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Scale 683: "Stogimic"

Scale 683: Stogimic, 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
Stogimic
Dozenal
Elfian

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

{0,1,3,5,7,9}

Forte Number

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

6-34

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: 2729

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 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.

yes

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.

[1, 2, 2, 2, 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.

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

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 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".

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)

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 TriadsF{5,9,0}221
Minor Triadscm{0,3,7}131.5
Augmented TriadsC♯+{1,5,9}131.5
Diminished Triads{9,0,3}221
Parsimonious Voice Leading Between Common Triads of Scale 683. Created by Ian Ring ©2019 cm cm cm->a° C#+ C#+ F F C#+->F F->a°

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 VerticesF, a°
Peripheral Verticescm, C♯+

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: {0, 3, 7}
Augmented: {1, 5, 9}

Modes

Modes are the rotational transformation of this scale. Scale 683 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2389
Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
3rd mode:
Scale 1621
Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus
4th mode:
Scale 1429
Scale 1429: Bythimic, Ian Ring Music TheoryBythimic
5th mode:
Scale 1381
Scale 1381: Padimic, Ian Ring Music TheoryPadimic
6th mode:
Scale 1369
Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [683, 2389, 1621, 1429, 1381, 1369] (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 683 is 2729

Scale 2729Scale 2729: Aeragimic, Ian Ring Music TheoryAeragimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 683 is chiral, and its enantiomorph is scale 2729

Scale 2729Scale 2729: Aeragimic, Ian Ring Music TheoryAeragimic

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

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 681Scale 681: Kyemyonjo, Ian Ring Music TheoryKyemyonjo
Scale 685Scale 685: Raga Suddha Bangala, Ian Ring Music TheoryRaga Suddha Bangala
Scale 687Scale 687: Aeolythian, Ian Ring Music TheoryAeolythian
Scale 675Scale 675: Altered Pentatonic, Ian Ring Music TheoryAltered Pentatonic
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 691Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga Kalavati
Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian
Scale 651Scale 651: Golitonic, Ian Ring Music TheoryGolitonic
Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic
Scale 715Scale 715: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2
Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian
Scale 555Scale 555: Aeolycritonic, Ian Ring Music TheoryAeolycritonic
Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic
Scale 811Scale 811: Radimic, Ian Ring Music TheoryRadimic
Scale 939Scale 939: Mela Senavati, Ian Ring Music TheoryMela Senavati
Scale 171Scale 171: Pruian, Ian Ring Music TheoryPruian
Scale 427Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha Simantini
Scale 1195Scale 1195: Raga Gandharavam, Ian Ring Music TheoryRaga Gandharavam
Scale 1707Scale 1707: Dorian Flat 2, Ian Ring Music TheoryDorian Flat 2
Scale 2731Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major

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.