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Scale 675: "Altered Pentatonic"

Scale 675: Altered Pentatonic, 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).

Common Names

Western Modern
Altered Pentatonic
Dozenal
Emsian
Carnatic
Raga Manaranjani II
Zeitler
Zyditonic

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,1,5,7,9}

Forte Number

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

5-30

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: 2217

Hemitonia

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

1 (unhemitonic)

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.

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.

4

Prime Form

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

no
prime: 339

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, 4, 2, 2, 3]

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.

<1, 2, 1, 3, 2, 1>

Interval Spectrum

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

p2m3ns2dt

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

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.

2.049

Polygon Perimeter

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

5.664

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".

Proper

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.

(0, 5, 34)

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}110.5
Augmented TriadsC♯+{1,5,9}110.5

The following pitch classes are not present in any of the common triads: {7}

Parsimonious Voice Leading Between Common Triads of Scale 675. Created by Ian Ring ©2019 C#+ C#+ F F C#+->F

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 675 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2385
Scale 2385: Aeolanitonic, Ian Ring Music TheoryAeolanitonic
3rd mode:
Scale 405
Scale 405: Raga Bhupeshwari, Ian Ring Music TheoryRaga Bhupeshwari
4th mode:
Scale 1125
Scale 1125: Ionaritonic, Ian Ring Music TheoryIonaritonic
5th mode:
Scale 1305
Scale 1305: Dynitonic, Ian Ring Music TheoryDynitonic

Prime

The prime form of this scale is Scale 339

Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic

Complement

The pentatonic modal family [675, 2385, 405, 1125, 1305] (Forte: 5-30) is the complement of the heptatonic modal family [855, 1395, 1485, 1845, 2475, 2745, 3285] (Forte: 7-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 675 is 2217

Scale 2217Scale 2217: Kagitonic, Ian Ring Music TheoryKagitonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 675 is chiral, and its enantiomorph is scale 2217

Scale 2217Scale 2217: Kagitonic, Ian Ring Music TheoryKagitonic

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> 675       T0I <11,0> 2217
T1 <1,1> 1350      T1I <11,1> 339
T2 <1,2> 2700      T2I <11,2> 678
T3 <1,3> 1305      T3I <11,3> 1356
T4 <1,4> 2610      T4I <11,4> 2712
T5 <1,5> 1125      T5I <11,5> 1329
T6 <1,6> 2250      T6I <11,6> 2658
T7 <1,7> 405      T7I <11,7> 1221
T8 <1,8> 810      T8I <11,8> 2442
T9 <1,9> 1620      T9I <11,9> 789
T10 <1,10> 3240      T10I <11,10> 1578
T11 <1,11> 2385      T11I <11,11> 3156
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2595      T0MI <7,0> 2187
T1M <5,1> 1095      T1MI <7,1> 279
T2M <5,2> 2190      T2MI <7,2> 558
T3M <5,3> 285      T3MI <7,3> 1116
T4M <5,4> 570      T4MI <7,4> 2232
T5M <5,5> 1140      T5MI <7,5> 369
T6M <5,6> 2280      T6MI <7,6> 738
T7M <5,7> 465      T7MI <7,7> 1476
T8M <5,8> 930      T8MI <7,8> 2952
T9M <5,9> 1860      T9MI <7,9> 1809
T10M <5,10> 3720      T10MI <7,10> 3618
T11M <5,11> 3345      T11MI <7,11> 3141

The transformations that map this set to itself are: T0

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 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 677Scale 677: Scottish Pentatonic, Ian Ring Music TheoryScottish Pentatonic
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic
Scale 691Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga Kalavati
Scale 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 659Scale 659: Raga Rasika Ranjani, Ian Ring Music TheoryRaga Rasika Ranjani
Scale 707Scale 707: Ehoian, Ian Ring Music TheoryEhoian
Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic
Scale 547Scale 547: Pyrric, Ian Ring Music TheoryPyrric
Scale 611Scale 611: Anchihoye, Ian Ring Music TheoryAnchihoye
Scale 803Scale 803: Loritonic, Ian Ring Music TheoryLoritonic
Scale 931Scale 931: Raga Kalakanthi, Ian Ring Music TheoryRaga Kalakanthi
Scale 163Scale 163: Bapian, Ian Ring Music TheoryBapian
Scale 419Scale 419: Hon-kumoi-joshi, Ian Ring Music TheoryHon-kumoi-joshi
Scale 1187Scale 1187: Kokin-joshi, Ian Ring Music TheoryKokin-joshi
Scale 1699Scale 1699: Raga Rasavali, Ian Ring Music TheoryRaga Rasavali
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika

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.