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Scale 885: "Sathian"

Scale 885: Sathian, 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
Sathian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

7 (heptatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,2,4,5,6,8,9}

Forte Number

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

7-26

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: 1497

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

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.

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.

6

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 699

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[2, 2, 1, 1, 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.

<3, 4, 4, 5, 3, 2>

Interval Spectrum

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

p3m5n4s4d3t2

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

Spectra Variation

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

2.286

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.

2.549

Polygon Perimeter

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

5.967

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

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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}242
F{5,9,0}421.25
Minor Triadsdm{2,5,9}331.5
fm{5,8,0}331.5
am{9,0,4}231.75
Augmented TriadsC+{0,4,8}242
Diminished Triads{2,5,8}231.75
f♯°{6,9,0}231.75
Parsimonious Voice Leading Between Common Triads of Scale 885. Created by Ian Ring ©2019 C+ C+ fm fm C+->fm am am C+->am dm dm d°->dm d°->fm D D dm->D F F dm->F f#° f#° D->f#° fm->F F->f#° F->am

view full size

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesF
Peripheral VerticesC+, D

Modes

Modes are the rotational transformation of this scale. Scale 885 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1245
Scale 1245: Lathian, Ian Ring Music TheoryLathian
3rd mode:
Scale 1335
Scale 1335: Elephant Scale, Ian Ring Music TheoryElephant Scale
4th mode:
Scale 2715
Scale 2715: Kynian, Ian Ring Music TheoryKynian
5th mode:
Scale 3405
Scale 3405: Stynian, Ian Ring Music TheoryStynian
6th mode:
Scale 1875
Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale
7th mode:
Scale 2985
Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian

Prime

The prime form of this scale is Scale 699

Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian

Complement

The heptatonic modal family [885, 1245, 1335, 2715, 3405, 1875, 2985] (Forte: 7-26) is the complement of the pentatonic modal family [309, 849, 1101, 1299, 2697] (Forte: 5-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 885 is 1497

Scale 1497Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 885 is chiral, and its enantiomorph is scale 1497

Scale 1497Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini

Transformations:

T0 885  T0I 1497
T1 1770  T1I 2994
T2 3540  T2I 1893
T3 2985  T3I 3786
T4 1875  T4I 3477
T5 3750  T5I 2859
T6 3405  T6I 1623
T7 2715  T7I 3246
T8 1335  T8I 2397
T9 2670  T9I 699
T10 1245  T10I 1398
T11 2490  T11I 2796

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 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic
Scale 881Scale 881: Aerothimic, Ian Ring Music TheoryAerothimic
Scale 883Scale 883: Ralian, Ian Ring Music TheoryRalian
Scale 889Scale 889: Borian, Ian Ring Music TheoryBorian
Scale 893Scale 893: Dadyllic, Ian Ring Music TheoryDadyllic
Scale 869Scale 869: Kothimic, Ian Ring Music TheoryKothimic
Scale 877Scale 877: Moravian Pistalkova, Ian Ring Music TheoryMoravian Pistalkova
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 821Scale 821: Aeranimic, Ian Ring Music TheoryAeranimic
Scale 949Scale 949: Mela Mararanjani, Ian Ring Music TheoryMela Mararanjani
Scale 1013Scale 1013: Stydyllic, Ian Ring Music TheoryStydyllic
Scale 629Scale 629: Aeronimic, Ian Ring Music TheoryAeronimic
Scale 757Scale 757: Ionyptian, Ian Ring Music TheoryIonyptian
Scale 373Scale 373: Epagimic, Ian Ring Music TheoryEpagimic
Scale 1397Scale 1397: Major Locrian, Ian Ring Music TheoryMajor Locrian
Scale 1909Scale 1909: Epicryllic, Ian Ring Music TheoryEpicryllic
Scale 2933Scale 2933, Ian Ring Music Theory

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. 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.