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Scale 2881: "Satian"

Scale 2881: Satian, 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




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

5 (pentatonic)

Pitch Class Set

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


Forte Number

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 91


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

2 (dihemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)


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.



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.


Prime Form

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

prime: 91


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

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


Interval Structure

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

[6, 2, 1, 2, 1]

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, 3, 1, 1, 1>

Interval Spectrum

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


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> = {1,2,6}
<2> = {3,7,8}
<3> = {4,5,9}
<4> = {6,10,11}

Spectra Variation

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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(14, 1, 30)

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
Diminished Triadsf♯°{6,9,0}000

The following pitch classes are not present in any of the common triads: {8,11}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 2881 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 109
Scale 109: Amsian, Ian Ring Music TheoryAmsian
3rd mode:
Scale 1051
Scale 1051: Gifian, Ian Ring Music TheoryGifian
4th mode:
Scale 2573
Scale 2573: Pulian, Ian Ring Music TheoryPulian
5th mode:
Scale 1667
Scale 1667: Kekian, Ian Ring Music TheoryKekian


The prime form of this scale is Scale 91

Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian


The pentatonic modal family [2881, 109, 1051, 2573, 1667] (Forte: 5-10) is the complement of the heptatonic modal family [607, 761, 1993, 2351, 3223, 3659, 3877] (Forte: 7-10)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2881 is 91

Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian


Only scales that are chiral will have an enantiomorph. Scale 2881 is chiral, and its enantiomorph is scale 91

Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian


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> 2881       T0I <11,0> 91
T1 <1,1> 1667      T1I <11,1> 182
T2 <1,2> 3334      T2I <11,2> 364
T3 <1,3> 2573      T3I <11,3> 728
T4 <1,4> 1051      T4I <11,4> 1456
T5 <1,5> 2102      T5I <11,5> 2912
T6 <1,6> 109      T6I <11,6> 1729
T7 <1,7> 218      T7I <11,7> 3458
T8 <1,8> 436      T8I <11,8> 2821
T9 <1,9> 872      T9I <11,9> 1547
T10 <1,10> 1744      T10I <11,10> 3094
T11 <1,11> 3488      T11I <11,11> 2093
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 721      T0MI <7,0> 361
T1M <5,1> 1442      T1MI <7,1> 722
T2M <5,2> 2884      T2MI <7,2> 1444
T3M <5,3> 1673      T3MI <7,3> 2888
T4M <5,4> 3346      T4MI <7,4> 1681
T5M <5,5> 2597      T5MI <7,5> 3362
T6M <5,6> 1099      T6MI <7,6> 2629
T7M <5,7> 2198      T7MI <7,7> 1163
T8M <5,8> 301      T8MI <7,8> 2326
T9M <5,9> 602      T9MI <7,9> 557
T10M <5,10> 1204      T10MI <7,10> 1114
T11M <5,11> 2408      T11MI <7,11> 2228

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 2883Scale 2883: Savian, Ian Ring Music TheorySavian
Scale 2885Scale 2885: Byrimic, Ian Ring Music TheoryByrimic
Scale 2889Scale 2889: Thoptimic, Ian Ring Music TheoryThoptimic
Scale 2897Scale 2897: Rycrimic, Ian Ring Music TheoryRycrimic
Scale 2913Scale 2913: Senian, Ian Ring Music TheorySenian
Scale 2817Scale 2817, Ian Ring Music Theory
Scale 2849Scale 2849: Rubian, Ian Ring Music TheoryRubian
Scale 2945Scale 2945: Sihian, Ian Ring Music TheorySihian
Scale 3009Scale 3009: Suvian, Ian Ring Music TheorySuvian
Scale 2625Scale 2625, Ian Ring Music Theory
Scale 2753Scale 2753: Ritian, Ian Ring Music TheoryRitian
Scale 2369Scale 2369: Offian, Ian Ring Music TheoryOffian
Scale 3393Scale 3393: Venian, Ian Ring Music TheoryVenian
Scale 3905Scale 3905: Yusian, Ian Ring Music TheoryYusian
Scale 833Scale 833: Febian, Ian Ring Music TheoryFebian
Scale 1857Scale 1857: Liwian, Ian Ring Music TheoryLiwian

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 ( 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.