The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2787: "Zyrian"

Scale 2787: Zyrian, 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
Zyrian
Dozenal
Ropian

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

{0,1,5,6,7,9,11}

Forte Number

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

7-15

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.

[3]

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.

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: 471

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

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

Interval Spectrum

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

p4m4n2s4d4t3

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

Spectra Variation

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

2.286

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.

[6]

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.

(16, 35, 96)

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 TriadsF{5,9,0}221
Minor Triadsf♯m{6,9,1}221
Augmented TriadsC♯+{1,5,9}221
Diminished Triadsf♯°{6,9,0}221

The following pitch classes are not present in any of the common triads: {7,11}

Parsimonious Voice Leading Between Common Triads of Scale 2787. Created by Ian Ring ©2019 C#+ C#+ F F C#+->F f#m f#m C#+->f#m f#° f#° F->f#° f#°->f#m

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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2787 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3441
Scale 3441: Thacrian, Ian Ring Music TheoryThacrian
3rd mode:
Scale 471
Scale 471: Dodian, Ian Ring Music TheoryDodianThis is the prime mode
4th mode:
Scale 2283
Scale 2283: Aeolyptian, Ian Ring Music TheoryAeolyptian
5th mode:
Scale 3189
Scale 3189: Aeolonian, Ian Ring Music TheoryAeolonian
6th mode:
Scale 1821
Scale 1821: Aeradian, Ian Ring Music TheoryAeradian
7th mode:
Scale 1479
Scale 1479: Mela Jalarnava, Ian Ring Music TheoryMela Jalarnava

Prime

The prime form of this scale is Scale 471

Scale 471Scale 471: Dodian, Ian Ring Music TheoryDodian

Complement

The heptatonic modal family [2787, 3441, 471, 2283, 3189, 1821, 1479] (Forte: 7-15) is the complement of the pentatonic modal family [327, 453, 1137, 2211, 3153] (Forte: 5-15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2787 is 2283

Scale 2283Scale 2283: Aeolyptian, Ian Ring Music TheoryAeolyptian

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> 2787       T0I <11,0> 2283
T1 <1,1> 1479      T1I <11,1> 471
T2 <1,2> 2958      T2I <11,2> 942
T3 <1,3> 1821      T3I <11,3> 1884
T4 <1,4> 3642      T4I <11,4> 3768
T5 <1,5> 3189      T5I <11,5> 3441
T6 <1,6> 2283      T6I <11,6> 2787
T7 <1,7> 471      T7I <11,7> 1479
T8 <1,8> 942      T8I <11,8> 2958
T9 <1,9> 1884      T9I <11,9> 1821
T10 <1,10> 3768      T10I <11,10> 3642
T11 <1,11> 3441      T11I <11,11> 3189
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2787       T0MI <7,0> 2283
T1M <5,1> 1479      T1MI <7,1> 471
T2M <5,2> 2958      T2MI <7,2> 942
T3M <5,3> 1821      T3MI <7,3> 1884
T4M <5,4> 3642      T4MI <7,4> 3768
T5M <5,5> 3189      T5MI <7,5> 3441
T6M <5,6> 2283      T6MI <7,6> 2787
T7M <5,7> 471      T7MI <7,7> 1479
T8M <5,8> 942      T8MI <7,8> 2958
T9M <5,9> 1884      T9MI <7,9> 1821
T10M <5,10> 3768      T10MI <7,10> 3642
T11M <5,11> 3441      T11MI <7,11> 3189

The transformations that map this set to itself are: T0, T6I, T0M, 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 2785Scale 2785: Ronian, Ian Ring Music TheoryRonian
Scale 2789Scale 2789: Zolian, Ian Ring Music TheoryZolian
Scale 2791Scale 2791: Mixothyllic, Ian Ring Music TheoryMixothyllic
Scale 2795Scale 2795: Van der Horst Octatonic, Ian Ring Music TheoryVan der Horst Octatonic
Scale 2803Scale 2803: Raga Bhatiyar, Ian Ring Music TheoryRaga Bhatiyar
Scale 2755Scale 2755: Rivian, Ian Ring Music TheoryRivian
Scale 2771Scale 2771: Marva That, Ian Ring Music TheoryMarva That
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 2659Scale 2659: Katynimic, Ian Ring Music TheoryKatynimic
Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian
Scale 3043Scale 3043: Ionayllic, Ian Ring Music TheoryIonayllic
Scale 2275Scale 2275: Messiaen Mode 5, Ian Ring Music TheoryMessiaen Mode 5
Scale 2531Scale 2531: Danian, Ian Ring Music TheoryDanian
Scale 3299Scale 3299: Syptian, Ian Ring Music TheorySyptian
Scale 3811Scale 3811: Epogyllic, Ian Ring Music TheoryEpogyllic
Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic
Scale 1763Scale 1763: Katalian, Ian Ring Music TheoryKatalian

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.