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Scale 1693: "Dogian"

Scale 1693: Dogian, 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
Dogian

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

{0,2,3,4,7,9,10}

Forte Number

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

7-29

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: 1837

Hemitonia

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

3 (trihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

1 (uncohemitonic)

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.

6

Prime Form

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

no
prime: 727

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.

[2, 1, 1, 3, 2, 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.

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

Interval Spectrum

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

p5m3n4s4d3t2

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

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.

2.549

Polygon Perimeter

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

5.967

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}331.43
D♯{3,7,10}331.43
Minor Triadscm{0,3,7}321.29
gm{7,10,2}142.14
am{9,0,4}241.86
Diminished Triads{4,7,10}231.57
{9,0,3}231.71
Parsimonious Voice Leading Between Common Triads of Scale 1693. Created by Ian Ring ©2019 cm cm C C cm->C D# D# cm->D# cm->a° C->e° am am C->am D#->e° gm gm D#->gm a°->am

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
Radius2
Self-Centeredno
Central Verticescm
Peripheral Verticesgm, am

Modes

Modes are the rotational transformation of this scale. Scale 1693 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1447
Scale 1447: Mela Ratnangi, Ian Ring Music TheoryMela Ratnangi
3rd mode:
Scale 2771
Scale 2771: Marva That, Ian Ring Music TheoryMarva That
4th mode:
Scale 3433
Scale 3433: Thonian, Ian Ring Music TheoryThonian
5th mode:
Scale 941
Scale 941: Mela Jhankaradhvani, Ian Ring Music TheoryMela Jhankaradhvani
6th mode:
Scale 1259
Scale 1259: Stadian, Ian Ring Music TheoryStadian
7th mode:
Scale 2677
Scale 2677: Thodian, Ian Ring Music TheoryThodian

Prime

The prime form of this scale is Scale 727

Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian

Complement

The heptatonic modal family [1693, 1447, 2771, 3433, 941, 1259, 2677] (Forte: 7-29) is the complement of the pentatonic modal family [331, 709, 1201, 1577, 2213] (Forte: 5-29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1693 is 1837

Scale 1837Scale 1837: Dalian, Ian Ring Music TheoryDalian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1693 is chiral, and its enantiomorph is scale 1837

Scale 1837Scale 1837: Dalian, Ian Ring Music TheoryDalian

Transformations:

T0 1693  T0I 1837
T1 3386  T1I 3674
T2 2677  T2I 3253
T3 1259  T3I 2411
T4 2518  T4I 727
T5 941  T5I 1454
T6 1882  T6I 2908
T7 3764  T7I 1721
T8 3433  T8I 3442
T9 2771  T9I 2789
T10 1447  T10I 1483
T11 2894  T11I 2966

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 1695Scale 1695: Phrodyllic, Ian Ring Music TheoryPhrodyllic
Scale 1689Scale 1689: Lorimic, Ian Ring Music TheoryLorimic
Scale 1691Scale 1691: Kathian, Ian Ring Music TheoryKathian
Scale 1685Scale 1685: Zeracrimic, Ian Ring Music TheoryZeracrimic
Scale 1677Scale 1677: Raga Manavi, Ian Ring Music TheoryRaga Manavi
Scale 1709Scale 1709: Dorian, Ian Ring Music TheoryDorian
Scale 1725Scale 1725: Minor Bebop, Ian Ring Music TheoryMinor Bebop
Scale 1757Scale 1757, Ian Ring Music Theory
Scale 1565Scale 1565, Ian Ring Music Theory
Scale 1629Scale 1629: Synian, Ian Ring Music TheorySynian
Scale 1821Scale 1821: Aeradian, Ian Ring Music TheoryAeradian
Scale 1949Scale 1949: Mathyllic, Ian Ring Music TheoryMathyllic
Scale 1181Scale 1181: Katagimic, Ian Ring Music TheoryKatagimic
Scale 1437Scale 1437: Sabach ascending, Ian Ring Music TheorySabach ascending
Scale 669Scale 669: Gycrimic, Ian Ring Music TheoryGycrimic
Scale 2717Scale 2717: Epygian, Ian Ring Music TheoryEpygian
Scale 3741Scale 3741: Zydyllic, Ian Ring Music TheoryZydyllic

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.