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Scale 1569: "Jocian"

Scale 1569: Jocian, 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.

4 (tetratonic)

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: 141


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

1 (unhemitonic)


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: 141


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.

[5, 4, 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, 1, 1, 1, 2, 0>

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

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.

(4, 2, 18)

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}000

The following pitch classes are not present in any of the common triads: {10}

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 1569 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 177
Scale 177: Bexian, Ian Ring Music TheoryBexian
3rd mode:
Scale 267
Scale 267: Bobian, Ian Ring Music TheoryBobian
4th mode:
Scale 2181
Scale 2181: Nemian, Ian Ring Music TheoryNemian


The prime form of this scale is Scale 141

Scale 141Scale 141: Babian, Ian Ring Music TheoryBabian


The tetratonic modal family [1569, 177, 267, 2181] (Forte: 4-14) is the complement of the octatonic modal family [759, 1839, 1977, 2427, 2967, 3261, 3531, 3813] (Forte: 8-14)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1569 is 141

Scale 141Scale 141: Babian, Ian Ring Music TheoryBabian


Only scales that are chiral will have an enantiomorph. Scale 1569 is chiral, and its enantiomorph is scale 141

Scale 141Scale 141: Babian, Ian Ring Music TheoryBabian


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> 1569       T0I <11,0> 141
T1 <1,1> 3138      T1I <11,1> 282
T2 <1,2> 2181      T2I <11,2> 564
T3 <1,3> 267      T3I <11,3> 1128
T4 <1,4> 534      T4I <11,4> 2256
T5 <1,5> 1068      T5I <11,5> 417
T6 <1,6> 2136      T6I <11,6> 834
T7 <1,7> 177      T7I <11,7> 1668
T8 <1,8> 354      T8I <11,8> 3336
T9 <1,9> 708      T9I <11,9> 2577
T10 <1,10> 1416      T10I <11,10> 1059
T11 <1,11> 2832      T11I <11,11> 2118
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 519      T0MI <7,0> 3081
T1M <5,1> 1038      T1MI <7,1> 2067
T2M <5,2> 2076      T2MI <7,2> 39
T3M <5,3> 57      T3MI <7,3> 78
T4M <5,4> 114      T4MI <7,4> 156
T5M <5,5> 228      T5MI <7,5> 312
T6M <5,6> 456      T6MI <7,6> 624
T7M <5,7> 912      T7MI <7,7> 1248
T8M <5,8> 1824      T8MI <7,8> 2496
T9M <5,9> 3648      T9MI <7,9> 897
T10M <5,10> 3201      T10MI <7,10> 1794
T11M <5,11> 2307      T11MI <7,11> 3588

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 1571Scale 1571: Lagitonic, Ian Ring Music TheoryLagitonic
Scale 1573Scale 1573: Raga Guhamanohari, Ian Ring Music TheoryRaga Guhamanohari
Scale 1577Scale 1577: Raga Chandrakauns (Kafi), Ian Ring Music TheoryRaga Chandrakauns (Kafi)
Scale 1585Scale 1585: Raga Khamaji Durga, Ian Ring Music TheoryRaga Khamaji Durga
Scale 1537Scale 1537: Jijian, Ian Ring Music TheoryJijian
Scale 1553Scale 1553: Josian, Ian Ring Music TheoryJosian
Scale 1601Scale 1601: Juwian, Ian Ring Music TheoryJuwian
Scale 1633Scale 1633: Kapian, Ian Ring Music TheoryKapian
Scale 1697Scale 1697: Raga Kuntvarali, Ian Ring Music TheoryRaga Kuntvarali
Scale 1825Scale 1825: Lecian, Ian Ring Music TheoryLecian
Scale 1057Scale 1057: Sansagari, Ian Ring Music TheorySansagari
Scale 1313Scale 1313: Iplian, Ian Ring Music TheoryIplian
Scale 545Scale 545: Dewian, Ian Ring Music TheoryDewian
Scale 2593Scale 2593: Puxian, Ian Ring Music TheoryPuxian
Scale 3617Scale 3617: Wovian, Ian Ring Music TheoryWovian

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.