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Scale 1405: "Goryllic"

Scale 1405: Goryllic, 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
Goryllic
Dozenal
Itlian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,2,3,4,5,6,8,10}

Forte Number

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

8-21

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.

[4]

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.

no

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.

3 (tricohemitonic)

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.

7

Prime Form

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

no
prime: 1375

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.

[2, 1, 1, 1, 1, 2, 2, 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, 7, 4, 6, 4, 3>

Interval Spectrum

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

p4m6n4s7d4t3

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

Spectra Variation

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

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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.

[8]

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.

(28, 74, 147)

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 TriadsG♯{8,0,3}242
A♯{10,2,5}242
Minor Triadsd♯m{3,6,10}242
fm{5,8,0}242
Augmented TriadsC+{0,4,8}242
D+{2,6,10}242
Diminished Triads{0,3,6}242
{2,5,8}242
Parsimonious Voice Leading Between Common Triads of Scale 1405. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ fm fm C+->fm C+->G# d°->fm A# A# d°->A# D+ D+ D+->d#m D+->A#

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1405 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1375
Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllicThis is the prime mode
3rd mode:
Scale 2735
Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic
4th mode:
Scale 3415
Scale 3415: Ionaptyllic, Ian Ring Music TheoryIonaptyllic
5th mode:
Scale 3755
Scale 3755: Phryryllic, Ian Ring Music TheoryPhryryllic
6th mode:
Scale 3925
Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic
7th mode:
Scale 2005
Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
8th mode:
Scale 1525
Scale 1525: Sodyllic, Ian Ring Music TheorySodyllic

Prime

The prime form of this scale is Scale 1375

Scale 1375Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllic

Complement

The octatonic modal family [1405, 1375, 2735, 3415, 3755, 3925, 2005, 1525] (Forte: 8-21) is the complement of the tetratonic modal family [85, 1045, 1285, 1345] (Forte: 4-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1405 is 2005

Scale 2005Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic

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> 1405       T0I <11,0> 2005
T1 <1,1> 2810      T1I <11,1> 4010
T2 <1,2> 1525      T2I <11,2> 3925
T3 <1,3> 3050      T3I <11,3> 3755
T4 <1,4> 2005      T4I <11,4> 3415
T5 <1,5> 4010      T5I <11,5> 2735
T6 <1,6> 3925      T6I <11,6> 1375
T7 <1,7> 3755      T7I <11,7> 2750
T8 <1,8> 3415      T8I <11,8> 1405
T9 <1,9> 2735      T9I <11,9> 2810
T10 <1,10> 1375      T10I <11,10> 1525
T11 <1,11> 2750      T11I <11,11> 3050
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1375      T0MI <7,0> 3925
T1M <5,1> 2750      T1MI <7,1> 3755
T2M <5,2> 1405       T2MI <7,2> 3415
T3M <5,3> 2810      T3MI <7,3> 2735
T4M <5,4> 1525      T4MI <7,4> 1375
T5M <5,5> 3050      T5MI <7,5> 2750
T6M <5,6> 2005      T6MI <7,6> 1405
T7M <5,7> 4010      T7MI <7,7> 2810
T8M <5,8> 3925      T8MI <7,8> 1525
T9M <5,9> 3755      T9MI <7,9> 3050
T10M <5,10> 3415      T10MI <7,10> 2005
T11M <5,11> 2735      T11MI <7,11> 4010

The transformations that map this set to itself are: T0, T8I, T2M, T6MI

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 1407Scale 1407: Tharygic, Ian Ring Music TheoryTharygic
Scale 1401Scale 1401: Pagian, Ian Ring Music TheoryPagian
Scale 1403Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
Scale 1397Scale 1397: Major Locrian, Ian Ring Music TheoryMajor Locrian
Scale 1389Scale 1389: Minor Locrian, Ian Ring Music TheoryMinor Locrian
Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian
Scale 1341Scale 1341: Madian, Ian Ring Music TheoryMadian
Scale 1469Scale 1469: Epiryllic, Ian Ring Music TheoryEpiryllic
Scale 1533Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
Scale 1149Scale 1149: Bydian, Ian Ring Music TheoryBydian
Scale 1277Scale 1277: Zadyllic, Ian Ring Music TheoryZadyllic
Scale 1661Scale 1661: Gonyllic, Ian Ring Music TheoryGonyllic
Scale 1917Scale 1917: Sacrygic, Ian Ring Music TheorySacrygic
Scale 381Scale 381: Kogian, Ian Ring Music TheoryKogian
Scale 893Scale 893: Dadyllic, Ian Ring Music TheoryDadyllic
Scale 2429Scale 2429: Kadyllic, Ian Ring Music TheoryKadyllic
Scale 3453Scale 3453: Katarygic, Ian Ring Music TheoryKatarygic

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.