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Scale 1881: "Katorian"

Scale 1881: Katorian, 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

41161837294116105072918310504116183
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
Katorian

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

{0,3,4,6,8,9,10}

Forte Number

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

7-28

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: 861

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.

6

Prime Form

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

no
prime: 747

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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, 4, 3, 3]

Interval Spectrum

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

p3m4n4s4d3t3

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

Harmonic Chords

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 TriadsG♯{8,0,3}331.63
Minor Triadsd♯m{3,6,10}231.88
am{9,0,4}331.63
Augmented TriadsC+{0,4,8}231.75
Diminished Triads{0,3,6}231.75
d♯°{3,6,9}231.88
f♯°{6,9,0}231.75
{9,0,3}231.75
Parsimonious Voice Leading Between Common Triads of Scale 1881. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ C+->G# am am C+->am d#° d#° d#°->d#m f#° f#° d#°->f#° f#°->am G#->a° 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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1881 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 747
Scale 747: Lynian, Ian Ring Music TheoryLynianThis is the prime mode
3rd mode:
Scale 2421
Scale 2421: Malian, Ian Ring Music TheoryMalian
4th mode:
Scale 1629
Scale 1629: Synian, Ian Ring Music TheorySynian
5th mode:
Scale 1431
Scale 1431: Phragian, Ian Ring Music TheoryPhragian
6th mode:
Scale 2763
Scale 2763: Mela Suvarnangi, Ian Ring Music TheoryMela Suvarnangi
7th mode:
Scale 3429
Scale 3429: Marian, Ian Ring Music TheoryMarian

Prime

The prime form of this scale is Scale 747

Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian

Complement

The heptatonic modal family [1881, 747, 2421, 1629, 1431, 2763, 3429] (Forte: 7-28) is the complement of the pentatonic modal family [333, 837, 1107, 1233, 2601] (Forte: 5-28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1881 is 861

Scale 861Scale 861: Rylian, Ian Ring Music TheoryRylian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1881 is chiral, and its enantiomorph is scale 861

Scale 861Scale 861: Rylian, Ian Ring Music TheoryRylian

Transformations:

T0 1881  T0I 861
T1 3762  T1I 1722
T2 3429  T2I 3444
T3 2763  T3I 2793
T4 1431  T4I 1491
T5 2862  T5I 2982
T6 1629  T6I 1869
T7 3258  T7I 3738
T8 2421  T8I 3381
T9 747  T9I 2667
T10 1494  T10I 1239
T11 2988  T11I 2478

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 1883Scale 1883, Ian Ring Music Theory
Scale 1885Scale 1885: Saptyllic, Ian Ring Music TheorySaptyllic
Scale 1873Scale 1873: Dathimic, Ian Ring Music TheoryDathimic
Scale 1877Scale 1877: Aeroptian, Ian Ring Music TheoryAeroptian
Scale 1865Scale 1865: Thagimic, Ian Ring Music TheoryThagimic
Scale 1897Scale 1897: Ionopian, Ian Ring Music TheoryIonopian
Scale 1913Scale 1913, Ian Ring Music Theory
Scale 1817Scale 1817: Phrythimic, Ian Ring Music TheoryPhrythimic
Scale 1849Scale 1849: Chromatic Hypodorian Inverse, Ian Ring Music TheoryChromatic Hypodorian Inverse
Scale 1945Scale 1945: Zarian, Ian Ring Music TheoryZarian
Scale 2009Scale 2009: Stacryllic, Ian Ring Music TheoryStacryllic
Scale 1625Scale 1625: Lythimic, Ian Ring Music TheoryLythimic
Scale 1753Scale 1753: Hungarian Major, Ian Ring Music TheoryHungarian Major
Scale 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic
Scale 857Scale 857: Aeolydimic, Ian Ring Music TheoryAeolydimic
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1
Scale 3929Scale 3929: Aeolothyllic, Ian Ring Music TheoryAeolothyllic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.