 The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

# Scale 1045 ### 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).

## Analysis

#### Cardinality

Cardinality is the count of how many pitches are in the scale.

4 (tetratonic)

#### Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,2,4,10}

#### Forte Number

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

4-21

#### 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.



#### 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.

no

#### Hemitonia

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

0 (anhemitonic)

#### 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.

3

#### Prime Form

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

no
prime: 85

#### 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.

[0, 3, 0, 2, 0, 1]

#### Interval Spectrum

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

m2s3t

#### 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> = {2,6}
<2> = {4,8}
<3> = {6,10}

#### Spectra Variation

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

3

#### 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.

1.299

#### Polygon Perimeter

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

5

#### Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

yes

#### 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.



#### 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

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

## Modes

Modes are the rotational transformation of this scale. Scale 1045 can be rotated to make 3 other scales. The 1st mode is itself.

 2nd mode:Scale 1285 3rd mode:Scale 1345 4th mode:Scale 85 This is the prime mode

## Prime

The prime form of this scale is Scale 85

 Scale 85 ## Complement

The tetratonic modal family [1045, 1285, 1345, 85] (Forte: 4-21) is the complement of the octatonic modal family [1375, 1405, 1525, 2005, 2735, 3415, 3755, 3925] (Forte: 8-21)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1045 is 1285

 Scale 1285 ## Transformations:

 T0 1045 T0I 1285 T1 2090 T1I 2570 T2 85 T2I 1045 T3 170 T3I 2090 T4 340 T4I 85 T5 680 T5I 170 T6 1360 T6I 340 T7 2720 T7I 680 T8 1345 T8I 1360 T9 2690 T9I 2720 T10 1285 T10I 1345 T11 2570 T11I 2690

## 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 1047 Scale 1041 Scale 1043 Scale 1049 Scale 1053 Scale 1029 Scale 1037 Warao Tetratonic Scale 1061 Scale 1077 Scale 1109 Kataditonic Scale 1173 Dominant Pentatonic Scale 1301 Koditonic Scale 1557 Scale 21 Scale 533 Scale 2069 Scale 3093 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.