The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1621: "Scriabin's Prometheus"

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.

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.

Notes

Name Sources

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.

 {0,2,4,6,9,10}

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.

[0222426391102](31)

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: 1357

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.

 4

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: 683

Forte Class

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

 6-34

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.

 [2, 2, 2, 3, 1, 2]

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, 4, 2, 4, 2, 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.

 p2m4n2s4dt2

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.

<0.2, 0.667, 0.4, 0.5, 0.4, 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.

2.482

Polygon Perimeter

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

5.932

Myhill's Property

A scale has Myhill's 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.

 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, 4, 45)

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.

0.938

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.

0.4

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.

31

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.

 AA6

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
FC3 = 1.414214
FC4 = 1.732051
FC5 = 1.931852
FC6 = 4

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 TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsD{2,6,9}221
Minor Triadsam{9,0,4}131.5
Augmented TriadsD+{2,6,10}131.5
Diminished Triadsf♯°{6,9,0}221
Parsimonious Voice Leading Between Common Triads of Scale 1621. Created by Ian Ring ©2019 D D D+ D+ D->D+ f#° f#° D->f#° am am f#°->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.

Diameter3
Radius2
Self-Centeredno
Central VerticesD, f♯°
Peripheral VerticesD+, am

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.


Augmented: {2, 6, 10}
Minor: {9, 0, 4}

Modes

Modes are the rotational transformation of this scale. Scale 1621 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1429		Bythimic
3rd mode:
Scale 1381		Padimic
4th mode:
Scale 1369		Boptimic
5th mode:
Scale 683		StogimicThis is the prime mode
6th mode:
Scale 2389		Inuit Hexatonic II

Prime

The prime form of this scale is Scale 683

Scale 683Stogimic

Complement

The hexatonic modal family [1621, 1429, 1381, 1369, 683, 2389] (Forte: 6-34) is the complement of the hexatonic modal family [683, 1369, 1381, 1429, 1621, 2389] (Forte: 6-34)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1621 is 1357

Scale 1357Takemitsu Linea II

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(4)

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.

kHierarchizabilityBreakdown PatternDiagram
111010101001101621k = 1h = 1
22(10)(10)(10)(10)01(10)1621k = 2h = 2
32(10)(10)(10)(10)01(10)1621k = 3h = 2
42([10][10])([10][10])01101621k = 4h = 2
52([10][10])([10][10])01101621k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1621 is chiral, and its enantiomorph is scale 1357

Scale 1357Takemitsu Linea II

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

(-0.022329, -0.083333)
Distance from Center0.086273
Angle in degrees

measured clockwise starting from the root.

345
Angle in cents

100 cents = 1 semitone.

1150

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> 1621       T0I <11,0> 1357
T1 <1,1> 3242      T1I <11,1> 2714
T2 <1,2> 2389      T2I <11,2> 1333
T3 <1,3> 683      T3I <11,3> 2666
T4 <1,4> 1366      T4I <11,4> 1237
T5 <1,5> 2732      T5I <11,5> 2474
T6 <1,6> 1369      T6I <11,6> 853
T7 <1,7> 2738      T7I <11,7> 1706
T8 <1,8> 1381      T8I <11,8> 3412
T9 <1,9> 2762      T9I <11,9> 2729
T10 <1,10> 1429      T10I <11,10> 1363
T11 <1,11> 2858      T11I <11,11> 2726
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1861      T0MI <7,0> 1117
T1M <5,1> 3722      T1MI <7,1> 2234
T2M <5,2> 3349      T2MI <7,2> 373
T3M <5,3> 2603      T3MI <7,3> 746
T4M <5,4> 1111      T4MI <7,4> 1492
T5M <5,5> 2222      T5MI <7,5> 2984
T6M <5,6> 349      T6MI <7,6> 1873
T7M <5,7> 698      T7MI <7,7> 3746
T8M <5,8> 1396      T8MI <7,8> 3397
T9M <5,9> 2792      T9MI <7,9> 2699
T10M <5,10> 1489      T10MI <7,10> 1303
T11M <5,11> 2978      T11MI <7,11> 2606

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 1623Alt 63
Scale 1617Phronitonic
Scale 1619Prometheus Neapolitan
Scale 1625Hungarian Major No5
Scale 1629Dorian Blues
Scale 1605Rāga Shubravarni
Scale 1613Thylimic
Scale 1637Syptimic
Scale 1653Minor Romani Inverse
Scale 1557JOVian
Scale 1589Rāga Rageshwari
Scale 1685Rāga Janasamohini
Scale 1749Acoustic
Scale 1877Lydian 7+
Scale 1109Five-Note Prometheus
Scale 1365Whole Tone
Scale 597Kung
Scale 2645Rāga Raj Kalyan
Scale 3669Lydian 65

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.