The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 1559: "Jowian"

Scale 1559: Jowian, 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.

6 (hexatonic)

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: 3341


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

3 (trihemitonic)


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

1 (uncohemitonic)


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: 187


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.

[1, 1, 2, 5, 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.

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

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.

(29, 7, 55)

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 TriadsA{9,1,4}210.67
Minor Triadsam{9,0,4}121
Diminished Triadsa♯°{10,1,4}121

The following pitch classes are not present in any of the common triads: {2}

Parsimonious Voice Leading Between Common Triads of Scale 1559. Created by Ian Ring ©2019 am am A A am->A a#° a#° A->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.

Central VerticesA
Peripheral Verticesam, a♯°


Modes are the rotational transformation of this scale. Scale 1559 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2827
Scale 2827: Runian, Ian Ring Music TheoryRunian
3rd mode:
Scale 3461
Scale 3461: Vodian, Ian Ring Music TheoryVodian
4th mode:
Scale 1889
Scale 1889: Loqian, Ian Ring Music TheoryLoqian
5th mode:
Scale 187
Scale 187: Bedian, Ian Ring Music TheoryBedianThis is the prime mode
6th mode:
Scale 2141
Scale 2141: Nanian, Ian Ring Music TheoryNanian


The prime form of this scale is Scale 187

Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian


The hexatonic modal family [1559, 2827, 3461, 1889, 187, 2141] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1559 is 3341

Scale 3341Scale 3341: Vahian, Ian Ring Music TheoryVahian


Only scales that are chiral will have an enantiomorph. Scale 1559 is chiral, and its enantiomorph is scale 3341

Scale 3341Scale 3341: Vahian, Ian Ring Music TheoryVahian


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> 1559       T0I <11,0> 3341
T1 <1,1> 3118      T1I <11,1> 2587
T2 <1,2> 2141      T2I <11,2> 1079
T3 <1,3> 187      T3I <11,3> 2158
T4 <1,4> 374      T4I <11,4> 221
T5 <1,5> 748      T5I <11,5> 442
T6 <1,6> 1496      T6I <11,6> 884
T7 <1,7> 2992      T7I <11,7> 1768
T8 <1,8> 1889      T8I <11,8> 3536
T9 <1,9> 3778      T9I <11,9> 2977
T10 <1,10> 3461      T10I <11,10> 1859
T11 <1,11> 2827      T11I <11,11> 3718
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1829      T0MI <7,0> 1181
T1M <5,1> 3658      T1MI <7,1> 2362
T2M <5,2> 3221      T2MI <7,2> 629
T3M <5,3> 2347      T3MI <7,3> 1258
T4M <5,4> 599      T4MI <7,4> 2516
T5M <5,5> 1198      T5MI <7,5> 937
T6M <5,6> 2396      T6MI <7,6> 1874
T7M <5,7> 697      T7MI <7,7> 3748
T8M <5,8> 1394      T8MI <7,8> 3401
T9M <5,9> 2788      T9MI <7,9> 2707
T10M <5,10> 1481      T10MI <7,10> 1319
T11M <5,11> 2962      T11MI <7,11> 2638

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 1557Scale 1557: Jovian, Ian Ring Music TheoryJovian
Scale 1555Scale 1555: Jotian, Ian Ring Music TheoryJotian
Scale 1563Scale 1563: Joyian, Ian Ring Music TheoryJoyian
Scale 1567Scale 1567: Jobian, Ian Ring Music TheoryJobian
Scale 1543Scale 1543: Jomian, Ian Ring Music TheoryJomian
Scale 1551Scale 1551: Jorian, Ian Ring Music TheoryJorian
Scale 1575Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic
Scale 1591Scale 1591: Rodian, Ian Ring Music TheoryRodian
Scale 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian
Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian
Scale 1815Scale 1815: Godian, Ian Ring Music TheoryGodian
Scale 1047Scale 1047: Gician, Ian Ring Music TheoryGician
Scale 1303Scale 1303: Epolimic, Ian Ring Music TheoryEpolimic
Scale 535Scale 535: Dejian, Ian Ring Music TheoryDejian
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 3607Scale 3607: Wopian, Ian Ring Music TheoryWopian

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.