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# Scale 1881: "Katorian" ### 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
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 Allen 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

#### 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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[3, 1, 2, 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, 4, 4, 4, 3, 3>

#### Interval Spectrum

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

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

#### 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.

(6, 33, 96)

## 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

am{9,0,4}331.63
d♯°{3,6,9}231.88
f♯°{6,9,0}231.75
{9,0,3}231.75

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 3 3 yes

## 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 Lynian This is the prime mode 3rd mode:Scale 2421 Malian 4th mode:Scale 1629 Synian 5th mode:Scale 1431 Phragian 6th mode:Scale 2763 Mela Suvarnangi 7th mode:Scale 3429 Marian

## Prime

The prime form of this scale is Scale 747

 Scale 747 Lynian

## 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 861 Rylian

## Enantiomorph

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

 Scale 861 Rylian

## 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> 1881       T0I <11,0> 861
T1 <1,1> 3762      T1I <11,1> 1722
T2 <1,2> 3429      T2I <11,2> 3444
T3 <1,3> 2763      T3I <11,3> 2793
T4 <1,4> 1431      T4I <11,4> 1491
T5 <1,5> 2862      T5I <11,5> 2982
T6 <1,6> 1629      T6I <11,6> 1869
T7 <1,7> 3258      T7I <11,7> 3738
T8 <1,8> 2421      T8I <11,8> 3381
T9 <1,9> 747      T9I <11,9> 2667
T10 <1,10> 1494      T10I <11,10> 1239
T11 <1,11> 2988      T11I <11,11> 2478
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 861      T0MI <7,0> 1881
T1M <5,1> 1722      T1MI <7,1> 3762
T2M <5,2> 3444      T2MI <7,2> 3429
T3M <5,3> 2793      T3MI <7,3> 2763
T4M <5,4> 1491      T4MI <7,4> 1431
T5M <5,5> 2982      T5MI <7,5> 2862
T6M <5,6> 1869      T6MI <7,6> 1629
T7M <5,7> 3738      T7MI <7,7> 3258
T8M <5,8> 3381      T8MI <7,8> 2421
T9M <5,9> 2667      T9MI <7,9> 747
T10M <5,10> 1239      T10MI <7,10> 1494
T11M <5,11> 2478      T11MI <7,11> 2988

The transformations that map this set to itself are: T0, T0MI

## 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 1883 Scale 1885 Saptyllic Scale 1873 Dathimic Scale 1877 Aeroptian Scale 1865 Thagimic Scale 1897 Ionopian Scale 1913 Scale 1817 Phrythimic Scale 1849 Chromatic Hypodorian Inverse Scale 1945 Zarian Scale 2009 Stacryllic Scale 1625 Lythimic Scale 1753 Hungarian Major Scale 1369 Boptimic Scale 857 Aeolydimic Scale 2905 Aeolian Flat 1 Scale 3929 Aeolothyllic

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.