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Scale 1691: "Kathian"

Scale 1691: Kathian, 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
Kathian

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

{0,1,3,4,7,9,10}

Forte Number

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

7-31

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: 2861

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.

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: 731

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, 3, 6, 3, 3, 3]

Interval Spectrum

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

p3m3n6s3d3t3

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

Spectra Variation

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

1.714

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 TriadsC{0,4,7}431.6
D♯{3,7,10}331.8
A{9,1,4}331.8
Minor Triadscm{0,3,7}331.7
am{9,0,4}331.7
Diminished Triadsc♯°{1,4,7}231.9
{4,7,10}231.9
{7,10,1}232
{9,0,3}232
a♯°{10,1,4}232
Parsimonious Voice Leading Between Common Triads of Scale 1691. Created by Ian Ring ©2019 cm cm C C cm->C D# D# cm->D# cm->a° c#° c#° C->c#° C->e° am am C->am A A c#°->A D#->e° D#->g° a#° a#° g°->a#° a°->am am->A A->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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1691 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2893
Scale 2893: Lylian, Ian Ring Music TheoryLylian
3rd mode:
Scale 1747
Scale 1747: Mela Ramapriya, Ian Ring Music TheoryMela Ramapriya
4th mode:
Scale 2921
Scale 2921: Pogian, Ian Ring Music TheoryPogian
5th mode:
Scale 877
Scale 877: Moravian Pistalkova, Ian Ring Music TheoryMoravian Pistalkova
6th mode:
Scale 1243
Scale 1243: Epylian, Ian Ring Music TheoryEpylian
7th mode:
Scale 2669
Scale 2669: Jeths' Mode, Ian Ring Music TheoryJeths' Mode

Prime

The prime form of this scale is Scale 731

Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode

Complement

The heptatonic modal family [1691, 2893, 1747, 2921, 877, 1243, 2669] (Forte: 7-31) is the complement of the pentatonic modal family [587, 601, 713, 1609, 2341] (Forte: 5-31)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1691 is 2861

Scale 2861Scale 2861: Katothian, Ian Ring Music TheoryKatothian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1691 is chiral, and its enantiomorph is scale 2861

Scale 2861Scale 2861: Katothian, Ian Ring Music TheoryKatothian

Transformations:

T0 1691  T0I 2861
T1 3382  T1I 1627
T2 2669  T2I 3254
T3 1243  T3I 2413
T4 2486  T4I 731
T5 877  T5I 1462
T6 1754  T6I 2924
T7 3508  T7I 1753
T8 2921  T8I 3506
T9 1747  T9I 2917
T10 3494  T10I 1739
T11 2893  T11I 3478

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 1689Scale 1689: Lorimic, Ian Ring Music TheoryLorimic
Scale 1693Scale 1693: Dogian, Ian Ring Music TheoryDogian
Scale 1695Scale 1695: Phrodyllic, Ian Ring Music TheoryPhrodyllic
Scale 1683Scale 1683: Raga Malayamarutam, Ian Ring Music TheoryRaga Malayamarutam
Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian
Scale 1675Scale 1675: Raga Salagavarali, Ian Ring Music TheoryRaga Salagavarali
Scale 1707Scale 1707: Dorian Flat 2, Ian Ring Music TheoryDorian Flat 2
Scale 1723Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic
Scale 1755Scale 1755: Octatonic, Ian Ring Music TheoryOctatonic
Scale 1563Scale 1563, Ian Ring Music Theory
Scale 1627Scale 1627: Zyptian, Ian Ring Music TheoryZyptian
Scale 1819Scale 1819: Pydian, Ian Ring Music TheoryPydian
Scale 1947Scale 1947: Byptyllic, Ian Ring Music TheoryByptyllic
Scale 1179Scale 1179: Sonimic, Ian Ring Music TheorySonimic
Scale 1435Scale 1435: Makam Huzzam, Ian Ring Music TheoryMakam Huzzam
Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic
Scale 2715Scale 2715: Kynian, Ian Ring Music TheoryKynian
Scale 3739Scale 3739: Epanyllic, Ian Ring Music TheoryEpanyllic

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.