The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 741: "Gathimic"

Scale 741: Gathimic, 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

Zeitler
Gathimic
Dozenal
Enoian

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

{0,2,5,6,7,9}

Forte Number

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

6-Z47

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: 1257

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.

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.

2

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: 663

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.

[2, 3, 1, 1, 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.

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

Interval Spectrum

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

p4m2n3s3d2t

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

Spectra Variation

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

2.333

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.

2.366

Polygon Perimeter

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

5.864

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

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.

(10, 16, 62)

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 TriadsD{2,6,9}221
F{5,9,0}221
Minor Triadsdm{2,5,9}221
Diminished Triadsf♯°{6,9,0}221

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

Parsimonious Voice Leading Between Common Triads of Scale 741. Created by Ian Ring ©2019 dm dm D D dm->D F F dm->F f#° f#° D->f#° F->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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 741 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1209
Scale 1209: Raga Bhanumanjari, Ian Ring Music TheoryRaga Bhanumanjari
3rd mode:
Scale 663
Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimicThis is the prime mode
4th mode:
Scale 2379
Scale 2379: Raga Gurjari Todi, Ian Ring Music TheoryRaga Gurjari Todi
5th mode:
Scale 3237
Scale 3237: Raga Brindabani Sarang, Ian Ring Music TheoryRaga Brindabani Sarang
6th mode:
Scale 1833
Scale 1833: Ionacrimic, Ian Ring Music TheoryIonacrimic

Prime

The prime form of this scale is Scale 663

Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic

Complement

The hexatonic modal family [741, 1209, 663, 2379, 3237, 1833] (Forte: 6-Z47) is the complement of the hexatonic modal family [363, 1419, 1581, 1713, 2229, 2757] (Forte: 6-Z25)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 741 is 1257

Scale 1257Scale 1257: Blues Scale, Ian Ring Music TheoryBlues Scale

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 741 is chiral, and its enantiomorph is scale 1257

Scale 1257Scale 1257: Blues Scale, Ian Ring Music TheoryBlues Scale

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> 741       T0I <11,0> 1257
T1 <1,1> 1482      T1I <11,1> 2514
T2 <1,2> 2964      T2I <11,2> 933
T3 <1,3> 1833      T3I <11,3> 1866
T4 <1,4> 3666      T4I <11,4> 3732
T5 <1,5> 3237      T5I <11,5> 3369
T6 <1,6> 2379      T6I <11,6> 2643
T7 <1,7> 663      T7I <11,7> 1191
T8 <1,8> 1326      T8I <11,8> 2382
T9 <1,9> 2652      T9I <11,9> 669
T10 <1,10> 1209      T10I <11,10> 1338
T11 <1,11> 2418      T11I <11,11> 2676
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3651      T0MI <7,0> 2127
T1M <5,1> 3207      T1MI <7,1> 159
T2M <5,2> 2319      T2MI <7,2> 318
T3M <5,3> 543      T3MI <7,3> 636
T4M <5,4> 1086      T4MI <7,4> 1272
T5M <5,5> 2172      T5MI <7,5> 2544
T6M <5,6> 249      T6MI <7,6> 993
T7M <5,7> 498      T7MI <7,7> 1986
T8M <5,8> 996      T8MI <7,8> 3972
T9M <5,9> 1992      T9MI <7,9> 3849
T10M <5,10> 3984      T10MI <7,10> 3603
T11M <5,11> 3873      T11MI <7,11> 3111

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 743Scale 743: Chromatic Hypophrygian Inverse, Ian Ring Music TheoryChromatic Hypophrygian Inverse
Scale 737Scale 737: Truian, Ian Ring Music TheoryTruian
Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic
Scale 745Scale 745: Kolimic, Ian Ring Music TheoryKolimic
Scale 749Scale 749: Aeologian, Ian Ring Music TheoryAeologian
Scale 757Scale 757: Ionyptian, Ian Ring Music TheoryIonyptian
Scale 709Scale 709: Raga Shri Kalyan, Ian Ring Music TheoryRaga Shri Kalyan
Scale 725Scale 725: Raga Yamuna Kalyani, Ian Ring Music TheoryRaga Yamuna Kalyani
Scale 677Scale 677: Scottish Pentatonic, Ian Ring Music TheoryScottish Pentatonic
Scale 613Scale 613: Phralitonic, Ian Ring Music TheoryPhralitonic
Scale 869Scale 869: Kothimic, Ian Ring Music TheoryKothimic
Scale 997Scale 997: Rycrian, Ian Ring Music TheoryRycrian
Scale 229Scale 229: Bidian, Ian Ring Music TheoryBidian
Scale 485Scale 485: Stoptimic, Ian Ring Music TheoryStoptimic
Scale 1253Scale 1253: Zolimic, Ian Ring Music TheoryZolimic
Scale 1765Scale 1765: Lonian, Ian Ring Music TheoryLonian
Scale 2789Scale 2789: Zolian, Ian Ring Music TheoryZolian

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.