The Exciting Universe Of Music Theory

presents

more than you ever wanted to know about...

Scale 843: "Molimic"

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 reflective 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).

Keyboard Diagram

Other diagrams coming soon!

Common Names

Names are messy, inconsistent, polysemic, and non-bijective. If you see a name with lots of citations beside it, that's a good measure of credulity.

Compound Chords
- 6^dim + 2^sus4[0]
- 5^dim + 6^sus4[0]
- 5^min + 6^maj[0]
Dozenal
- FEHian[1]
Zeitler
- Molimic[2]

Name Sources

[0] Ring, Ian: The Exciting Universe Of Music Theory retrieved 2024-11-26
[1] Pecot, Justin: The Dozenal Standard, 2nd editon. Available from justinpecot.com.
[2] Zeitler, William: "all the scales". https://allthescales.org. Retrieved April 2024

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale. Cardinalities can be expressed as an adjective, e.g. pentatonic, hexatonic, heptatonic, and so on.	6 (hexatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11.	{0,1,3,6,8,9}
Leonard Notation As practiced in the theoretical work of B P Leonard, this notation for describing a pitch class set replaces commas with subscripted numbers indicating the interval distance between adjacent tones. Convenient when you are doing certain kinds of analysis. The superscript in parentheses is the sonority's Brightness.	[0₁1₂3₃6₂8₁9₃]⁽²⁷⁾
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.	[4.5]
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.	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.	3
Modes Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.	6
Prime Form Describes if this scale is in prime form, using the Starr/Rahn algorithm.	no prime: 717
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-Z29
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, an indicator of maximum hierarchization.	no
Interval Structure Defines the scale as the sequence of intervals between one tone and the next.	[1, 2, 3, 2, 1, 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, 2, 4, 2, 3, 2>
Hanson's Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson. Hanson categorizes all intervals as being one of six classes, and gives each a letter: p m n s d t, ordered from most consonant (p) to most dissonant (t). When an interval appears more than once in a sonority, it is superscripted with a number, like p².	p³m²n⁴s²d²t²
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.	<0.4, 0.333, 0.8, 0, 0.6, 0.667>
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> = {3,4,5} <3> = {5,6,7} <4> = {7,8,9} <5> = {9,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	1.667
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 Distribution Spectra have exactly two specific intervals for every generic interval.	no
Centre of Gravity Distance When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.	0.086273
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.	[9]
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, 12, 57)
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.	0.815
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.	0.24
Brightness Based on the theories of B P Leonard, Brightness is a measurement of interval content calculated by taking the sum of each tone's distance from the root. Scales with more tones at higher pitches will have a greater Brightness than those with fewer, lower pitches. Typically used to compare pitch class sets with the same cardinality.	27
Xenome A naming schema invented by Qid Love, and published in their book The Book Of Xenomes. A xenome is a succinct encoding of the pitch class set as three hexadecimal characters, with reversed bits such that they read left to right as ascending pitches.	D2C
Lewin-Quinn FC Components Conceived by David Lewin in 1959, and generalized by Ian Quinn; FC is a function that transforms the set by multiplying by a given number of semitones, then gives the total distance of the sum of the vectors produced by each tone when mapped to a circular plane. The result is a discrete Fourier transform that quantifies harmonic qualities. A more thorough and sensible explanation is coming in the book. FC Components can be used to analyze other tuning systems besides 12-TET. FC0 is the cardinality of the scale.	FC0 = 6 FC1 = 0.517638 FC2 = 1.732051 FC3 = 1.414214 FC4 = 3 FC5 = 1.931852 FC6 = 0

Generator

This scale has no generator.

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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Major Triads	G♯	{8,0,3}	2	3	1.5
Minor Triads	f♯m	{6,9,1}	2	3	1.5
Diminished Triads	c°	{0,3,6}	2	3	1.5
	d♯°	{3,6,9}	2	3	1.5
	f♯°	{6,9,0}	2	3	1.5
	a°	{9,0,3}	2	3	1.5

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.

Diameter	3
Radius	3
Self-Centered	yes

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.

Minor: {6, 9, 1}
Major: {8, 0, 3}

Modes

Modes are the rotational transformation of this scale. Scale 843 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode: Scale 2469	Rāga Bhinna Pancama
3rd mode: Scale 1641	Bocrimic
4th mode: Scale 717	Rāga Shivawanti	This is the prime mode
5th mode: Scale 1203	Rāga Gaula
6th mode: Scale 2649	Aeolythimic

Prime

The prime form of this scale is Scale 717

Scale 717

Rāga Shivawanti

Complement

The hexatonic modal family [843, 2469, 1641, 717, 1203, 2649] (Forte: 6-Z29) is the complement of the hexatonic modal family [723, 813, 1227, 1689, 2409, 2661] (Forte: 6-Z50)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 843 is 2649

Scale 2649

Aeolythimic

Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

Contradictions (0)

Ambiguities(12)

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

k	Hierarchizability	Breakdown Pattern
1	1	`110100101100`
2	1	`110100101100`
3	1	`110100101100`
4	2	`10(1001)01(1001)`
5	2	`10(1001)01(1001)`

Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center	(-0.061004, -0.061004)
Distance from Center	0.086273
Angle in degrees measured clockwise starting from the root.	315
Angle in cents 100 cents = 1 semitone.	1050

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev	Operation	Result	Abbrev	Operation	Result
T₀	<1,0>	843	T₀I	<11,0>	2649
T₁	<1,1>	1686	T₁I	<11,1>	1203
T₂	<1,2>	3372	T₂I	<11,2>	2406
T₃	<1,3>	2649	T₃I	<11,3>	717
T₄	<1,4>	1203	T₄I	<11,4>	1434
T₅	<1,5>	2406	T₅I	<11,5>	2868
T₆	<1,6>	717	T₆I	<11,6>	1641
T₇	<1,7>	1434	T₇I	<11,7>	3282
T₈	<1,8>	2868	T₈I	<11,8>	2469
T₉	<1,9>	1641	T₉I	<11,9>	843
T₁₀	<1,10>	3282	T₁₀I	<11,10>	1686
T₁₁	<1,11>	2469	T₁₁I	<11,11>	3372
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	633	T₀MI	<7,0>	969
T₁M	<5,1>	1266	T₁MI	<7,1>	1938
T₂M	<5,2>	2532	T₂MI	<7,2>	3876
T₃M	<5,3>	969	T₃MI	<7,3>	3657
T₄M	<5,4>	1938	T₄MI	<7,4>	3219
T₅M	<5,5>	3876	T₅MI	<7,5>	2343
T₆M	<5,6>	3657	T₆MI	<7,6>	591
T₇M	<5,7>	3219	T₇MI	<7,7>	1182
T₈M	<5,8>	2343	T₈MI	<7,8>	2364
T₉M	<5,9>	591	T₉MI	<7,9>	633
T₁₀M	<5,10>	1182	T₁₀MI	<7,10>	1266
T₁₁M	<5,11>	2364	T₁₁MI	<7,11>	2532

The transformations that map this set to itself are: T₀, T₉I

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 841				Phrothitonic
Scale 845				Rāga Neelangi
Scale 847				Alt 347
Scale 835				FECian
Scale 839				Ionathimic
Scale 851				Rāga Hejjajji
Scale 859				Ultralocrian
Scale 875				Locrian 7
Scale 779				ETRian
Scale 811				Radimic
Scale 907				Tholimic
Scale 971				Mela Gavambodhi
Scale 587				Major Pentatonic 3
Scale 715				T4 Prime Mode
Scale 331				Rāga Chaya Todi
Scale 1355				Rāga Mangal Gujari
Scale 1867				Lydian 7+23
Scale 2891				Lydian +23

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by 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 were invented by living persons, and used here with permission where required: notably collections of names by William Zeitler, Justin Pecot, Rich Cochrane, and Robert Bedwell.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Gratitudes to Qid Love for the Xenomes. Thanks to B P Leonard for the Brightness metrics. Thanks to u/howaboot for inventing the Center of Gravity metrics.

Scale 843: "Molimic"

Bracelet Diagram

Tonnetz Diagram

Keyboard Diagram

Common Names

Name Sources

Analysis

Cardinality

Pitch Class Set

Leonard Notation

Proportional Saturation Vector

Polygon Perimeter

Centre of Gravity Distance

Heteromorphic Profile

Coherence Quotient

Sameness Quotient

Brightness

Xenome

Lewin-Quinn FC Components