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Scale 1643: "Locrian Natural 6"

Scale 1643: Locrian Natural 6, 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 Altered
Locrian Natural 6
Locrian Sharp 6
Arabic
Maqam Tarznauyn
Zeitler
Thyptian

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

{0,1,3,5,6,9,10}

Forte Number

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

7-32

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: 2765

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.

0 (ancohemitonic)

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: 859

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 2, 1, 3, 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, 3, 5, 4, 4, 2>

Interval Spectrum

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

p4m4n5s3d3t2

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

Spectra Variation

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

1.429

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.

2.549

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".

Proper

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.

(0, 18, 82)

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}331.8
F♯{6,10,1}331.7
Minor Triadsd♯m{3,6,10}331.8
f♯m{6,9,1}431.6
a♯m{10,1,5}232
Augmented TriadsC♯+{1,5,9}331.7
Diminished Triads{0,3,6}232
d♯°{3,6,9}231.9
f♯°{6,9,0}231.9
{9,0,3}232
Parsimonious Voice Leading Between Common Triads of Scale 1643. Created by Ian Ring ©2019 d#m d#m c°->d#m c°->a° C#+ C#+ F F C#+->F f#m f#m C#+->f#m a#m a#m C#+->a#m d#° d#° d#°->d#m d#°->f#m F# F# d#m->F# f#° f#° F->f#° F->a° f#°->f#m f#m->F# F#->a#m

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 1643 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2869
Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
3rd mode:
Scale 1741
Scale 1741: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
4th mode:
Scale 1459
Scale 1459: Phrygian Dominant, Ian Ring Music TheoryPhrygian Dominant
5th mode:
Scale 2777
Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic
6th mode:
Scale 859
Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrianThis is the prime mode
7th mode:
Scale 2477
Scale 2477: Harmonic Minor, Ian Ring Music TheoryHarmonic Minor

Prime

The prime form of this scale is Scale 859

Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian

Complement

The heptatonic modal family [1643, 2869, 1741, 1459, 2777, 859, 2477] (Forte: 7-32) is the complement of the pentatonic modal family [595, 665, 805, 1225, 2345] (Forte: 5-32)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1643 is 2765

Scale 2765Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1643 is chiral, and its enantiomorph is scale 2765

Scale 2765Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished

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> 1643       T0I <11,0> 2765
T1 <1,1> 3286      T1I <11,1> 1435
T2 <1,2> 2477      T2I <11,2> 2870
T3 <1,3> 859      T3I <11,3> 1645
T4 <1,4> 1718      T4I <11,4> 3290
T5 <1,5> 3436      T5I <11,5> 2485
T6 <1,6> 2777      T6I <11,6> 875
T7 <1,7> 1459      T7I <11,7> 1750
T8 <1,8> 2918      T8I <11,8> 3500
T9 <1,9> 1741      T9I <11,9> 2905
T10 <1,10> 3482      T10I <11,10> 1715
T11 <1,11> 2869      T11I <11,11> 3430
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 623      T0MI <7,0> 3785
T1M <5,1> 1246      T1MI <7,1> 3475
T2M <5,2> 2492      T2MI <7,2> 2855
T3M <5,3> 889      T3MI <7,3> 1615
T4M <5,4> 1778      T4MI <7,4> 3230
T5M <5,5> 3556      T5MI <7,5> 2365
T6M <5,6> 3017      T6MI <7,6> 635
T7M <5,7> 1939      T7MI <7,7> 1270
T8M <5,8> 3878      T8MI <7,8> 2540
T9M <5,9> 3661      T9MI <7,9> 985
T10M <5,10> 3227      T10MI <7,10> 1970
T11M <5,11> 2359      T11MI <7,11> 3940

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 1641Scale 1641: Bocrimic, Ian Ring Music TheoryBocrimic
Scale 1645Scale 1645: Dorian Flat 5, Ian Ring Music TheoryDorian Flat 5
Scale 1647Scale 1647: Polyllic, Ian Ring Music TheoryPolyllic
Scale 1635Scale 1635: Sygimic, Ian Ring Music TheorySygimic
Scale 1639Scale 1639: Aeolothian, Ian Ring Music TheoryAeolothian
Scale 1651Scale 1651: Asian, Ian Ring Music TheoryAsian
Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban
Scale 1611Scale 1611: Dacrimic, Ian Ring Music TheoryDacrimic
Scale 1627Scale 1627: Zyptian, Ian Ring Music TheoryZyptian
Scale 1579Scale 1579: Sagimic, Ian Ring Music TheorySagimic
Scale 1707Scale 1707: Dorian Flat 2, Ian Ring Music TheoryDorian Flat 2
Scale 1771Scale 1771, Ian Ring Music Theory
Scale 1899Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic
Scale 2667Scale 2667: Byrian, Ian Ring Music TheoryByrian
Scale 3691Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic

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.