A Study of Scales

Where we discuss every possible combination of notes

This exploration of scales is based on work by William Zeitler1. In fact much of the material on this page repeats Zeitler's findings, presented here along with additional observations and/or PHP code used to generate the scales. This exploration also owes a debt to Andrew Duncan's work on combinatorial music theory2. Most of the code in this treatise is the result of continuous exploration and development of PHPMusicTools, an open-source project at GitHub. The section related to scale-chord relationships is indebted to Enrico Dell'Aquila, and his video about Mode Colours

Assumptions

This exploration of scales is based in a musical universe founded on two assumptions:

  • Octave Equivalence
    We assume that for the purpose of defining a scale, a pitch is functionally equivalent to another pitch separated by an octave. So it follows that if you're playing a scale in one octave, if you continue the pattern into the next octave you will play pitches with the same name.

  • 12 tone equal temperament
    We're using the 12 tones of an equally-tempered tuning system, as you'd find on a piano. Equal temperament asserts that the perceptual (or functional) relationship between two pitches is the same as the relationship between two other pitches with the same chromatic interval distance. Other tuning systems have the notion of scales, but they're not being considered in this study.

Representing a scale

When I began piano lessons as a child, I learned that a scale was made up of whole- and half-steps. The scale was essentially a stack of intervals piled up, that happened to add up to an octave. While this view of the scale is useful for understanding diatonic melodies, it's not a convenient paradigm for analysis. Instead, we should regard a scale as being a set of 12 possibilities, and each one is either on or off.

The major scale, in lights.

What we have in the 12-tone system is a binary "word" made of 12 bits. We can assign one bit to each degree of the chromatic scale, and use the power of binary arithmetic and logic to do some pretty awesome analysis with them. When represented as bits it reads from right to left - the lowest bit is the root, and each bit going from right to left ascends by one semitone.

The total number of possible combinations of on and off bits is called the "power set". The number of sets in a power set of size n is (2^n). Using a word of 12 bits, the power set (2^12) is equal to 4096. The fun thing about binary power sets is that we can produce every possible combination, by merely invoking the integers from 0 (no tones) to 4095 (all 12 tones).

This means that every possible combination of tones in the 12-tone set can be represented by a number between 0 and 4095. We don't need to remember the fancy names like "phrygian", we can just call it scale number 1451. Convenient!

$allscales = range(0, 4095);
decimalbinary
0 000000000000 no notes in the scale
1 000000000001 just the root tone
1365 010101010101 whole tone scale
2741 101010110101 major scale
4095 111111111111 chromatic scale

An important concept here is that any set of tones can be represented by a number. This number is not "ordinal" - it's not merely describing the position of the set in an indexed scheme; it's also not "cardinal" because it's not describing an amount of something. This is a nominal number because the number *is* the scale. You can do binary arithmetic with it, and you are adding and subtracting scales with no need to convert the scale into some other representation.


Scale Finder

Not sure what scale you've got? Check the boxes below tones that appear in your scale. Then follow the link to learn more about it!


Binary:
Decimal:

Because scales are cyclical - they repeat and continue beyond a single octave - it is intuitive to represent them in "bracelet" notation. This visual representation is useful because it helps conceptualize modes and rotation (which we'll get into further below).

Here is the major scale, represented as a bracelet. Black beads are on bits, white beads are off bits.

If you imagine the bracelet with clock numbers, the topmost bead - at 12 o'clock - is the root, or 1st degree of the scale. Beware that in some of the math, this is the "0" position, as if we're counting semitones above the root. To play a scale ascending, start from the top and move around it clockwise. To play the scale descending, start at 12 o'clock and go around counterclockwise.

Interval Pattern

Another popular way of representing a scale is by its interval pattern. When I was learning the major scale, I was taught to say aloud: "tone, tone, semitone, tone, tone, tone, semitone". Many music theorists like to represent a scale this way because it's accurate and easy to understand: "TTSTTTS". Having a scale's interval pattern has merit as an intermediary step can make some kinds of analysis simpler. Expressed numerically - which is more convenient for computation - the major scale is [2,2,1,2,2,2,1].

Pitch Class Sets

Yet another way to represent a scale is as a "pitch class set", where the tones are assigned numbers 0 to 11 (sometimes using "T" and "E" for 10 and 11), and the set enumerates the ones present in the scale. A pitch class set for the major scale is notated like this: {0,2,4,5,7,9,11}. The "scales" we'll study here are a subset of Pitch Classes (ie those that have a root, and obey Zeitler's Rules3) and we can use many of the same mathematical tricks to manipulate them.

What is a scale?

Or more importantly, what is *not* a scale?

Now that we have the superset of all possible sets of tones, we can whittle it down to exclude ones that we don't consider to be a legitimate "scale". We can do this with just two rules.

  • A scale starts on the root tone.

    This means any set of notes that doesn't have that first bit turned on is not eligible to be called a scale. This cuts our power set in exactly half, leaving 2048 sets.

    In binary, it's easy to see that the first bit is on or off. In decimal, you can easily tell this by whether the number is odd or even. All even numbers have the first bit off; therefore all scales are represented by an odd number.

    We could have cut some corners in our discussion of scales by omitting the root tone (always assumed to be on) to work with 11 bits instead of 12, but there are compelling reasons for keeping the 12-bit number for our scales, such as simplifying the analysis of symmetry, simplifying the calculation of modes, and performing analysis of sonorities that do not include the root, where an even number is appropriate.

    function hasRootTone($scale) { // returns true if the first bit is not a zero return (1 & $scale) != 0; } $allscales = array_filter($allscales, 'hasRootTone');

    scales remaining: 2048

  • A scale does not have any leaps greater than n semitones.

    For the purposes of this exercise we are saying n = 4, a.k.a. a major third. Any collection of tones that has an interval greater than a major third is not considered a "scale". This configuration is consistent with Zeitler's constant used to generate his comprehensive list of scales.

    function doesNotHaveFourConsecutiveOffBits($scale) { $c = 0; for ($i=0; $i<12; $i++) { if (!($scale & (1 << ($i)))) { $c++; if ($c >= 4) { return false; } } else { $c = 0; } } return true; } $allscales = array_filter($allscales, 'doesNotHaveFourConsecutiveOffBits');

    scales remaining: 1490

Now that we've whittled our set of tones to only the ones we'd call a "scale", let's count how many there are with each number of tones.

function countOnBits($scale) { $c = 0; for ($i=0; $i<12; $i++) { if ($scale & (1 << ($i))) { $c++; } } return $c; }
number of tones how many scales
10
20
31
431
5155
6336
7413
8322
9165
1055
1111
121

Heptatonics

The equal temperament system of 12 tones has a special affinity for heptatonic scales. Of the 1490 scales, 413 of them are heptatonic, comprising 28% of the total. The predominant scales used in popular music are heptatonic, including those derived most closely from the circle of fifths.

Some heptatonics are diatonic. To be diatonic means that it includes only notes in the prevailing key; so the set of diatonic scales are very limited to just the major scale and its 6 modes - and only when used in a context where the modes are offset in respect to the tonic key.

Hepatonia prima, secunda, tertia

(coming soon)

Symmetry

There are two kinds of symmetry of interest to scale analysis. They are rotational symmetry and reflective symmetry.

Rotational Symmetry

Rotational symmetry is the symmetry that occurs by transposing a scale up or down by an interval, and observing whether the transposition and the original have an identical pattern of steps.

The set of 12 tones has 5 axes of symmetry. The twelve can be divided by 1, 2, 3, 4, and 6.

Any scale containing this kind of symmetry can reproduce its own tones by transposition, and is also called a "mode of limited transposition" (Messaien)

function symmetries($scale) { $rotateme = $scale; $symmetries = array(); for ($i = 0; $i < 12; $i++) { $rotateme = rotate_bitmask($rotateme); if ($rotateme == $scale) { $symmetries[] = $i; } } return $symmetries; }

Below are all the scales that have rotational symmetry.

axes of symmetryinterval of repetitionscales
1,2,3,4,5,6,7,8,9,10,11 semitone
2,4,6,8,10 whole tone
3,6,9 minor thirds

585 
4,8 major thirds

273 

819 
6 tritones

325 

455 

715 

845 

975 

number of notes in scale Placement of rotational symmetries
1234567891011
300010001000
400100300100
500000000000
6010301003010
700000000000
8002001000200
900030003000
1000000500000
1100000000000
1211111111111

A curious numeric pattern

You'll notice that in the list of scales exhibiting rotational symmetry, many have a decimal representation that ends with the digit 5. A closer look shows that the pattern exists for all scales that have symmetry on axes of 6, 4, and 2 - where all the notes in the scale are in pairs a tritone apart. In the binary schema, each pair of notes that are a tritone apart add up to a multiple of 10, except for the root pair that adds up to 65. Therefore, all scales with rotational symmetry on the tritone will end with the digit 5.


1 + 64
= 65

2 + 128
= 130

4 + 256
= 260

8 + 512
= 520

16 + 1024
= 1040

32 + 2048
= 2080

Messiaen's Modes - and their truncations

The French composer Messiaen studied scales with rotational symmetry, because they are useful in creating sonorities with an ambiguous tonic. He famously named seven such modes, and asserted that they are the only ones that may exist; all other symmetrical scales are truncations of those. We can prove that Messiaen was correct, by putting every symmetrical scale into a chart of their truncation relationships.

In all of Messiaen's modes, the modes are regarded as identical, which makes sense because the whole point was to create ambiguous tonality. Therefore when we speak of one of the Messiaenic modes, we're talking about a whole modal family of scales, like this one which he labeled Mode 7:

Truncation is the process of removing notes from a scale, without affecting its symmetry. The concept is best explained by example. Let's take the mode which Messiaen called the Second Mode:

That scale is symmetrical along the axis of a 3-semitone interval, i.e. by transposing it up a minor third. When we remove tones to create a truncation, we must preserve that symmetry, by removing four notes from the scale, like this:


585 

In addition to the 3-semitone symmetry, that scale is also symmetrical along the axis of a 6-semitone interval. We can create two different truncations of 1755  that preserve that symmetry:

Technically, all of Messiaen's modes are truncated forms of 4095 , the 12-tone scale, which is symmetrical at all intervals.

Hierarchy of truncations

This diagram illustrates that all symmetrical scales are truncations of the Messaienic modes (labeled "M" plus their number as assigned by Messaien). M1, M2, M3, and M7 are all truncations of the chromatic scale 4095; M2, M4, and M6 are truncations of M7; M1 and M5 are truncaions of M6. M1 is also a truncation of M3, and M5 is also a truncation of M4. All the other rotationally symmetrical scales are labeled "T#" and are shown where they belong in the hierarchy of truncations.

Modal FamilyScalesis truncation of
*  
Messiaen's Modes of Limited Transposition
M1 *, M3, M6
M2 *, M7
M3 *
M4 M7
M5 M6, M4
M6 M7
M7 *
Truncations
T1

585 
M2, T5, T4
T2 M3
T3

273 
M1, T2
T4 M2, M6, M4
T5 M2, M6, M4
T6 T4, T5, M5, M1

Going Further

  • In this study we are ignoring sonorities that aren't like a "scale" by disallowing large interval leaps (see the first section). If we ignore those rules, what other truncated sets exist? (spoiler hint: not very many!)
  • Does the naming of certain scales by Messiaen seem haphazard? Why did he name M5, which is merely a truncation of both M6 and M4, which are in turn truncations of M7?

Reflective Symmetry

A scale can be said to have reflective symmetry if it has the same interval pattern whether ascending or descending. A reflectively symmetrical scale, symmetrical on the axis of the root tone, is called a palindromic scale.

function isPalindromic($scale) { for ($i=1; $i<=5; $i++) { if ( (bool)($scale & (1 << $i)) !== (bool)($scale & (1 << (12 - $i))) ) { return false; } } return true; }

Here are all the scales that are palindromic:


273 

337 

433 

497 

585 

681 

745 

793 

857 

953 

Chirality

An object is chiral if it is distinguishable from its mirror image, and can't be transformed into its mirror image by rotation. Chirality is an important concept in knot theory, and also has applications in molecular chemistry.

The concept of chirality is apparent when you consider "handedness" of a shape, such as a glove. A pair of gloves has a left hand and a right hand, and you can't transform a left glove into a right one by flipping it over or rotating it. The glove, therefore, is a chiral object. Conversely, a sock can be worn by the right or left foot; the mirror image of a sock would still be the same sock. Therefore a sock is not a chiral object.

Palindromic scales are achiral. But not all non-palindromic scales are chiral. For example, consider the scales 1105  and 325  (below). One is the mirror image (reflection) of the other. They are not palindromic scales. They are also achiral, because one can transform into the other by rotation.

The major scale is achiral. If you reflect the major scale, you get the phrygian scale; these two scales are modes (rotations) of each other. Therefore neither of them are chiral. Similarly, none of the other ecclesiastical major modes are chiral.

What makes chirality interesting? For one thing chirality indicates that a scale does not have any symmetry along any axis. If you examine all achiral scales, they will all have some axis of reflective symmetry - it's just not necessarily the root tone (which would make it palindromic).

Some achiral scales, and their axes of symmetry
function isChiral($scale) { $reflected = reflect_bitmask($scale); for ($i = 0; $i < 12; $i++) { $reflected = rotate_bitmask($reflected, 1, 1); if ($reflected == $scale) { return false; } } return true; }

A chiral object and its mirror image are called enantiomorphs. (source) function enantiomorph($scale) { $scale = reflect_bitmask($scale); $scale = rotate_bitmask($scale, -1, 1); return $scale; }
ScaleNameChirality / Enantiomorph

273 
Augmented Triadachiral

585 
Diminished Seventhachiral

661 
Major Pentatonicachiral

859 
Ultralocrian
Minor Pentatonicachiral
Blues scale

741 
Whole-toneachiral
Superlocrianachiral
Locrianachiral
Minor Locrianachiral
Major Locrianachiral
Phrygianachiral
Aeolianachiral
Phrygian Dominant
Minor Romani
Lydian Minorachiral
Bebop Locrian
Prometheus (Scriabin)
Locrian natural 6
Dorianachiral
Mixolydianachiral
Minor Bebopachiral
Altered Dorian
Acousticachiral
Hungarian Major

877 
Octatonicachiral
Lydian Pentatonic

355 
Messiaen Mode 5achiral
Major Augmentedachiral
Neapolitan Minor
Harmonic Minor
Double Harmonicachiral
Double Harmonic Minorachiral
Messiaen mode 4achiral
Neapolitan Majorachiral
Melodic Minor ascendingachiral
Majorachiral
Lydianachiral
Aeolian Harmonic

875 
Ionian Augmented
Lydian Augmentedachiral
Diminishedachiral
Bebop Minor
Major Bebopachiral
Messiaen mode 7achiral
Enigmatic
Messiaen mode 6 inverseachiral
Messiaen mode 3 inverseachiral
Dominant Bebopachiral
Chromaticachiral


Going Further

  • Reflective symmetry can occur with an axis on other notes of the scale. Are those interesting?
  • The reflection axis can be on a tone, or between two tones. Is that interesting?
  • Is there a more optimal or elegant way to find the reflective symmetry axes of a scale?
  • Are there chiral enantiomorph pairs that are both named scales?

Combined Symmetry

Using similar methods to those above, we can identify all the scales that are both palindromic and have rotational symmetry. Here they are:


273 

585 

Interval Spectrum / Richness / Interval Vector

Howard Hanson, in the book "Harmonic Materials"4, posits that the character of a sonority is defined by the intervals that are within it. This definition is also called an "Interval Vector"5 or "Interval Class Vector" in Pitch Class Set theory. Furthermore, in his system an interval is equivalent to its inverse, i.e. the sonority of a perfect 5th is the same as the perfect 4th. Hanson categorizes all intervals as being one of six classes, and gives each a letter: p m n s d t, ordered from most consonant (p) to most dissonant (t). When an interval appears more than once in a sonority, it is superscripted with a number, like p2.


P - the Perfects (5 or 7)

This is the interval of a perfect 5th, or perfect 4th.


M - The Major Third (4 or 8)

This is the interval of a major 3rd, or minor 6th


N - The Minor Third (3 or 9)

This is the interval of a minor 3rd, or a major 6th


S - the second (2 or 10)

This is the interval of a major 2nd, or minor 7th


D - the Diminished (1 or 11)

Intervals of a minor 2nd, or a major 7th


T - the Tritone (6 semitones)

For example, in Hanson's analysis, a C major triad has the sonority pmn, because it contains one perfect 5th between the C and G (p), a major third between C and E (m), and a minor third between the E and G (n). Interesting to note that a minor triad of A-C-E is also a pmn sonority, just with the arrangement of the minor and major intervals swapped. The diminished seventh chord 585  has the sonority n4t2 because it contains four different minor thirds, and two tritones.

We can count the appearances of an interval using a method called "cyclic autocorrelation". To find out how many of interval X appear in a scale, we rotate a copy of the scale by X degrees, and then count the number of positions where an "on" bit appears in both the original and copy.

function find_spectrum($scale) { $spectrum = array(); $rotateme = $scale; for ($i=0; $i<6; $i++) { $rotateme = rotate_bitmask($rotateme, $direction = 1, $amount = 1); $spectrum[$i] = countOnBits($scale & $rotateme); } // special rule: if there is a tritone in the sonority, it will show up twice, so we divide by 2 $spectrum[5] = $spectrum[5] / 2; return $spectrum; }

All members of the same modal family will have the same interval spectrum, because the spectrum doesn't care about rotation, and modes are merely rotations of each other. And there will be many examples of different families that have the same interval spectrum - for example, 281  and 275  both have the spectrum "pm3nd", but they are not modes of each other.

Below is a table of some of the named scales, and their interval spectrum. You'll notice that modal families will all have the same spectrum; hence every named mode of the diatonic scale (lydian, dorian, phrygian etc.) has the same spectrum of p6m3n4s5d2t

273 (Augmented Triad) m3
585 (Diminished Seventh) n4t2
661 (Major Pentatonic) p4mn2s3
859 (Ultralocrian) p4m4n5s3d3t2
1193 (Minor Pentatonic) p4mn2s3
1257 (Blues scale) p4m2n3s3d2t
1365 (Whole-tone) m6s6t3
1371 (Superlocrian) p4m4n4s5d2t2
1387 (Locrian) p6m3n4s5d2t
1389 (Minor Locrian) p4m4n4s5d2t2
1397 (Major Locrian) p2m6n2s6d2t3
1451 (Phrygian) p6m3n4s5d2t
1453 (Aeolian) p6m3n4s5d2t
1459 (Phrygian Dominant) p4m4n5s3d3t2
1485 (Minor Romani) p4m5n3s4d3t2
1493 (Lydian Minor) p2m6n2s6d2t3
1499 (Bebop Locrian) p5m5n6s5d4t3
1621 (Prometheus (Scriabin)) p2m4n2s4dt2
1643 (Locrian natural 6) p4m4n5s3d3t2
1709 (Dorian) p6m3n4s5d2t
1717 (Mixolydian) p6m3n4s5d2t
1725 (Minor Bebop) p7m4n5s6d4t2
1741 (Altered Dorian) p4m4n5s3d3t2
1749 (Acoustic) p4m4n4s5d2t2
1753 (Hungarian Major) p3m3n6s3d3t3
1755 (Octatonic) p4m4n8s4d4t4
2257 (Lydian Pentatonic) p3m2nsd2t
2275 (Messiaen Mode 5) p4m2s2d4t3
2457 (Major Augmented) p3m6n3d3
2475 (Neapolitan Minor) p4m5n3s4d3t2
2477 (Harmonic Minor) p4m4n5s3d3t2
2483 (Double Harmonic) p4m5n4s2d4t2
2509 (Double Harmonic Minor) p4m5n4s2d4t2
2535 (Messiaen mode 4) p6m4n4s4d6t4
2731 (Neapolitan Major) p2m6n2s6d2t3
2733 (Melodic Minor ascending) p4m4n4s5d2t2
2741 (Major) p6m3n4s5d2t
2773 (Lydian) p6m3n4s5d2t
2777 (Aeolian Harmonic) p4m4n5s3d3t2
2869 (Ionian Augmented) p4m4n5s3d3t2
2901 (Lydian Augmented) p4m4n4s5d2t2
2925 (Diminished) p4m4n8s4d4t4
2989 (Bebop Minor) p5m5n6s5d4t3
2997 (Major Bebop) p6m5n6s5d4t2
3055 (Messiaen mode 7) p8m8n8s8d8t5
3411 (Enigmatic) p4m4n3s5d3t2
3445 (Messiaen mode 6 inverse) p4m6n4s6d4t4
3549 (Messiaen mode 3 inverse) p6m9n6s6d6t3
3765 (Dominant Bebop) p7m4n5s6d4t2
4095 (Chromatic) p12m12n12s12d12t6

Going Further

  • Is there an optimal or elegant way to find all scales with a given spectrum?
  • What patterns appear in interval distribution?
  • Which are the most common, and least common spectra?

Evenness

Another interesting property of a scale is whether the notes are evenly spaced, or clumped together. The theory of musical scale evenness owes to "Diatonic Set Theory", the work of Richard Krantz and Jack Douhett6. In their paper, they explain how you can determine the "evenness" of a scale, first by establishing the intervals between each note and every other.

Generic interval is 2, Specific interval is 5
A crucial concept to understand in Diatonic Set Theory is the distinction between a generic interval and a specific interval. A specific interval is the number of semitones between two tones; for example between a C and a E, that's a specific interval of 4 semitones. The generic interval is the number of scale steps between two tones of a scale; for example in C major scale, the distance between C and E is 2.

To measure the evenness of the scale, the first step is to build the distribution spectra. The spectra shows the distinct specific intervals between notes, for each generic interval of the scale. Each spectrum is notated like this:

<generic interval> = { specific interval, specific interval, ...}

The number in angle brackets is the generic interval, ie we are asking "for notes that are this many steps away in the scale". The numbers in curly brackets are the specific intervals we find present for those steps, ie "between those steps we find notes that are this many semitones apart".

It's best explained with an example. Below is the scale bracelet diagram and distribution spectra for Scale 1449:

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

In line 1, the first spectrum, <1> indicates that we are looking at notes that are one scale step away from each other. We have notes that are one semitone apart (eg G and G#), two semitones (D# and F), and three semitones (C and D#). Duplicates of these are ignored; we merely want to know what intervals are present, not how many of them exist.

In line 2, the second spectrum, <2> indicates that we are looking at notes that are two scale steps away from each other. We see pairs that are three semitones apart (eg F and G#), four semitones (D# and G), and five semitones (C and F).

When there is more than one specific interval, the spectrum width is the difference between the largest and smallest value. For example for the <3> spectrum above, the specific intervals are {5,7} and so its width is 2, which is 7 minus 5.

The spectrum variation is the average of all those widths.

Once the distribution specta are built, we analyze them to discover interesting properties of the scale. For instance,

  • If all the spectra have just one specific interval, then the scale has exactly equal distribution
  • If the spectra have two intervals with a difference no greater than one, then the scale is maximally even - it's distributed as evenly as it can be with no room for improvement.
  • If the spectra has any widths greater than 1, then it's not maximally even.
  • If there are exactly two specific intervals in all the spectra, then the scale is said to have Myhill's property.

Ultimately, the measure of a scale's evenness is its Spectra Variation. We add up all the spectrum widths, and divide by the number of tones in the scale, to achieve an average width with respect to the scale size. If a scale has perfectly spaced notes with completely uniform evenness, then it has a spectra variation of zero. A higher variation means the scale distribution is less even.

The following four scales have a perfect score - a spectra variation of zero:


273 

585 

Obviously, it is possible to evenly distribute 6 tones around a 12-tone scale. But it is impossible to do that with a 5 tone (pentatonic) or 7 tone (heptatonic) scale. For such tone counts all we can hope to achieve is an optimally even distribution.

Below are all the prime scales (ie with rotations omitted), sorted from most even to least even. If you click to each scale detail page, you can read its spectra variation there.

3 tones


273 

4 tones


585 

325 

293 

297 

277 

291 

275 

5 tones


661 

597 

595 

341 

587 

403 

339 

333 

355 

331 

307 

327 

309 

301 

299 

283 

313 

295 

279 

6 tones


819 

715 

723 

717 

693 

691 

685 

683 

427 

679 

667 

619 

455 

663 

615 

423 

363 

603 

411 

347 

359 

365 

407 

605 

599 

371 

591 

349 

399 

343 

335 

315 

311 

303 

317 

287 

7 tones


859 

871 

855 

731 

823 

747 

743 

727 

699 

733 

719 

695 

471 

443 

439 

631 

701 

755 

463 

687 

375 

431 

671 

623 

607 

415 

379 

367 

381 

351 

319 

8 tones


975 

951 

943 

887 

879 

891 

927 

759 

751 

863 

495 

735 

831 

763 

765 

703 

479 

639 

447 

383 

9 tones


991 

959 

895 

767 

511 

10 tones

11 tones

12 tones

Myhill's Property

Myhill's Property is the quality of a pitch class set where the spectrum has exactly two specific intervals for every generic interval. There are 47 scales with Myhill's property:


341 

511 

661 

677 

Propriety / Coherence

A scale is said to have "Rothenberg propriety", a quality named after David Rothenberg7, if the scale has unambiguous relationship between generic intervals (scale degrees) and specific intervals. The same concept was discovered by Gerald Balzano, who named it coherence.

To be proper (aka coherent), every two-step generic interval must be specifically larger than every one-step generic interval, every three-step generic interval must be specifically larger than every two-step, and so on. This means that when hearing any interval from within a scale, you can unambiguously know what generic distance it is within the scale.

You can see the coherence of a scale by inspecting its distribution spectra. Look at this scale:

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

You can see that if you hear an interval of 2 semitones, that is unambiguously a one-step generic distance. We know this because the specific distance of 2 appears in the spectra only in the 1-step spectrum.

Contrarily, if we know that our scale is 1449 rooted on C, and we hear an interval of 5 semitones (a perfect fourth), this could be a generic distance of a two steps (between C and F), or it might be a generic distance of three steps (between D# and G#). That ambiguity means this scale is not coherent.

There are exactly 31 coherent scales. Here they are:


273 

291 

293 

297 

325 

329 

393 

549 

553 

561 

581 

585 

593 

649 

657 

661 

677 

819 

Deep Scales

A "deep" scale is one for which the evenness distribution spectrum of specific intervals consists of unique values. There are only 13 deep scales. Here they are:


693 

Hemitonia and Tritonia

One interesting approach to understanding scales is to look at the distribution of consonant and dissonant intervals in the scale. We did this already in the "spectrum analysis" above, but two intervals in particular are commonly regarded in more depth: tritones and semitones - which are deemed the most dissonant of intervals, and those that give a scale its spice and flavour.

A scale that contains semitones is called "hemitonic", and we will also refer to a hemitonic as the scale member that has the upper semitone neighbour (for example, mi-fa and ti-do in a major scale, the hemitones are mi and fa).

A scale that contains tritones is called "tritonic", and we will also refer to a tritonic as the scale member that has a tritone above it. This concept seems very close to the notion of "imperfection" (explained below)

Since hemitonia and tritonia are based on the interval spectrum, all the modes of a scale will have the same hemitonia. For example, the major scale is dihemitonic and ancohemitonic; thus so are dorian, phrygian, lydian, etc.

Number of tones# of Hemitonic Scales# of Tritonic Scales
300
41224
5140150
6335335
7413413
8322322
9165165
105555
111111
1211

It's well to know that so many scales are hemitonic and tritonic (having more than zero semitones or tritones), but obviously some scales are more hemitonic than others. A scale could have no hemitonics ("ahemitonic"), one ("unihemitonic"), two ("dihemitonic"), three ("trihemitonic") and so on. Let's look at the count of hemitones in our scales.

Number of hemitones found in all scales
Number of hemitones
tones in scale 0123456789101112
31000000000000
4191200000000000
51580600000000000
61301501401500000000
70021140210420000000
800007016884000000
9000000847290000
10000000004510000
1100000000001100
120000000000001

Fun fact: there are no scales with 11 hemitones. Do you understand why?

Number of tritones found in all scales
Number of tritones
tones in scale 0123456789101112
31000000000000
471680000000000
55407530500000000
61121021466960000000
70014112196847000000
800006216884800000
9000000847290000
10000000004510000
1100000000001100
120000000000001

Cohemitonia

Cohemitonia is the presense of two semitones consecutively in scale order. For instance, if a scale has three cromatically consecutive notes, then it is cohemitonic. We will also refer to the cehemitonic as the tone that has two semitone neighbours above it.

Heliotonia

A scale is heliotonic if it is heptatonic (has seven tones), and if it can be represented on a staff with a note head on each line and space, using no more than a full-tone alteration accidental (sharp, flat, double-sharp, double-flat).

Since this property is affected by the uneven interval distance between what we call "the white notes" (ie steps that are not altered with an accidental), heliotonia is dependent on the root. A scale that is heliotonic in C might not be when the root is F#.

Of the 413 heptatonic scales, 363 of them are heliotonic when the root is C.

At present, insufficient research has been done to correlate heliotonia with other properties such as evenness, cohemitonia, etc. It has also not been sufficiently investigated how the stats of heliotonia change for each root tone. As such, heliotonia is a subject with opportunities for further exploration.

Here are all the scales that are heliotonic, when C is the root:


687 

695 

699 

701 

719 

727 

731 

733 

743 

747 

749 

755 

757 

761 

815 

823 

827 

829 

847 

855 

859 

861 

871 

875 

877 

883 

885 

889 

911 

919 

923 

925 

935 

939 

941 

947 

949 

953 

967 

971 

973 

979 

981 

985 

995 

997