The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1267: "Katynian"

Scale 1267: Katynian, 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

Zeitler
Katynian

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

{0,1,4,5,6,7,10}

Forte Number

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

7-19

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: 2533

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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: 719

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, 3, 1, 1, 1, 3, 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.

<4, 3, 4, 3, 4, 3>

Interval Spectrum

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

p4m3n4s3d4t3

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> = {3,5,6}
<4> = {6,7,9}
<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.

2.286

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.

2.433

Polygon Perimeter

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

5.899

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

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 TriadsC{0,4,7}231.71
F♯{6,10,1}231.71
Minor Triadsa♯m{10,1,5}231.71
Diminished Triadsc♯°{1,4,7}231.71
{4,7,10}231.71
{7,10,1}231.71
a♯°{10,1,4}231.71
Parsimonious Voice Leading Between Common Triads of Scale 1267. Created by Ian Ring ©2019 C C c#° c#° C->c#° C->e° a#° a#° c#°->a#° e°->g° F# F# F#->g° a#m a#m F#->a#m a#°->a#m

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 1267 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2681
Scale 2681: Aerycrian, Ian Ring Music TheoryAerycrian
3rd mode:
Scale 847
Scale 847: Ganian, Ian Ring Music TheoryGanian
4th mode:
Scale 2471
Scale 2471: Mela Ganamurti, Ian Ring Music TheoryMela Ganamurti
5th mode:
Scale 3283
Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari
6th mode:
Scale 3689
Scale 3689: Katocrian, Ian Ring Music TheoryKatocrian
7th mode:
Scale 973
Scale 973: Mela Syamalangi, Ian Ring Music TheoryMela Syamalangi

Prime

The prime form of this scale is Scale 719

Scale 719Scale 719: Kanian, Ian Ring Music TheoryKanian

Complement

The heptatonic modal family [1267, 2681, 847, 2471, 3283, 3689, 973] (Forte: 7-19) is the complement of the pentatonic modal family [203, 707, 1561, 2149, 2401] (Forte: 5-19)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1267 is 2533

Scale 2533Scale 2533: Podian, Ian Ring Music TheoryPodian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1267 is chiral, and its enantiomorph is scale 2533

Scale 2533Scale 2533: Podian, Ian Ring Music TheoryPodian

Transformations:

T0 1267  T0I 2533
T1 2534  T1I 971
T2 973  T2I 1942
T3 1946  T3I 3884
T4 3892  T4I 3673
T5 3689  T5I 3251
T6 3283  T6I 2407
T7 2471  T7I 719
T8 847  T8I 1438
T9 1694  T9I 2876
T10 3388  T10I 1657
T11 2681  T11I 3314

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 1265Scale 1265: Pynimic, Ian Ring Music TheoryPynimic
Scale 1269Scale 1269: Katythian, Ian Ring Music TheoryKatythian
Scale 1271Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic
Scale 1275Scale 1275: Stagyllic, Ian Ring Music TheoryStagyllic
Scale 1251Scale 1251: Sylimic, Ian Ring Music TheorySylimic
Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian
Scale 1235Scale 1235: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2
Scale 1203Scale 1203: Pagimic, Ian Ring Music TheoryPagimic
Scale 1139Scale 1139: Aerygimic, Ian Ring Music TheoryAerygimic
Scale 1395Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
Scale 1523Scale 1523: Zothyllic, Ian Ring Music TheoryZothyllic
Scale 1779Scale 1779: Zynyllic, Ian Ring Music TheoryZynyllic
Scale 243Scale 243, Ian Ring Music Theory
Scale 755Scale 755: Phrythian, Ian Ring Music TheoryPhrythian
Scale 2291Scale 2291: Zydian, Ian Ring Music TheoryZydian
Scale 3315Scale 3315: Tcherepnin Octatonic Mode 1, Ian Ring Music TheoryTcherepnin Octatonic Mode 1

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. 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.