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Scale 1939: "Dathian"

Scale 1939: Dathian, 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
Dathian

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

{0,1,4,7,8,9,10}

Forte Number

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

7-16

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: 2365

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.

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: 623

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, 3, 3, 1, 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.

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

Interval Spectrum

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

p3m4n5s3d4t2

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> = {1,2,3}
<2> = {2,3,4,6}
<3> = {3,4,5,6,7}
<4> = {5,6,7,8,9}
<5> = {6,8,9,10}
<6> = {9,10,11}

Spectra Variation

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

2.857

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.

2.433

Polygon Perimeter

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

5.899

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

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 TriadsC{0,4,7}331.67
A{9,1,4}331.67
Minor Triadsc♯m{1,4,8}331.67
am{9,0,4}231.89
Augmented TriadsC+{0,4,8}331.67
Diminished Triadsc♯°{1,4,7}231.89
{4,7,10}231.89
{7,10,1}232
a♯°{10,1,4}231.89
Parsimonious Voice Leading Between Common Triads of Scale 1939. Created by Ian Ring ©2019 C C C+ C+ C->C+ c#° c#° C->c#° C->e° c#m c#m C+->c#m am am C+->am c#°->c#m A A c#m->A e°->g° a#° a#° g°->a#° am->A A->a#°

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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1939 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3017
Scale 3017: Gacrian, Ian Ring Music TheoryGacrian
3rd mode:
Scale 889
Scale 889: Borian, Ian Ring Music TheoryBorian
4th mode:
Scale 623
Scale 623: Sycrian, Ian Ring Music TheorySycrianThis is the prime mode
5th mode:
Scale 2359
Scale 2359: Gadian, Ian Ring Music TheoryGadian
6th mode:
Scale 3227
Scale 3227: Aeolocrian, Ian Ring Music TheoryAeolocrian
7th mode:
Scale 3661
Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian

Prime

The prime form of this scale is Scale 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

Complement

The heptatonic modal family [1939, 3017, 889, 623, 2359, 3227, 3661] (Forte: 7-16) is the complement of the pentatonic modal family [155, 865, 1555, 2125, 2825] (Forte: 5-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1939 is 2365

Scale 2365Scale 2365: Sythian, Ian Ring Music TheorySythian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1939 is chiral, and its enantiomorph is scale 2365

Scale 2365Scale 2365: Sythian, Ian Ring Music TheorySythian

Transformations:

T0 1939  T0I 2365
T1 3878  T1I 635
T2 3661  T2I 1270
T3 3227  T3I 2540
T4 2359  T4I 985
T5 623  T5I 1970
T6 1246  T6I 3940
T7 2492  T7I 3785
T8 889  T8I 3475
T9 1778  T9I 2855
T10 3556  T10I 1615
T11 3017  T11I 3230

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 1937Scale 1937: Galimic, Ian Ring Music TheoryGalimic
Scale 1941Scale 1941: Aeranian, Ian Ring Music TheoryAeranian
Scale 1943Scale 1943, Ian Ring Music Theory
Scale 1947Scale 1947: Byptyllic, Ian Ring Music TheoryByptyllic
Scale 1923Scale 1923, Ian Ring Music Theory
Scale 1931Scale 1931: Stogian, Ian Ring Music TheoryStogian
Scale 1955Scale 1955: Sonian, Ian Ring Music TheorySonian
Scale 1971Scale 1971: Aerynyllic, Ian Ring Music TheoryAerynyllic
Scale 2003Scale 2003: Podyllic, Ian Ring Music TheoryPodyllic
Scale 1811Scale 1811: Kyptimic, Ian Ring Music TheoryKyptimic
Scale 1875Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale
Scale 1683Scale 1683: Raga Malayamarutam, Ian Ring Music TheoryRaga Malayamarutam
Scale 1427Scale 1427: Lolimic, Ian Ring Music TheoryLolimic
Scale 915Scale 915: Raga Kalagada, Ian Ring Music TheoryRaga Kalagada
Scale 2963Scale 2963: Bygian, Ian Ring Music TheoryBygian
Scale 3987Scale 3987: Loryllic, Ian Ring Music TheoryLoryllic

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.