The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1515: "Phrygian/Locrian Mixed"

Scale 1515: Phrygian/Locrian Mixed, 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

Western Mixed
Phrygian/Locrian Mixed
Zeitler
Solyllic

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

{0,1,3,5,6,7,8,10}

Forte Number

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

8-23

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.

[0.5]

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.

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.

1

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: 1455

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 2, 1, 1, 1, 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, 6, 5, 4, 7, 2>

Interval Spectrum

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

p7m4n5s6d4t2

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}
<4> = {5,6,7}
<5> = {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.

1.5

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.

[1]

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

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}242.3
D♯{3,7,10}341.9
F♯{6,10,1}341.9
G♯{8,0,3}242.1
Minor Triadscm{0,3,7}341.9
d♯m{3,6,10}341.9
fm{5,8,0}242.3
a♯m{10,1,5}242.1
Diminished Triads{0,3,6}242.1
{7,10,1}242.1
Parsimonious Voice Leading Between Common Triads of Scale 1515. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m D# D# cm->D# G# G# cm->G# C# C# fm fm C#->fm a#m a#m C#->a#m d#m->D# F# F# d#m->F# D#->g° fm->G# F#->g° F#->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 1515 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2805
Scale 2805: Ishikotsucho, Ian Ring Music TheoryIshikotsucho
3rd mode:
Scale 1725
Scale 1725: Minor Bebop, Ian Ring Music TheoryMinor Bebop
4th mode:
Scale 1455
Scale 1455: Quartal Octamode, Ian Ring Music TheoryQuartal OctamodeThis is the prime mode
5th mode:
Scale 2775
Scale 2775: Godyllic, Ian Ring Music TheoryGodyllic
6th mode:
Scale 3435
Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
7th mode:
Scale 3765
Scale 3765: Dominant Bebop, Ian Ring Music TheoryDominant Bebop
8th mode:
Scale 1965
Scale 1965: Raga Mukhari, Ian Ring Music TheoryRaga Mukhari

Prime

The prime form of this scale is Scale 1455

Scale 1455Scale 1455: Quartal Octamode, Ian Ring Music TheoryQuartal Octamode

Complement

The octatonic modal family [1515, 2805, 1725, 1455, 2775, 3435, 3765, 1965] (Forte: 8-23) is the complement of the tetratonic modal family [165, 645, 1065, 1185] (Forte: 4-23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1515 is 2805

Scale 2805Scale 2805: Ishikotsucho, Ian Ring Music TheoryIshikotsucho

Transformations:

T0 1515  T0I 2805
T1 3030  T1I 1515
T2 1965  T2I 3030
T3 3930  T3I 1965
T4 3765  T4I 3930
T5 3435  T5I 3765
T6 2775  T6I 3435
T7 1455  T7I 2775
T8 2910  T8I 1455
T9 1725  T9I 2910
T10 3450  T10I 1725
T11 2805  T11I 3450

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 1513Scale 1513: Stathian, Ian Ring Music TheoryStathian
Scale 1517Scale 1517: Sagyllic, Ian Ring Music TheorySagyllic
Scale 1519Scale 1519: Locrian/Aeolian Mixed, Ian Ring Music TheoryLocrian/Aeolian Mixed
Scale 1507Scale 1507: Zynian, Ian Ring Music TheoryZynian
Scale 1511Scale 1511: Styptyllic, Ian Ring Music TheoryStyptyllic
Scale 1523Scale 1523: Zothyllic, Ian Ring Music TheoryZothyllic
Scale 1531Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic
Scale 1483Scale 1483: Mela Bhavapriya, Ian Ring Music TheoryMela Bhavapriya
Scale 1499Scale 1499: Bebop Locrian, Ian Ring Music TheoryBebop Locrian
Scale 1451Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian
Scale 1771Scale 1771, Ian Ring Music Theory
Scale 2027Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic
Scale 491Scale 491: Aeolyrian, Ian Ring Music TheoryAeolyrian
Scale 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 2539Scale 2539: Half-Diminished Bebop, Ian Ring Music TheoryHalf-Diminished Bebop
Scale 3563Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic

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. 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.