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Scale 1967: "Diatonic Dorian Mixed"

Scale 1967: Diatonic Dorian 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
Diatonic Dorian Mixed
Mixed Chromatic Dorian
Dozenal
Malian
Zeitler
Godygic

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

9 (enneatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,2,3,5,7,8,9,10}

Forte Number

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

9-9

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.

[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.

6 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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.

8

Prime Form

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

no
prime: 1519

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 5
origin: 9

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, 1, 1, 2, 2, 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.

<6, 7, 6, 6, 8, 3>

Interval Spectrum

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

p8m6n6s7d6t3

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

Spectra Variation

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

1.333

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.

2.799

Polygon Perimeter

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

6.106

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.

[10]

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.

(4, 83, 168)

Generator

This scale has a generator of 5, originating on 9.

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}342.21
D♯{3,7,10}242.57
F{5,9,0}342.21
G♯{8,0,3}342.43
A♯{10,2,5}342.29
Minor Triadscm{0,3,7}242.57
dm{2,5,9}342.21
fm{5,8,0}342.29
gm{7,10,2}342.43
a♯m{10,1,5}342.21
Augmented TriadsC♯+{1,5,9}442
Diminished Triads{2,5,8}242.57
{7,10,1}242.57
{9,0,3}242.57
Parsimonious Voice Leading Between Common Triads of Scale 1967. Created by Ian Ring ©2019 cm cm D# D# cm->D# G# G# cm->G# C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F a#m a#m C#+->a#m d°->dm A# A# dm->A# gm gm D#->gm fm->F fm->G# F->a° g°->gm g°->a#m gm->A# G#->a° a#m->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1967 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3031
Scale 3031: Epithygic, Ian Ring Music TheoryEpithygic
3rd mode:
Scale 3563
Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic
4th mode:
Scale 3829
Scale 3829: Taishikicho, Ian Ring Music TheoryTaishikicho
5th mode:
Scale 1981
Scale 1981: Houseini, Ian Ring Music TheoryHouseini
6th mode:
Scale 1519
Scale 1519: Locrian/Aeolian Mixed, Ian Ring Music TheoryLocrian/Aeolian MixedThis is the prime mode
7th mode:
Scale 2807
Scale 2807: Zylygic, Ian Ring Music TheoryZylygic
8th mode:
Scale 3451
Scale 3451: Garygic, Ian Ring Music TheoryGarygic
9th mode:
Scale 3773
Scale 3773: Raga Malgunji, Ian Ring Music TheoryRaga Malgunji

Prime

The prime form of this scale is Scale 1519

Scale 1519Scale 1519: Locrian/Aeolian Mixed, Ian Ring Music TheoryLocrian/Aeolian Mixed

Complement

The enneatonic modal family [1967, 3031, 3563, 3829, 1981, 1519, 2807, 3451, 3773] (Forte: 9-9) is the complement of the tritonic modal family [133, 161, 1057] (Forte: 3-9)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1967 is 3773

Scale 3773Scale 3773: Raga Malgunji, Ian Ring Music TheoryRaga Malgunji

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> 1967       T0I <11,0> 3773
T1 <1,1> 3934      T1I <11,1> 3451
T2 <1,2> 3773      T2I <11,2> 2807
T3 <1,3> 3451      T3I <11,3> 1519
T4 <1,4> 2807      T4I <11,4> 3038
T5 <1,5> 1519      T5I <11,5> 1981
T6 <1,6> 3038      T6I <11,6> 3962
T7 <1,7> 1981      T7I <11,7> 3829
T8 <1,8> 3962      T8I <11,8> 3563
T9 <1,9> 3829      T9I <11,9> 3031
T10 <1,10> 3563      T10I <11,10> 1967
T11 <1,11> 3031      T11I <11,11> 3934
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3647      T0MI <7,0> 3983
T1M <5,1> 3199      T1MI <7,1> 3871
T2M <5,2> 2303      T2MI <7,2> 3647
T3M <5,3> 511      T3MI <7,3> 3199
T4M <5,4> 1022      T4MI <7,4> 2303
T5M <5,5> 2044      T5MI <7,5> 511
T6M <5,6> 4088      T6MI <7,6> 1022
T7M <5,7> 4081      T7MI <7,7> 2044
T8M <5,8> 4067      T8MI <7,8> 4088
T9M <5,9> 4039      T9MI <7,9> 4081
T10M <5,10> 3983      T10MI <7,10> 4067
T11M <5,11> 3871      T11MI <7,11> 4039

The transformations that map this set to itself are: T0, T10I

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 1965Scale 1965: Raga Mukhari, Ian Ring Music TheoryRaga Mukhari
Scale 1963Scale 1963: Epocryllic, Ian Ring Music TheoryEpocryllic
Scale 1959Scale 1959: Katolyllic, Ian Ring Music TheoryKatolyllic
Scale 1975Scale 1975: Ionocrygic, Ian Ring Music TheoryIonocrygic
Scale 1983Scale 1983: Soryllian, Ian Ring Music TheorySoryllian
Scale 1935Scale 1935: Mycryllic, Ian Ring Music TheoryMycryllic
Scale 1951Scale 1951: Marygic, Ian Ring Music TheoryMarygic
Scale 1999Scale 1999: Zacrygic, Ian Ring Music TheoryZacrygic
Scale 2031Scale 2031: Gadyllian, Ian Ring Music TheoryGadyllian
Scale 1839Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
Scale 1903Scale 1903: Rocrygic, Ian Ring Music TheoryRocrygic
Scale 1711Scale 1711: Adonai Malakh, Ian Ring Music TheoryAdonai Malakh
Scale 1455Scale 1455: Quartal Octamode, Ian Ring Music TheoryQuartal Octamode
Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic
Scale 2991Scale 2991: Zanygic, Ian Ring Music TheoryZanygic
Scale 4015Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian

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.