The Exciting Universe Of Music Theory

presents

more than you ever wanted to know about...

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.

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).

Other diagrams coming soon!

Names are messy, inconsistent, polysemic, and non-bijective. If you see a name with lots of citations beside it, that's a good measure of credulity.

- [0] Bedwell, Robert: Modal Method of Music. 2023. Locrian Publishing, Brighton, East Sussex.
- [1] Cochrane, Rich: Arpeggio and Scale Resources, a Guitar Encyclopedia. Big Noise Publishing, London UK.
- [2] Pecot, Justin: The Dozenal Standard, 2nd editon. Available from justinpecot.com.

## CardinalityCardinality is the count of how many pitches are in the scale. Cardinalities can be expressed as an adjective, e.g. |
7 (heptatonic) |

## Pitch Class SetThe tones in this scale, expressed as numbers from 0 to 11. |
{0,1,3,4,5,10,11} |

## Leonard NotationAs practiced in the theoretical work of B P Leonard, this notation for describing a pitch class set replaces commas with subscripted numbers indicating the interval distance between adjacent tones. Convenient when you are doing certain kinds of analysis. The superscript in parentheses is the sonority's | [0_{1}1_{2}3_{1}4_{1}5_{5}10_{1}11_{1}]^{(34)} |

## Rotational SymmetrySome scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity. |
none |

## Reflection AxesIf 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 |

## PalindromicityA palindromic scale has the same pattern of intervals both ascending and descending. |
no |

## ChiralityA chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph. |
yes enantiomorph: 2951 |

## HemitoniaA hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist. |
5 (multihemitonic) |

## CohemitoniaA cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist. |
3 (tricohemitonic) |

## ImperfectionsAn 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. |
3 |

## ModesModes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes. |
7 |

## Prime FormDescribes if this scale is in prime form, using the Starr/Rahn algorithm. |
no prime: 239 |

## GeneratorIndicates if the scale can be constructed using a generator, and an origin. |
none |

## Deep ScaleA deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization. |
no |

## Interval StructureDefines the scale as the sequence of intervals between one tone and the next. |
[1, 2, 1, 1, 5, 1, 1] |

## Interval VectorDescribes 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. |
<5, 4, 3, 3, 4, 2> |

## Hanson's Interval SpectrumThe same as the Interval Vector, but expressed in a syntax used by Howard Hanson. Hanson categorizes all intervals as being one of six classes, and gives each a letter: |
p^{4}m^{3}n^{3}s^{4}d^{5}t^{2} |

## Proportional Saturation VectorFirst described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible. | <0.75, 0.5, 0.25, 0, 0.5, 0.5> |

## Distribution SpectraDescribes 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} <3> = {3,4,7} <4> = {5,8,9} <5> = {6,9,10} <6> = {7,10,11} |

## Spectra VariationDetermined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality. |
3.429 |

## Maximally EvenA scale is maximally even if the tones are optimally spaced apart from each other. |
no |

## Maximal Area SetA 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 AreaArea 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. |
1.933 |

## Polygon PerimeterPerimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle. |
5.52 |

## Myhill PropertyA scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval. |
no |

## Centre of Gravity DistanceWhen tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero. |
0.342024 |

## Ridge TonesRidge 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 |

## ProprietyAlso 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 |

## Heteromorphic ProfileDefined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where | (55, 20, 84) |

## Coherence QuotientThe Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale. | 0.464 |

## Sameness QuotientThe Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity. | 0.333 |

## BrightnessBased on the theories of B P Leonard, Brightness is a measurement of interval content calculated by taking the sum of each tone's distance from the root. Scales with more tones at higher pitches will have a greater Brightness than those with fewer, lower pitches. Typically used to compare pitch class sets with the same cardinality. | 34 |

## XenomeA naming schema invented by Qid Love, and published in their book The Book Of Xenomes. A xenome is a succinct encoding of the pitch class set as three hexadecimal characters, with reversed bits such that they read left to right as ascending pitches. |
DC3 |

## Forte NumberA code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations. |
7-5 |

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

This scale has no generator.

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 Type | Triad^{*} | Pitch Classes | Degree | Eccentricity | Closeness Centrality |
---|---|---|---|---|---|

Minor Triads | a♯m | {10,1,5} | 1 | 1 | 0.5 |

Diminished Triads | a♯° | {10,1,4} | 1 | 1 | 0.5 |

The following pitch classes are not present in any of the common triads: {0,3,11}

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.

Diameter | 1 |
---|---|

Radius | 1 |

Self-Centered | yes |

Modes are the rotational transformation of this scale. Scale 3131 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode: Scale 3613 | WOSian | ||||

3rd mode: Scale 1927 | LUNian | ||||

4th mode: Scale 3011 | SUWian | ||||

5th mode: Scale 3553 | WEHian | ||||

6th mode: Scale 239 | BIKian | This is the prime mode | |||

7th mode: Scale 2167 | NEDian |

The prime form of this scale is Scale 239

Scale 239 | Infra-Alt 67 |

The heptatonic modal family [3131, 3613, 1927, 3011, 3553, 239, 2167] (Forte: 7-5) is the complement of the pentatonic modal family [143, 481, 2119, 3107, 3601] (Forte: 5-5)

The inverse of a scale is a reflection using the root as its axis. The inverse of 3131 is 2951

Scale 2951 | Neapolitan Major 534 |

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

## Contradictions (55) | ## Ambiguities(20) |

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable *k* is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

k | Hierarchizability | Breakdown Pattern | Diagram |
---|---|---|---|

1 | 1 | 110111000011 | |

2 | 1 | 110111000011 | |

3 | 1 | 110111000011 | |

4 | 3 | 1(0[1][1][1])000(0[1][1][1]) | |

5 | 3 | 1(0[1][1][1])000(0[1][1][1]) |

Only scales that are chiral will have an enantiomorph. Scale 3131 is chiral, and its enantiomorph is scale 2951

Scale 2951 | Neapolitan Major 534 |

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center | (0.214286, -0.266575) |
---|---|

Distance from Center | 0.342024 |

Angle in degrees measured clockwise starting from the root. | 38.794 |

Angle in cents 100 cents = 1 semitone. | 129.313333 |

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`. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev | Operation | Result | Abbrev | Operation | Result | |||
---|---|---|---|---|---|---|---|---|

T_{0} | <1,0> | 3131 | T_{0}I | <11,0> | 2951 | |||

T_{1} | <1,1> | 2167 | T_{1}I | <11,1> | 1807 | |||

T_{2} | <1,2> | 239 | T_{2}I | <11,2> | 3614 | |||

T_{3} | <1,3> | 478 | T_{3}I | <11,3> | 3133 | |||

T_{4} | <1,4> | 956 | T_{4}I | <11,4> | 2171 | |||

T_{5} | <1,5> | 1912 | T_{5}I | <11,5> | 247 | |||

T_{6} | <1,6> | 3824 | T_{6}I | <11,6> | 494 | |||

T_{7} | <1,7> | 3553 | T_{7}I | <11,7> | 988 | |||

T_{8} | <1,8> | 3011 | T_{8}I | <11,8> | 1976 | |||

T_{9} | <1,9> | 1927 | T_{9}I | <11,9> | 3952 | |||

T_{10} | <1,10> | 3854 | T_{10}I | <11,10> | 3809 | |||

T_{11} | <1,11> | 3613 | T_{11}I | <11,11> | 3523 | |||

Abbrev | Operation | Result | Abbrev | Operation | Result | |||

T_{0}M | <5,0> | 431 | T_{0}MI | <7,0> | 3761 | |||

T_{1}M | <5,1> | 862 | T_{1}MI | <7,1> | 3427 | |||

T_{2}M | <5,2> | 1724 | T_{2}MI | <7,2> | 2759 | |||

T_{3}M | <5,3> | 3448 | T_{3}MI | <7,3> | 1423 | |||

T_{4}M | <5,4> | 2801 | T_{4}MI | <7,4> | 2846 | |||

T_{5}M | <5,5> | 1507 | T_{5}MI | <7,5> | 1597 | |||

T_{6}M | <5,6> | 3014 | T_{6}MI | <7,6> | 3194 | |||

T_{7}M | <5,7> | 1933 | T_{7}MI | <7,7> | 2293 | |||

T_{8}M | <5,8> | 3866 | T_{8}MI | <7,8> | 491 | |||

T_{9}M | <5,9> | 3637 | T_{9}MI | <7,9> | 982 | |||

T_{10}M | <5,10> | 3179 | T_{10}MI | <7,10> | 1964 | |||

T_{11}M | <5,11> | 2263 | T_{11}MI | <7,11> | 3928 |

The transformations that map this set to itself are: T_{0}

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 3129 | TOQian | |||

Scale 3133 | Melodic 465 | |||

Scale 3135 | Octatonic Chromatic 3 | |||

Scale 3123 | TOMian | |||

Scale 3127 | Alt 7635 | |||

Scale 3115 | TIHian | |||

Scale 3099 | TIXian | |||

Scale 3163 | Alt 76 | |||

Scale 3195 | Raryllic | |||

Scale 3259 | Loptyllic | |||

Scale 3387 | Aeryptyllic | |||

Scale 3643 | Kydyllic | |||

Scale 2107 | MUTian | |||

Scale 2619 | Alt 675 | |||

Scale 1083 | GOYian |

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages were invented by living persons, and used here with permission where required: notably collections of names by William Zeitler, Justin Pecot, Rich Cochrane, and Robert Bedwell.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Gratitudes to Qid Love for the Xenomes. Thanks to B P Leonard for the Brightness metrics. Thanks to u/howaboot for inventing the Center of Gravity metrics.