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Scale 3747: "Myrian"

Scale 3747: Myrian, 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
Myrian
Dozenal
Xuwian

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

{0,1,5,7,9,10,11}

Forte Number

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

7-9

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: 2223

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.

6

Prime Form

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

no
prime: 351

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, 2, 2, 1, 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.

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

Interval Spectrum

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

p3m4n3s5d4t2

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

Spectra Variation

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

3.429

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.

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.

(46, 40, 104)

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}131.5
Minor Triadsa♯m{10,1,5}221
Augmented TriadsC♯+{1,5,9}221
Diminished Triads{7,10,1}131.5

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

Parsimonious Voice Leading Between Common Triads of Scale 3747. Created by Ian Ring ©2019 C#+ C#+ F F C#+->F a#m a#m C#+->a#m g°->a#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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC♯+, a♯m
Peripheral VerticesF, g°

Modes

Modes are the rotational transformation of this scale. Scale 3747 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3921
Scale 3921: Pythian, Ian Ring Music TheoryPythian
3rd mode:
Scale 501
Scale 501: Katylian, Ian Ring Music TheoryKatylian
4th mode:
Scale 1149
Scale 1149: Bydian, Ian Ring Music TheoryBydian
5th mode:
Scale 1311
Scale 1311: Bynian, Ian Ring Music TheoryBynian
6th mode:
Scale 2703
Scale 2703: Galian, Ian Ring Music TheoryGalian
7th mode:
Scale 3399
Scale 3399: Zonian, Ian Ring Music TheoryZonian

Prime

The prime form of this scale is Scale 351

Scale 351Scale 351: Epanian, Ian Ring Music TheoryEpanian

Complement

The heptatonic modal family [3747, 3921, 501, 1149, 1311, 2703, 3399] (Forte: 7-9) is the complement of the pentatonic modal family [87, 1473, 1797, 2091, 3093] (Forte: 5-9)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3747 is 2223

Scale 2223Scale 2223: Konian, Ian Ring Music TheoryKonian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3747 is chiral, and its enantiomorph is scale 2223

Scale 2223Scale 2223: Konian, Ian Ring Music TheoryKonian

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> 3747       T0I <11,0> 2223
T1 <1,1> 3399      T1I <11,1> 351
T2 <1,2> 2703      T2I <11,2> 702
T3 <1,3> 1311      T3I <11,3> 1404
T4 <1,4> 2622      T4I <11,4> 2808
T5 <1,5> 1149      T5I <11,5> 1521
T6 <1,6> 2298      T6I <11,6> 3042
T7 <1,7> 501      T7I <11,7> 1989
T8 <1,8> 1002      T8I <11,8> 3978
T9 <1,9> 2004      T9I <11,9> 3861
T10 <1,10> 4008      T10I <11,10> 3627
T11 <1,11> 3921      T11I <11,11> 3159
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2727      T0MI <7,0> 3243
T1M <5,1> 1359      T1MI <7,1> 2391
T2M <5,2> 2718      T2MI <7,2> 687
T3M <5,3> 1341      T3MI <7,3> 1374
T4M <5,4> 2682      T4MI <7,4> 2748
T5M <5,5> 1269      T5MI <7,5> 1401
T6M <5,6> 2538      T6MI <7,6> 2802
T7M <5,7> 981      T7MI <7,7> 1509
T8M <5,8> 1962      T8MI <7,8> 3018
T9M <5,9> 3924      T9MI <7,9> 1941
T10M <5,10> 3753      T10MI <7,10> 3882
T11M <5,11> 3411      T11MI <7,11> 3669

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 3745Scale 3745: Xuvian, Ian Ring Music TheoryXuvian
Scale 3749Scale 3749: Raga Sorati, Ian Ring Music TheoryRaga Sorati
Scale 3751Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
Scale 3755Scale 3755: Phryryllic, Ian Ring Music TheoryPhryryllic
Scale 3763Scale 3763: Modyllic, Ian Ring Music TheoryModyllic
Scale 3715Scale 3715: Xician, Ian Ring Music TheoryXician
Scale 3731Scale 3731: Aeryrian, Ian Ring Music TheoryAeryrian
Scale 3779Scale 3779, Ian Ring Music Theory
Scale 3811Scale 3811: Epogyllic, Ian Ring Music TheoryEpogyllic
Scale 3619Scale 3619: Thanimic, Ian Ring Music TheoryThanimic
Scale 3683Scale 3683: Dycrian, Ian Ring Music TheoryDycrian
Scale 3875Scale 3875: Aeryptian, Ian Ring Music TheoryAeryptian
Scale 4003Scale 4003: Sadyllic, Ian Ring Music TheorySadyllic
Scale 3235Scale 3235: Pothimic, Ian Ring Music TheoryPothimic
Scale 3491Scale 3491: Tharian, Ian Ring Music TheoryTharian
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 1699Scale 1699: Raga Rasavali, Ian Ring Music TheoryRaga Rasavali

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.