s4d3t2

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

Spectra Variation

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

1.714

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.

2.549

Polygon Perimeter

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

5.967

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.

(2, 22, 86)

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 TriadsE{4,8,11}231.71
G♯{8,0,3}331.43
B{11,3,6}331.43
Minor Triadsd♯m{3,6,10}142.14
g♯m{8,11,3}321.29
Augmented TriadsC+{0,4,8}241.86
Diminished Triads{0,3,6}231.57
Parsimonious Voice Leading Between Common Triads of Scale 3417. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ E E C+->E C+->G# d#m d#m d#m->B g#m g#m E->g#m g#m->G# g#m->B

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.

Diameter4
Radius2
Self-Centeredno
Central Verticesg♯m
Peripheral VerticesC+, d♯m

Modes

Modes are the rotational transformation of this scale. Scale 3417 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 939
Scale 939: Mela Senavati, Ian Ring Music TheoryMela Senavati
3rd mode:
Scale 2517
Scale 2517: Harmonic Lydian, Ian Ring Music TheoryHarmonic Lydian
4th mode:
Scale 1653
Scale 1653: Minor Romani Inverse, Ian Ring Music TheoryMinor Romani Inverse
5th mode:
Scale 1437
Scale 1437: Sabach ascending, Ian Ring Music TheorySabach ascending
6th mode:
Scale 1383
Scale 1383: Pynian, Ian Ring Music TheoryPynian
7th mode:
Scale 2739
Scale 2739: Mela Suryakanta, Ian Ring Music TheoryMela Suryakanta

Prime

The prime form of this scale is Scale 855

Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian

Complement

The heptatonic modal family [3417, 939, 2517, 1653, 1437, 1383, 2739] (Forte: 7-30) is the complement of the pentatonic modal family [339, 789, 1221, 1329, 2217] (Forte: 5-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3417 is 855

Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3417 is chiral, and its enantiomorph is scale 855

Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian

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

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 3419Scale 3419: Magen Abot 1, Ian Ring Music TheoryMagen Abot 1
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 3409Scale 3409: Katanimic, Ian Ring Music TheoryKatanimic
Scale 3413Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone
Scale 3401Scale 3401: Palimic, Ian Ring Music TheoryPalimic
Scale 3433Scale 3433: Thonian, Ian Ring Music TheoryThonian
Scale 3449Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
Scale 3353Scale 3353: Phraptimic, Ian Ring Music TheoryPhraptimic
Scale 3385Scale 3385: Chromatic Phrygian, Ian Ring Music TheoryChromatic Phrygian
Scale 3481Scale 3481: Katathian, Ian Ring Music TheoryKatathian
Scale 3545Scale 3545: Thyptyllic, Ian Ring Music TheoryThyptyllic
Scale 3161Scale 3161: Kodimic, Ian Ring Music TheoryKodimic
Scale 3289Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6
Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian
Scale 3929Scale 3929: Aeolothyllic, Ian Ring Music TheoryAeolothyllic
Scale 2393Scale 2393: Zathimic, Ian Ring Music TheoryZathimic
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1
Scale 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic

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.