The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1915: "Thydygic"

Scale 1915: Thydygic, 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
Thydygic
Dozenal
Logian

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,1,3,4,5,6,8,9,10}

Forte Number

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

9-11

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: 3037

Hemitonia

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

6 (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.

2

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.

8

Prime Form

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

no
prime: 1775

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, 1, 1, 2, 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.

<6, 6, 7, 7, 7, 3>

Interval Spectrum

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

p7m7n7s6d6t3

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

Spectra Variation

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

1.111

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.

2.799

Polygon Perimeter

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

6.106

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".

Proper

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.

(0, 51, 138)

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 TriadsC♯{1,5,8}342.39
F{5,9,0}442.17
F♯{6,10,1}342.5
G♯{8,0,3}342.56
A{9,1,4}442.17
Minor Triadsc♯m{1,4,8}342.44
d♯m{3,6,10}342.67
fm{5,8,0}342.44
f♯m{6,9,1}442.28
am{9,0,4}442.22
a♯m{10,1,5}342.39
Augmented TriadsC+{0,4,8}442.33
C♯+{1,5,9}542
Diminished Triads{0,3,6}242.72
d♯°{3,6,9}242.72
f♯°{6,9,0}242.56
{9,0,3}242.67
a♯°{10,1,4}242.67
Parsimonious Voice Leading Between Common Triads of Scale 1915. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ c#m c#m C+->c#m fm fm C+->fm C+->G# am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F f#m f#m C#+->f#m C#+->A a#m a#m C#+->a#m d#° d#° d#°->d#m d#°->f#m F# F# d#m->F# fm->F f#° f#° F->f#° F->am f#°->f#m f#m->F# F#->a#m G#->a° a°->am am->A a#° a#° A->a#° a#°->a#m

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1915 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3005
Scale 3005: Gycrygic, Ian Ring Music TheoryGycrygic
3rd mode:
Scale 1775
Scale 1775: Lyrygic, Ian Ring Music TheoryLyrygicThis is the prime mode
4th mode:
Scale 2935
Scale 2935: Modygic, Ian Ring Music TheoryModygic
5th mode:
Scale 3515
Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian
6th mode:
Scale 3805
Scale 3805: Moptygic, Ian Ring Music TheoryMoptygic
7th mode:
Scale 1975
Scale 1975: Ionocrygic, Ian Ring Music TheoryIonocrygic
8th mode:
Scale 3035
Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic
9th mode:
Scale 3565
Scale 3565: Aeolorygic, Ian Ring Music TheoryAeolorygic

Prime

The prime form of this scale is Scale 1775

Scale 1775Scale 1775: Lyrygic, Ian Ring Music TheoryLyrygic

Complement

The enneatonic modal family [1915, 3005, 1775, 2935, 3515, 3805, 1975, 3035, 3565] (Forte: 9-11) is the complement of the tritonic modal family [137, 289, 529] (Forte: 3-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1915 is 3037

Scale 3037Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1915 is chiral, and its enantiomorph is scale 3037

Scale 3037Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale

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> 1915       T0I <11,0> 3037
T1 <1,1> 3830      T1I <11,1> 1979
T2 <1,2> 3565      T2I <11,2> 3958
T3 <1,3> 3035      T3I <11,3> 3821
T4 <1,4> 1975      T4I <11,4> 3547
T5 <1,5> 3950      T5I <11,5> 2999
T6 <1,6> 3805      T6I <11,6> 1903
T7 <1,7> 3515      T7I <11,7> 3806
T8 <1,8> 2935      T8I <11,8> 3517
T9 <1,9> 1775      T9I <11,9> 2939
T10 <1,10> 3550      T10I <11,10> 1783
T11 <1,11> 3005      T11I <11,11> 3566
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 895      T0MI <7,0> 4057
T1M <5,1> 1790      T1MI <7,1> 4019
T2M <5,2> 3580      T2MI <7,2> 3943
T3M <5,3> 3065      T3MI <7,3> 3791
T4M <5,4> 2035      T4MI <7,4> 3487
T5M <5,5> 4070      T5MI <7,5> 2879
T6M <5,6> 4045      T6MI <7,6> 1663
T7M <5,7> 3995      T7MI <7,7> 3326
T8M <5,8> 3895      T8MI <7,8> 2557
T9M <5,9> 3695      T9MI <7,9> 1019
T10M <5,10> 3295      T10MI <7,10> 2038
T11M <5,11> 2495      T11MI <7,11> 4076

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 1913Scale 1913: Lofian, Ian Ring Music TheoryLofian
Scale 1917Scale 1917: Sacrygic, Ian Ring Music TheorySacrygic
Scale 1919Scale 1919: Rocryllian, Ian Ring Music TheoryRocryllian
Scale 1907Scale 1907: Lynyllic, Ian Ring Music TheoryLynyllic
Scale 1911Scale 1911: Messiaen Mode 3, Ian Ring Music TheoryMessiaen Mode 3
Scale 1899Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic
Scale 1883Scale 1883: Lomian, Ian Ring Music TheoryLomian
Scale 1851Scale 1851: Zacryllic, Ian Ring Music TheoryZacryllic
Scale 1979Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic
Scale 2043Scale 2043: Maqam Tarzanuyn, Ian Ring Music TheoryMaqam Tarzanuyn
Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban
Scale 1787Scale 1787: Mycrygic, Ian Ring Music TheoryMycrygic
Scale 1403Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic
Scale 2939Scale 2939: Goptygic, Ian Ring Music TheoryGoptygic
Scale 3963Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian

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.