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Scale 2741: "Major"

Scale 2741: Major, 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
Ionian
Modern Ionian
Southeast Asia
Khmer Hepatatonic 4
Thailand
Thang Phiang Aw Bon
Ancient Greek
Greek Lydian
Medieval
Medieval Ionian
Medieval Hypolydian
Hindustani
Bilaval That
Bilaval Theta
That Bilaval
Carnatic
Mela Shankarabharanam
Mela Dhirasankarabharana
Raga Atana
Dhirashankarabharanam
Unknown / Unsorted
Ghana Heptatonic
Peruvian Major
Matzore
Rast Ascending
4th Plagal Byzantine
Ararai
Ajam Ashiran
Xin
DS2
Iranian
Dastgahi-e Rast Panjgah
Dastgah-e Rast Panjgah
Chinese
Zeitler
Ionian
Dozenal
IONian
Carnatic Melakarta
Dheerasankarabaranam
Carnatic Numbered Melakarta
29th Melakarta raga

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,2,4,5,7,9,11}

Forte Number

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

7-35

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.

2 (dihemitonic)

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.

6

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 1387

Generator

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

generator: 5
origin: 11

Deep Scale

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

yes

Interval Structure

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

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

<2, 5, 4, 3, 6, 1>

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.

p6m3n4s5d2t

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

Spectra Variation

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

0.857

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.

2.665

Polygon Perimeter

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

6.035

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".

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, 1, 56)

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.

0.993

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.

0.556

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

Generator

This scale has a generator of 5, originating on 11.

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}231.71
F{5,9,0}231.71
G{7,11,2}231.71
Minor Triadsdm{2,5,9}231.71
em{4,7,11}231.71
am{9,0,4}231.71
Diminished Triads{11,2,5}231.71
Parsimonious Voice Leading Between Common Triads of Scale 2741. Created by Ian Ring ©2019 C C em em C->em am am C->am dm dm F F dm->F dm->b° Parsimonious Voice Leading Between Common Triads of Scale 2741. Created by Ian Ring ©2019 G em->G F->am G->b°

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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2741 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1709
Scale 1709: Dorian, Ian Ring Music TheoryDorian
3rd mode:
Scale 1451
Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian
4th mode:
Scale 2773
Scale 2773: Lydian, Ian Ring Music TheoryLydian
5th mode:
Scale 1717
Scale 1717: Mixolydian, Ian Ring Music TheoryMixolydian
6th mode:
Scale 1453
Scale 1453: Aeolian, Ian Ring Music TheoryAeolian
7th mode:
Scale 1387
Scale 1387: Locrian, Ian Ring Music TheoryLocrianThis is the prime mode

Prime

The prime form of this scale is Scale 1387

Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian

Complement

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

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2741 is 1451

Scale 1451Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian

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> 2741       T0I <11,0> 1451
T1 <1,1> 1387      T1I <11,1> 2902
T2 <1,2> 2774      T2I <11,2> 1709
T3 <1,3> 1453      T3I <11,3> 3418
T4 <1,4> 2906      T4I <11,4> 2741
T5 <1,5> 1717      T5I <11,5> 1387
T6 <1,6> 3434      T6I <11,6> 2774
T7 <1,7> 2773      T7I <11,7> 1453
T8 <1,8> 1451      T8I <11,8> 2906
T9 <1,9> 2902      T9I <11,9> 1717
T10 <1,10> 1709      T10I <11,10> 3434
T11 <1,11> 3418      T11I <11,11> 2773
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3971      T0MI <7,0> 2111
T1M <5,1> 3847      T1MI <7,1> 127
T2M <5,2> 3599      T2MI <7,2> 254
T3M <5,3> 3103      T3MI <7,3> 508
T4M <5,4> 2111      T4MI <7,4> 1016
T5M <5,5> 127      T5MI <7,5> 2032
T6M <5,6> 254      T6MI <7,6> 4064
T7M <5,7> 508      T7MI <7,7> 4033
T8M <5,8> 1016      T8MI <7,8> 3971
T9M <5,9> 2032      T9MI <7,9> 3847
T10M <5,10> 4064      T10MI <7,10> 3599
T11M <5,11> 4033      T11MI <7,11> 3103

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 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic
Scale 2737Scale 2737: Raga Hari Nata, Ian Ring Music TheoryRaga Hari Nata
Scale 2739Scale 2739: Mela Suryakanta, Ian Ring Music TheoryMela Suryakanta
Scale 2745Scale 2745: Mela Sulini, Ian Ring Music TheoryMela Sulini
Scale 2749Scale 2749: Spanish Octamode 1st Rotation, Ian Ring Music TheorySpanish Octamode 1st Rotation
Scale 2725Scale 2725: Raga Nagagandhari, Ian Ring Music TheoryRaga Nagagandhari
Scale 2733Scale 2733: Melodic Minor Ascending, Ian Ring Music TheoryMelodic Minor Ascending
Scale 2709Scale 2709: Raga Kumud, Ian Ring Music TheoryRaga Kumud
Scale 2773Scale 2773: Lydian, Ian Ring Music TheoryLydian
Scale 2805Scale 2805: Ichikotsuchô, Ian Ring Music TheoryIchikotsuchô
Scale 2613Scale 2613: Raga Hamsa Vinodini, Ian Ring Music TheoryRaga Hamsa Vinodini
Scale 2677Scale 2677: Thodian, Ian Ring Music TheoryThodian
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
Scale 2997Scale 2997: Major Bebop, Ian Ring Music TheoryMajor Bebop
Scale 2229Scale 2229: Raga Nalinakanti, Ian Ring Music TheoryRaga Nalinakanti
Scale 2485Scale 2485: Harmonic Major, Ian Ring Music TheoryHarmonic Major
Scale 3253Scale 3253: Mela Naganandini, Ian Ring Music TheoryMela Naganandini
Scale 3765Scale 3765: Dominant Bebop, Ian Ring Music TheoryDominant Bebop
Scale 693Scale 693: Arezzo Major Diatonic Hexachord, Ian Ring Music TheoryArezzo Major Diatonic Hexachord
Scale 1717Scale 1717: Mixolydian, Ian Ring Music TheoryMixolydian

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.