The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1621: "Scriabin's Prometheus"

Scale 1621: Scriabin's Prometheus, 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

Named After Composers
Scriabin's Prometheus
Mystic
Western
Prometheus
Carnatic
Raga Barbara
Zeitler
Aeolathimic
Dozenal
KAHian

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

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

Forte Number

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

 6-34

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

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>

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>

Interval Spectrum

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

 p2m4n2s4dt2

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 Property

A scale has Myhill Property if the Distribution Spectra have 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".

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

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		Scale 1429: Bythimic, Ian Ring Music TheoryBythimic
3rd mode:
Scale 1381		Scale 1381: Padimic, Ian Ring Music TheoryPadimic
4th mode:
Scale 1369		Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic
5th mode:
Scale 683		Scale 683: Stogimic, Ian Ring Music TheoryStogimicThis is the prime mode
6th mode:
Scale 2389		Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2

Prime

The prime form of this scale is Scale 683

Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic

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 1357Scale 1357: Takemitsu Linea Mode 2, Ian Ring Music TheoryTakemitsu Linea Mode 2

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 1357Scale 1357: Takemitsu Linea Mode 2, Ian Ring Music TheoryTakemitsu Linea Mode 2

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 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian
Scale 1617Scale 1617: Phronitonic, Ian Ring Music TheoryPhronitonic
Scale 1619Scale 1619: Prometheus Neapolitan, Ian Ring Music TheoryPrometheus Neapolitan
Scale 1625Scale 1625: Hungarian Major No5, Ian Ring Music TheoryHungarian Major No5
Scale 1629Scale 1629: Synian, Ian Ring Music TheorySynian
Scale 1605Scale 1605: Zanitonic, Ian Ring Music TheoryZanitonic
Scale 1613Scale 1613: Thylimic, Ian Ring Music TheoryThylimic
Scale 1637Scale 1637: Syptimic, Ian Ring Music TheorySyptimic
Scale 1653Scale 1653: Minor Romani Inverse, Ian Ring Music TheoryMinor Romani Inverse
Scale 1557Scale 1557: JOVian, Ian Ring Music TheoryJOVian
Scale 1589Scale 1589: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
Scale 1685Scale 1685: Zeracrimic, Ian Ring Music TheoryZeracrimic
Scale 1749Scale 1749: Acoustic, Ian Ring Music TheoryAcoustic
Scale 1877Scale 1877: Aeroptian, Ian Ring Music TheoryAeroptian
Scale 1109Scale 1109: Kataditonic, Ian Ring Music TheoryKataditonic
Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone
Scale 597Scale 597: Kung, Ian Ring Music TheoryKung
Scale 2645Scale 2645: Raga Mruganandana, Ian Ring Music TheoryRaga Mruganandana
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian

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.