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# Scale 3467: "Katonian" ### 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).

Zeitler
Katonian
Dozenal
Vuhian

## 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

{0,1,3,7,8,10,11}

#### Forte Number

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

7-11

#### 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: 2615

#### 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: 379

#### 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.

[1, 2, 4, 1, 2, 1, 1]

#### 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, 4, 4, 4, 4, 1>

#### Interval Spectrum

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

p4m4n4s4d4t

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

#### Spectra Variation

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

3.143

#### 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.

2.299

#### Polygon Perimeter

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

5.803

#### 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.

(43, 27, 92)

## 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. 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

G♯{8,0,3}241.83
g♯m{8,11,3}231.5

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.

Diameter 4 2 no D♯+ g°, G♯

## Modes

Modes are the rotational transformation of this scale. Scale 3467 can be rotated to make 6 other scales. The 1st mode is itself.

 2nd mode:Scale 3781 Gyphian 3rd mode:Scale 1969 Stylian 4th mode:Scale 379 Aeragian This is the prime mode 5th mode:Scale 2237 Epothian 6th mode:Scale 1583 Salian 7th mode:Scale 2839 Lyptian

## Prime

The prime form of this scale is Scale 379

 Scale 379 Aeragian

## Complement

The heptatonic modal family [3467, 3781, 1969, 379, 2237, 1583, 2839] (Forte: 7-11) is the complement of the pentatonic modal family [157, 929, 1063, 2579, 3337] (Forte: 5-11)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3467 is 2615

 Scale 2615 Thoptian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3467 is chiral, and its enantiomorph is scale 2615

 Scale 2615 Thoptian

## 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> 3467       T0I <11,0> 2615
T1 <1,1> 2839      T1I <11,1> 1135
T2 <1,2> 1583      T2I <11,2> 2270
T3 <1,3> 3166      T3I <11,3> 445
T4 <1,4> 2237      T4I <11,4> 890
T5 <1,5> 379      T5I <11,5> 1780
T6 <1,6> 758      T6I <11,6> 3560
T7 <1,7> 1516      T7I <11,7> 3025
T8 <1,8> 3032      T8I <11,8> 1955
T9 <1,9> 1969      T9I <11,9> 3910
T10 <1,10> 3938      T10I <11,10> 3725
T11 <1,11> 3781      T11I <11,11> 3355
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2237      T0MI <7,0> 1955
T1M <5,1> 379      T1MI <7,1> 3910
T2M <5,2> 758      T2MI <7,2> 3725
T3M <5,3> 1516      T3MI <7,3> 3355
T4M <5,4> 3032      T4MI <7,4> 2615
T5M <5,5> 1969      T5MI <7,5> 1135
T6M <5,6> 3938      T6MI <7,6> 2270
T7M <5,7> 3781      T7MI <7,7> 445
T8M <5,8> 3467       T8MI <7,8> 890
T9M <5,9> 2839      T9MI <7,9> 1780
T10M <5,10> 1583      T10MI <7,10> 3560
T11M <5,11> 3166      T11MI <7,11> 3025

The transformations that map this set to itself are: T0, T8M

## 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 3465 Katathimic Scale 3469 Monian Scale 3471 Gyryllic Scale 3459 Vocian Scale 3463 Vofian Scale 3475 Kylian Scale 3483 Mixotharyllic Scale 3499 Hamel Scale 3531 Neveseri Scale 3339 Smuian Scale 3403 Bylian Scale 3211 Epacrimic Scale 3723 Myptian Scale 3979 Dynyllic Scale 2443 Panimic Scale 2955 Thorian Scale 1419 Raga Kashyapi

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.