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Scale 1185: "Genus Primum Inverse"

Scale 1185: Genus Primum Inverse, 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

Ancient Greek
Genus Primum Inverse



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.



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

0 (anhemitonic)


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: 165


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

generator: 5
origin: 7

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, 2, 3, 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.

<0, 2, 1, 0, 3, 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> = {2,3,5}
<2> = {5,7}
<3> = {7,9,10}

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.

(0, 4, 14)


This scale has a generator of 5, originating on 7.

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 1185 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 165
Scale 165: Genus Primum, Ian Ring Music TheoryGenus PrimumThis is the prime mode
3rd mode:
Scale 1065
Scale 1065: Gonian, Ian Ring Music TheoryGonian
4th mode:
Scale 645
Scale 645: Duyian, Ian Ring Music TheoryDuyian


The prime form of this scale is Scale 165

Scale 165Scale 165: Genus Primum, Ian Ring Music TheoryGenus Primum


The tetratonic modal family [1185, 165, 1065, 645] (Forte: 4-23) is the complement of the octatonic modal family [1455, 1515, 1725, 1965, 2775, 2805, 3435, 3765] (Forte: 8-23)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1185 is 165

Scale 165Scale 165: Genus Primum, Ian Ring Music TheoryGenus Primum


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> 1185       T0I <11,0> 165
T1 <1,1> 2370      T1I <11,1> 330
T2 <1,2> 645      T2I <11,2> 660
T3 <1,3> 1290      T3I <11,3> 1320
T4 <1,4> 2580      T4I <11,4> 2640
T5 <1,5> 1065      T5I <11,5> 1185
T6 <1,6> 2130      T6I <11,6> 2370
T7 <1,7> 165      T7I <11,7> 645
T8 <1,8> 330      T8I <11,8> 1290
T9 <1,9> 660      T9I <11,9> 2580
T10 <1,10> 1320      T10I <11,10> 1065
T11 <1,11> 2640      T11I <11,11> 2130
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2055      T0MI <7,0> 3075
T1M <5,1> 15      T1MI <7,1> 2055
T2M <5,2> 30      T2MI <7,2> 15
T3M <5,3> 60      T3MI <7,3> 30
T4M <5,4> 120      T4MI <7,4> 60
T5M <5,5> 240      T5MI <7,5> 120
T6M <5,6> 480      T6MI <7,6> 240
T7M <5,7> 960      T7MI <7,7> 480
T8M <5,8> 1920      T8MI <7,8> 960
T9M <5,9> 3840      T9MI <7,9> 1920
T10M <5,10> 3585      T10MI <7,10> 3840
T11M <5,11> 3075      T11MI <7,11> 3585

The transformations that map this set to itself are: T0, T5I

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 1187Scale 1187: Kokin-joshi, Ian Ring Music TheoryKokin-joshi
Scale 1189Scale 1189: Suspended Pentatonic, Ian Ring Music TheorySuspended Pentatonic
Scale 1193Scale 1193: Minor Pentatonic, Ian Ring Music TheoryMinor Pentatonic
Scale 1201Scale 1201: Mixolydian Pentatonic, Ian Ring Music TheoryMixolydian Pentatonic
Scale 1153Scale 1153: Choian, Ian Ring Music TheoryChoian
Scale 1169Scale 1169: Raga Mahathi, Ian Ring Music TheoryRaga Mahathi
Scale 1217Scale 1217: Hician, Ian Ring Music TheoryHician
Scale 1249Scale 1249: Howian, Ian Ring Music TheoryHowian
Scale 1057Scale 1057: Sansagari, Ian Ring Music TheorySansagari
Scale 1121Scale 1121: Guwian, Ian Ring Music TheoryGuwian
Scale 1313Scale 1313: Iplian, Ian Ring Music TheoryIplian
Scale 1441Scale 1441: Jabian, Ian Ring Music TheoryJabian
Scale 1697Scale 1697: Raga Kuntvarali, Ian Ring Music TheoryRaga Kuntvarali
Scale 161Scale 161: Raga Sarvasri, Ian Ring Music TheoryRaga Sarvasri
Scale 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 2209Scale 2209: Nidian, Ian Ring Music TheoryNidian
Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian

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.