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Scale 4005

Scale 4005, 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).

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,2,5,7,8,9,10,11}

Forte Number

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

8-10

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.

[3.5]

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.

5 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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.

7

Prime Form

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

no
prime: 765

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.

[2, 3, 2, 1, 1, 1, 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.

<5, 6, 6, 4, 5, 2>

Interval Spectrum

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

p5m4n6s6d5t2

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

Spectra Variation

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

2.75

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.

2.616

Polygon Perimeter

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

6.002

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.

[7]

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 TriadsF{5,9,0}242.1
G{7,11,2}342.1
A♯{10,2,5}341.9
Minor Triadsdm{2,5,9}341.9
fm{5,8,0}342.1
gm{7,10,2}242.1
Diminished Triads{2,5,8}242.1
{5,8,11}242.1
g♯°{8,11,2}242.1
{11,2,5}242.1
Parsimonious Voice Leading Between Common Triads of Scale 4005. Created by Ian Ring ©2019 dm dm d°->dm fm fm d°->fm F F dm->F A# A# dm->A# f°->fm g#° g#° f°->g#° fm->F gm gm Parsimonious Voice Leading Between Common Triads of Scale 4005. Created by Ian Ring ©2019 G gm->G gm->A# G->g#° G->b° A#->b°

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 4005 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2025
Scale 2025, Ian Ring Music Theory
3rd mode:
Scale 765
Scale 765, Ian Ring Music TheoryThis is the prime mode
4th mode:
Scale 1215
Scale 1215, Ian Ring Music Theory
5th mode:
Scale 2655
Scale 2655, Ian Ring Music Theory
6th mode:
Scale 3375
Scale 3375, Ian Ring Music Theory
7th mode:
Scale 3735
Scale 3735, Ian Ring Music Theory
8th mode:
Scale 3915
Scale 3915, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 765

Scale 765Scale 765, Ian Ring Music Theory

Complement

The octatonic modal family [4005, 2025, 765, 1215, 2655, 3375, 3735, 3915] (Forte: 8-10) is the complement of the tetratonic modal family [45, 1035, 1665, 2565] (Forte: 4-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4005 is 1215

Scale 1215Scale 1215, Ian Ring Music Theory

Transformations:

T0 4005  T0I 1215
T1 3915  T1I 2430
T2 3735  T2I 765
T3 3375  T3I 1530
T4 2655  T4I 3060
T5 1215  T5I 2025
T6 2430  T6I 4050
T7 765  T7I 4005
T8 1530  T8I 3915
T9 3060  T9I 3735
T10 2025  T10I 3375
T11 4050  T11I 2655

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 4007Scale 4007: Doptygic, Ian Ring Music TheoryDoptygic
Scale 4001Scale 4001, Ian Ring Music Theory
Scale 4003Scale 4003: Sadyllic, Ian Ring Music TheorySadyllic
Scale 4009Scale 4009: Phranyllic, Ian Ring Music TheoryPhranyllic
Scale 4013Scale 4013: Raga Pilu, Ian Ring Music TheoryRaga Pilu
Scale 4021Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
Scale 3973Scale 3973, Ian Ring Music Theory
Scale 3989Scale 3989: Sythyllic, Ian Ring Music TheorySythyllic
Scale 4037Scale 4037: Ionyllic, Ian Ring Music TheoryIonyllic
Scale 4069Scale 4069: Starygic, Ian Ring Music TheoryStarygic
Scale 3877Scale 3877: Thanian, Ian Ring Music TheoryThanian
Scale 3941Scale 3941: Stathyllic, Ian Ring Music TheoryStathyllic
Scale 3749Scale 3749: Raga Sorati, Ian Ring Music TheoryRaga Sorati
Scale 3493Scale 3493: Rathian, Ian Ring Music TheoryRathian
Scale 2981Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
Scale 1957Scale 1957: Pyrian, Ian Ring Music TheoryPyrian

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.