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Scale 1889: "Loqian"

Scale 1889: Loqian, 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

Dozenal
Loqian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,5,6,8,9,10}

Forte Number

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

6-Z10

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: 221

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.

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.

5

Prime Form

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

no
prime: 187

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.

[5, 1, 2, 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.

<3, 3, 3, 3, 2, 1>

Interval Spectrum

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

p2m3n3s3d3t

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

Spectra Variation

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

3.667

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.

1.866

Polygon Perimeter

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

5.485

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.

(29, 7, 55)

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}210.67
Minor Triadsfm{5,8,0}121
Diminished Triadsf♯°{6,9,0}121

The following pitch classes are not present in any of the common triads: {10}

Parsimonious Voice Leading Between Common Triads of Scale 1889. Created by Ian Ring ©2019 fm fm F F fm->F f#° f#° F->f#°

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesF
Peripheral Verticesfm, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 1889 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 187
Scale 187: Bedian, Ian Ring Music TheoryBedianThis is the prime mode
3rd mode:
Scale 2141
Scale 2141: Nanian, Ian Ring Music TheoryNanian
4th mode:
Scale 1559
Scale 1559: Jowian, Ian Ring Music TheoryJowian
5th mode:
Scale 2827
Scale 2827: Runian, Ian Ring Music TheoryRunian
6th mode:
Scale 3461
Scale 3461: Vodian, Ian Ring Music TheoryVodian

Prime

The prime form of this scale is Scale 187

Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian

Complement

The hexatonic modal family [1889, 187, 2141, 1559, 2827, 3461] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1889 is 221

Scale 221Scale 221: Biyian, Ian Ring Music TheoryBiyian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1889 is chiral, and its enantiomorph is scale 221

Scale 221Scale 221: Biyian, Ian Ring Music TheoryBiyian

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> 1889       T0I <11,0> 221
T1 <1,1> 3778      T1I <11,1> 442
T2 <1,2> 3461      T2I <11,2> 884
T3 <1,3> 2827      T3I <11,3> 1768
T4 <1,4> 1559      T4I <11,4> 3536
T5 <1,5> 3118      T5I <11,5> 2977
T6 <1,6> 2141      T6I <11,6> 1859
T7 <1,7> 187      T7I <11,7> 3718
T8 <1,8> 374      T8I <11,8> 3341
T9 <1,9> 748      T9I <11,9> 2587
T10 <1,10> 1496      T10I <11,10> 1079
T11 <1,11> 2992      T11I <11,11> 2158
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 599      T0MI <7,0> 3401
T1M <5,1> 1198      T1MI <7,1> 2707
T2M <5,2> 2396      T2MI <7,2> 1319
T3M <5,3> 697      T3MI <7,3> 2638
T4M <5,4> 1394      T4MI <7,4> 1181
T5M <5,5> 2788      T5MI <7,5> 2362
T6M <5,6> 1481      T6MI <7,6> 629
T7M <5,7> 2962      T7MI <7,7> 1258
T8M <5,8> 1829      T8MI <7,8> 2516
T9M <5,9> 3658      T9MI <7,9> 937
T10M <5,10> 3221      T10MI <7,10> 1874
T11M <5,11> 2347      T11MI <7,11> 3748

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 1891Scale 1891: Thalian, Ian Ring Music TheoryThalian
Scale 1893Scale 1893: Ionylian, Ian Ring Music TheoryIonylian
Scale 1897Scale 1897: Ionopian, Ian Ring Music TheoryIonopian
Scale 1905Scale 1905: Katacrian, Ian Ring Music TheoryKatacrian
Scale 1857Scale 1857: Liwian, Ian Ring Music TheoryLiwian
Scale 1873Scale 1873: Dathimic, Ian Ring Music TheoryDathimic
Scale 1825Scale 1825: Lecian, Ian Ring Music TheoryLecian
Scale 1953Scale 1953: Macian, Ian Ring Music TheoryMacian
Scale 2017Scale 2017: Meqian, Ian Ring Music TheoryMeqian
Scale 1633Scale 1633: Kapian, Ian Ring Music TheoryKapian
Scale 1761Scale 1761: Kuqian, Ian Ring Music TheoryKuqian
Scale 1377Scale 1377: Insian, Ian Ring Music TheoryInsian
Scale 865Scale 865: Jahian, Ian Ring Music TheoryJahian
Scale 2913Scale 2913: Senian, Ian Ring Music TheorySenian
Scale 3937Scale 3937: Zalian, Ian Ring Music TheoryZalian

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.