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Scale 661: "Major Pentatonic"

Scale 661: Major 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
Major Pentatonic
Pentatonic Major
Major Pentatonic Type 1
Korean
Kyemyonjo
Japanese
Yo Yonanuki sempû
Ryosen
Yona Nuki Major
Yo Yonanuki sempû
Unknown / Unsorted
Ghana Pent.2
Ethiopian
Tezeta Major
Tizita Major
Chinese
Man Jue
Gong
Gōngdiào
Carnatic
Raga Bhopali
Raga Bhupali
Raga Bhup
Bhoop
Mohanam
Raag Deskar
Bilahari
Kokila
Jait Kalyan
Exoticisms
Peruvian Pentatonic 1
African
Ghana Pentatonic 2
Myanmar
Paksabou Palé
Leibauk auk Pyan
Ngabauk Myinsain
Southeast Asia
Akha Khmer Pentatonic 1
Kmhmu 5 Tone Type 3
Lahuzu 5 Tone Type 5
MiaoYao 5 Tone Type 1
Zeitler
Pentatonic
Dozenal
ULTian

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

{0,2,4,7,9}

Forte Number

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

5-35

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.

[2]

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.

1

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 Starr/Rahn algorithm.

yes

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 5
origin: 4

Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.

no

Interval Structure

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

[2, 2, 3, 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.

<0, 3, 2, 1, 4, 0>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0, 0.75, 0.5, 0, 1, 0>

Interval Spectrum

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

p4mn2s3

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

Spectra Variation

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

0.8

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

yes

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.

yes

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.

2.299

Polygon Perimeter

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

5.828

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

[4]

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".

Strictly 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, 0, 20)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

1

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.5

Generator

This scale has a generator of 5, originating on 4.

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 TriadsC{0,4,7}110.5
Minor Triadsam{9,0,4}110.5

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

Parsimonious Voice Leading Between Common Triads of Scale 661. Created by Ian Ring ©2019 C C am am C->am

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

2nd mode:
Scale 1189
Scale 1189: Suspended Pentatonic, Ian Ring Music TheorySuspended Pentatonic
3rd mode:
Scale 1321
Scale 1321: Blues Minor, Ian Ring Music TheoryBlues Minor
4th mode:
Scale 677
Scale 677: Scottish Pentatonic, Ian Ring Music TheoryScottish Pentatonic
5th mode:
Scale 1193
Scale 1193: Minor Pentatonic, Ian Ring Music TheoryMinor Pentatonic

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [661, 1189, 1321, 677, 1193] (Forte: 5-35) is the complement of the heptatonic modal family [1387, 1451, 1453, 1709, 1717, 2741, 2773] (Forte: 7-35)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 661 is 1321

Scale 1321Scale 1321: Blues Minor, Ian Ring Music TheoryBlues Minor

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> 661       T0I <11,0> 1321
T1 <1,1> 1322      T1I <11,1> 2642
T2 <1,2> 2644      T2I <11,2> 1189
T3 <1,3> 1193      T3I <11,3> 2378
T4 <1,4> 2386      T4I <11,4> 661
T5 <1,5> 677      T5I <11,5> 1322
T6 <1,6> 1354      T6I <11,6> 2644
T7 <1,7> 2708      T7I <11,7> 1193
T8 <1,8> 1321      T8I <11,8> 2386
T9 <1,9> 2642      T9I <11,9> 677
T10 <1,10> 1189      T10I <11,10> 1354
T11 <1,11> 2378      T11I <11,11> 2708
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3841      T0MI <7,0> 31
T1M <5,1> 3587      T1MI <7,1> 62
T2M <5,2> 3079      T2MI <7,2> 124
T3M <5,3> 2063      T3MI <7,3> 248
T4M <5,4> 31      T4MI <7,4> 496
T5M <5,5> 62      T5MI <7,5> 992
T6M <5,6> 124      T6MI <7,6> 1984
T7M <5,7> 248      T7MI <7,7> 3968
T8M <5,8> 496      T8MI <7,8> 3841
T9M <5,9> 992      T9MI <7,9> 3587
T10M <5,10> 1984      T10MI <7,10> 3079
T11M <5,11> 3968      T11MI <7,11> 2063

The transformations that map this set to itself are: T0, T4I

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 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic
Scale 657Scale 657: Lahuzu 4 Tone Type 3, Ian Ring Music TheoryLahuzu 4 Tone Type 3
Scale 659Scale 659: Raga Rasika Ranjani, Ian Ring Music TheoryRaga Rasika Ranjani
Scale 665Scale 665: Raga Mohanangi, Ian Ring Music TheoryRaga Mohanangi
Scale 669Scale 669: Gycrimic, Ian Ring Music TheoryGycrimic
Scale 645Scale 645: DUYian, Ian Ring Music TheoryDUYian
Scale 653Scale 653: Dorian Pentatonic, Ian Ring Music TheoryDorian Pentatonic
Scale 677Scale 677: Scottish Pentatonic, Ian Ring Music TheoryScottish Pentatonic
Scale 693Scale 693: Arezzo Major Diatonic Hexachord, Ian Ring Music TheoryArezzo Major Diatonic Hexachord
Scale 725Scale 725: Raga Yamuna Kalyani, Ian Ring Music TheoryRaga Yamuna Kalyani
Scale 533Scale 533: DEHian, Ian Ring Music TheoryDEHian
Scale 597Scale 597: Kung, Ian Ring Music TheoryKung
Scale 789Scale 789: Zogitonic, Ian Ring Music TheoryZogitonic
Scale 917Scale 917: Dygimic, Ian Ring Music TheoryDygimic
Scale 149Scale 149: Eskimo Tetratonic, Ian Ring Music TheoryEskimo Tetratonic
Scale 405Scale 405: Raga Bhupeshwari, Ian Ring Music TheoryRaga Bhupeshwari
Scale 1173Scale 1173: Dominant Pentatonic, Ian Ring Music TheoryDominant Pentatonic
Scale 1685Scale 1685: Zeracrimic, Ian Ring Music TheoryZeracrimic
Scale 2709Scale 2709: Raga Kumud, Ian Ring Music TheoryRaga Kumud

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.