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Scale 1805

Scale 1805, 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).

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,2,3,8,9,10}

Forte Number

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

6-Z12

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: 1565

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.

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.

5

Prime Form

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

no
prime: 215

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, 2, 2, 3, 2]

Interval Spectrum

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

p3m2n2s3d3t2

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

Spectra Variation

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

3.333

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.

1.866

Polygon Perimeter

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

5.485

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

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 TriadsG♯{8,0,3}110.5
Diminished Triads{9,0,3}110.5

The following pitch classes are not present in any of the common triads: {2,10}

Parsimonious Voice Leading Between Common Triads of Scale 1805. Created by Ian Ring ©2019 G# G# G#->a°

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1805 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1475
Scale 1475, Ian Ring Music Theory
3rd mode:
Scale 2785
Scale 2785, Ian Ring Music Theory
4th mode:
Scale 215
Scale 215, Ian Ring Music TheoryThis is the prime mode
5th mode:
Scale 2155
Scale 2155, Ian Ring Music Theory
6th mode:
Scale 3125
Scale 3125, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 215

Scale 215Scale 215, Ian Ring Music Theory

Complement

The hexatonic modal family [1805, 1475, 2785, 215, 2155, 3125] (Forte: 6-Z12) is the complement of the hexatonic modal family [335, 965, 1265, 2215, 3155, 3625] (Forte: 6-Z41)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1805 is 1565

Scale 1565Scale 1565, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1805 is chiral, and its enantiomorph is scale 1565

Scale 1565Scale 1565, Ian Ring Music Theory

Transformations:

T0 1805  T0I 1565
T1 3610  T1I 3130
T2 3125  T2I 2165
T3 2155  T3I 235
T4 215  T4I 470
T5 430  T5I 940
T6 860  T6I 1880
T7 1720  T7I 3760
T8 3440  T8I 3425
T9 2785  T9I 2755
T10 1475  T10I 1415
T11 2950  T11I 2830

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 1807Scale 1807, Ian Ring Music Theory
Scale 1801Scale 1801, Ian Ring Music Theory
Scale 1803Scale 1803, Ian Ring Music Theory
Scale 1797Scale 1797, Ian Ring Music Theory
Scale 1813Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
Scale 1821Scale 1821: Aeradian, Ian Ring Music TheoryAeradian
Scale 1837Scale 1837: Dalian, Ian Ring Music TheoryDalian
Scale 1869Scale 1869: Katyrian, Ian Ring Music TheoryKatyrian
Scale 1933Scale 1933: Mocrian, Ian Ring Music TheoryMocrian
Scale 1549Scale 1549, Ian Ring Music Theory
Scale 1677Scale 1677: Raga Manavi, Ian Ring Music TheoryRaga Manavi
Scale 1293Scale 1293, Ian Ring Music Theory
Scale 781Scale 781, Ian Ring Music Theory
Scale 2829Scale 2829, Ian Ring Music Theory
Scale 3853Scale 3853, Ian Ring Music Theory

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.