A Study of Scales

Where we discuss every possible combination of notes

This exploration of scales is based on work by William Zeitler, as published at http://allthescales.org/. In fact much of the material on this page merely repeats Zeitler's findings, presented here along with PHP code I used to generate the scales. This exploration also owes a debt to Andrew Duncan's work on combinatorial music theory. Most of the code in this treatise is the result of continuous exploration and development of PHPMusicTools, an open-source project at GitHub.

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.

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; } $howmany = array_fill(1, 12, 0); foreach ($allscales as $scale) { $howmany[countOnBits($scale)]++; }
number of tones how many scales
10
20
31
431
5155
6336
7413
8322
9165
1055
1111
121

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
4,8 major thirds
6 tritones

number of notes in scale Placement of rotational symmetries
1234567891011
100000000000
200000000000
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:

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

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 M2, T5, T4
T2 M3
T3 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 pattern of steps 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:

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; }
Named ScaleChirality / Enantiomorphs
273 (augmented triad)achiral
585 (diminished seventh)achiral
1123 (iwato)11232245
1186 (insen)11862212
1365 (whole tone, second mode of limited transposition)achiral
1371 (altered)achiral
1387 (locrian)achiral
1389 (half diminished)achiral
1451 (phrygian)achiral
1453 (aeolian, natural minor)achiral
1459 (phrygian dominant, spanish romani)14592485
1485 (aolian #4, romani scale)14851653
1709 (dorian)achiral
1717 (mixolydian)achiral
1741 (ukranian dorian, romanian scale, altered dorian)17411645
1749 (acoustic, lydian dominant)achiral
2257 (hirajoshi)2257355
2733 (heptatonia seconda, ascending melodic minor, jazz minor)achiral
2741 (major, ionian)achiral
2773 (lydian)achiral
2483 (double harmonic)achiral
2457 (augmented)achiral
2477 (harmonic minor)24771715
2509 (hungarian minor)achiral
2901 (lydian augmented)achiral
2731 (major neapolitan)achiral
2475 (minor neapolitan)24752739
3669 (prometheus)36691359
1235 (tritone scale)12352405
1755 (octatonic, second mode of limited transposition)achiral
3549 (third mode of limited transposition)achiral
2535 (fourth mode of limited transposition)achiral
2275 (fifth mode of limited transposition)achiral
3445 (sixth mode of limited transposition)achiral
3055 (seventh mode of limited transposition)achiral
3765 (bebop dominant)achiral
4095 (chromatic 12-tone)achiral


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:

Imperfection

Imperfection is a concept invented (so far as I can tell) by William Zeitler, to describe the presence or absense of perfect fifths in the scale tones. Any tone in the scale that does not have the perfect fifth above it represented in the scale is an "imperfect" tone. The number of imperfections is a metric that plausibly correlates with the perception of dissonance in a sonority.

The only scale that has no imperfections is the 12-tone chromatic scale.

This table differs from the one at allthescales.org, because this script does not de-duplicate modes. If an imperfection exists in a scale, it will also exist in all the sibling modes of that scale. Hence the single imperfect tone in the 11-tone scale is found 11 times (in 11 scales that are all modally related), whereas Zeitler only counts it as one.

number of notes in scale # of Imperfections
0123456
10000000
20000000
30001000
400816700
50530754050
60669146102121
70784196112140
808841686200
9097284000
10010450000
1101100000
121000000

Going Further

  • Looking at the table, there are 5 scales of 5 tones with one imperfection. There are 6 scales of 6 tones with only one imperfection. The same patten continues right up to 11 tones. What causes this pattern?
  • The 7-note scale with only one imperfection is the major diatonic scale (plus all its modes). Is it a coincidence that we find this scale sonically pleasing?

Interval Spectrum / Richness

Howard Hanson, in the book "Harmonic Materials", posits that the character of a sonority is defined by the intervals that are within it. 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. He 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.

Below is a table of all 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
1123 (iwato)p3m2nsd2t
1186 (insen)pmn2st
1365 (whole tone, second mode of limited transposition)m6s6t3
1371 (altered)p4m4n4s5d2t2
1387 (locrian)p6m3n4s5d2t
1389 (half diminished)p4m4n4s5d2t2
1451 (phrygian)p6m3n4s5d2t
1453 (aeolian, natural minor)p6m3n4s5d2t
1459 (phrygian dominant, spanish romani)p4m4n5s3d3t2
1485 (aolian #4, romani scale)p4m5n3s4d3t2
1709 (dorian)p6m3n4s5d2t
1717 (mixolydian)p6m3n4s5d2t
1741 (ukranian dorian, romanian scale, altered dorian)p4m4n5s3d3t2
1749 (acoustic, lydian dominant)p4m4n4s5d2t2
2257 (hirajoshi)p3m2nsd2t
2733 (heptatonia seconda, ascending melodic minor, jazz minor)p4m4n4s5d2t2
2741 (major, ionian)p6m3n4s5d2t
2773 (lydian)p6m3n4s5d2t
2483 (double harmonic)p4m5n4s2d4t2
2457 (augmented)p3m6n3d3
2477 (harmonic minor)p4m4n5s3d3t2
2509 (hungarian minor)p4m5n4s2d4t2
2901 (lydian augmented)p4m4n4s5d2t2
2731 (major neapolitan)p2m6n2s6d2t3
2475 (minor neapolitan)p4m5n3s4d3t2
3669 (prometheus)p4m4n3s5d3t2
1235 (tritone scale)p2m2n4s2d2t3
1755 (octatonic, second mode of limited transposition)p4m4n8s4d4t4
3549 (third mode of limited transposition)p6m9n6s6d6t3
2535 (fourth mode of limited transposition)p6m4n4s4d6t4
2275 (fifth mode of limited transposition)p4m2s2d4t3
3445 (sixth mode of limited transposition)p4m6n4s6d4t4
3055 (seventh mode of limited transposition)p8m8n8s8d8t5
3765 (bebop dominant)p7m4n5s6d4t2
4095 (chromatic 12-tone)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?

Proximity

We would consider two scales to be "near" each other, if they bear many similarities; sharing many of the same notes, such that it would take few transformations to turn one into the other.

This distance measured by transformation is almost the same thing as a "Levenshtein Distance". The normal definition of a Levenshtein Distance on a string of characters allows three kinds of transformation: insertion, deletion, and substitution. Our scale transformations are different from a string transformation, in that scales can't do substitution, instead we move tones up or down with rules for handling collision. This difference means we can't employ all the elegant formulas bequeathed to us by Vladimir Levenshtein.

We can transform a scale in three ways:

  • Move a tone up or down by a semitone
  • Remove a tone
  • Add a tone

It is simple to generate all the scales at a distance of 1, just by performing all possible transformations to every interval above the root.

function findNearby($scale) { $near = array(); for ($i=1; $i<12; $i++) { if ($scale & (1 << ($i))) { $copy = $scale; $off = $copy ^ 1 << ($i); $near[] = $off; $copy = $off | 1 << ($i - 1); $near[] = $copy; $copy = $off | 1 << ($i + 1); $near[] = $copy; } else { $copy = $scale; $copy = $copy | 1 << ($i); $near[] = $copy; } } return $near; }

Example

Here are all the scales that are a distance of 1 from the major scale, aka 2741, shown here as a simple C major scale:

add a tone at C#2743
lower the D to C#2739
raise the D to D#2745
delete the D2737
add a tone at D#2749
lower the E to D#2733 (heptatonia seconda, ascending melodic minor, jazz minor)
raise the E to Fsame as deleting E
delete the E2725
lower the F to Esame as deleting F
raise the F to F#2773 (lydian)
delete the F2709
add a tone at F#2805
lower the G to F#2677
raise the G to G#2869
delete the G2613
add a tone at G#2997
lower the A to G#2485
raise the A to A#3253
delete the A2229
add a tone at A#3765 (bebop dominant)
lower the B to A#1717 (mixolydian)
raise the B to Csame as deleting B
delete the B693

Evidently since every scale will have a collection of neighbours similar to this one, a complete graph of scale proximity is large, but not unfathomably so.

We can also easily calculate the Levenshtein distance between two scales. Apply an XOR operation to expose changed bits between two scales. Adjacent on bits in the XOR where the bits were different in the original scale identify a "move" operation. Any bits remaining are either an addition or deletion.

function levenshtein_scale($scale1, $scale2) { $distance = 0; $d = $scale1 ^ $scale2; for ($i=0; $i<12; $i++) { if ( ($d & (1 << ($i))) && ($d &(1 << ($i+1))) && ($scale1 & (1 << ($i))) != ($scale1 & (1 << ($i+1))) ) { $distance++; $d = $d & ( ~ (1 << ($i))); $d = $d & ( ~ (1 << ($i+1))); } } for ($i=0; $i<12; $i++) { if (($d & (1 << ($i)))) { $distance++; } } return $distance; }

Modes

There is a lot of confusion about what is a "mode", chiefly because the word is used slightly differently in various contexts.

When we say "C major", the word "major" refers to a specific pattern of whole- and half-steps. The "C" tells us to begin that pattern on the root tone of "C".

Modes are created when you use the same patterns of whole- and half-steps, but you begin on a different step. For instance, the "D Dorian" mode uses all the same notes as C major (the white keys on a piano), but it begins with D. Compared with the Major (also known as "Ionian" mode), the Dorian sounds different, because relative to the root note D, it has a minor third and a minor seventh.

The best way to understand modes is to think of a toy piano where the black keys are just painted on - all you have are the white keys: C D E F G A B. Can you play a song that sounds like it's in a minor key? You can't play a song in C minor, because that would require three flats. So instead you play the song with A as the root (A B C D E F G). That scale is a mode of the major scale, called the Aeolian Mode.

When you play that minor scale, you're not playing "C minor", you're playing the relative minor of C, which is "A minor". Modes are relatives of each other if they have the same pattern of steps, starting on different steps.

To compute a mode of the current scale, we "rotate" all the notes down one semitone. then if the rotated notes have an on bit in the root, then it is a mode of the original scale.

			101010110101 = 2741 - major scale, "ionian" mode
			110101011010 = 3418 - rotated down 1 semitone - not a scale
			011010101101 = 1709 - rotated down 2 semitones - "dorian"
			101101010110 = 2902 - rotated down 3 semitones - not a scale
			010110101011 = 1451 - rotated down 4 semitones - "phrygian"
			101011010101 = 2773 - rotated down 5 semitones - "lydian"
			110101101010 = 3434 - rotated down 6 semitones - not a scale
			011010110101 = 1717 - rotated down 7 semitones - "mixolydian"
			101101011010 = 2906 - rotated down 8 semitones - not a scale
			010110101101 = 1453 - rotated down 9 semitones - "aeolian"
			101011010110 = 2774 - rotated down 10 semitones - not a scale
			010101101011 = 1387 - rotated down 11 semitones - "locrian"
			
function modes($scale) { $rotateme = $scale; $modes = array(); for ($i = 0; $i < 12; $i++) { $rotateme = rotate_bitmask($rotateme); if (($rotateme & 1) == 0) { continue; } $modes[] = $rotateme; } return $modes; } function rotate_bitmask($bits, $direction = 1, $amount = 1) { for ($i = 0; $i < $amount; $i++) { if ($direction == 1) { $firstbit = $bits & 1; $bits = $bits >> 1; $bits = $bits | ($firstbit << 11); } else { $firstbit = $bits & (1 << 11); $bits = $bits << 1; $bits = $bits & ~(1 << 12); $bits = $bits | ($firstbit >> 11); } } return $bits; }

When we do this to every scale, we see modal relationships between scales, and we also discover symmetries when a scale is a mode of itself on another degree.

Modal Siblings

We can present the list of all modes grouped in sets of modal siblings

Number of tones in the scale Number of modal families
10
20
31
49
531
659
759
842
919
106
111
121

A complete list of all modal families

Modal families with 3 tones

m3273 (augmented triad)

Modal families with 4 tones (tetratonic)

pm3nd2814015472321
p2m2nd2913935612193
pm3nd2753057852185
m3s2t27733710931297
pmn2st3295535811169
p2mn2s2975496571161
m2s2t23251105
pmn2st2935936491097
n4t2585 (diminished seventh)

Modal families with 5 tones (pentatonic)

pm3ns2d2t285465109525953345
p2m2n2sd2t45756958323393217
p2m3n2sd231355191323233209
p2m2s2d2t2327453113722113153
p2m2n2sd2t29562590521953145
pm3ns2d2t279369180921873141
p2m3n2sd2283433157121892833
p3m2n2s2d425565116513152705
pm3n2s2dt309849110112992697
p3mn2s2dt421653112911872641
pmn4sdt258961784111712633
p2mn3s2dt361557116316812629
p3m2nsd2t3974191123 (iwato)2257 (hirajoshi)2609
pm2n2s2dt2333837110712332601
p2mn3s2dt301721109916732597
p2m4n2d240956380323292449
p2m4n2d230778781722012441
p2m3ns2dt405675112513052385
p2m2n3sdt61365980911772377
pm2n2s2dt2357651111316172373
p2m3n2d2t40361179322492353
p2m2n3sdt59566580512252345
pmn4sdt258760171316092341
pm3n2s2dt345555142516052325
p3m2nsd2t355395158522252245
p2m3ns2dt339789122113292217
p3mn2s2dt331709120115772213
p3m2n2s2d299689141715732197
m4s4t23411109130113491361
p2m2n2s3t597681117313171353
p4mn2s3661677118911931321

Modal families with 6 tones (hexatonic)

p2m3n2s3d4t2874972191314336193857
p3m2n3s3d3t4895731167263133633729
p2m3n3s3d3t3179771103259933473721
p3m2n2s3d3t24856551145237532353665
p2m2n4s2d3t2591633969234332193657
p2m3n3s3d3t3775591937232732113653
p4m2ns2d4t23994832247228931713633
p3m2n2s3d3t23359651265221531553625
p3m2n3s3d3t3037531929219931473621
p2m4n3s2d3t4735711607233328513473
p3m4n3s2d33159451575220528353465
p2m4ns4d2t246911411309135127233409
p3m3n3s3d2t6299371181131927073401
pm4n2s4d2t237311171303187326993397
p3m3n2s2d3t24677971223228126593377
p4m2n3s3d2t6699331191125726433369
p2m2n4s3d2t26057451175186526353365
p3m4n2s2d3t4139311127251326113353
pm4n2s4d2t234911111489186126033349
p3m3n2s2d3t24618391139246726173281
p3m4n3sd3t627807921236124513273
p3m4n2s2d3t3717911841223324433269
p4m2n2s2d3t24597111593227724033249
p4m3n2s3d2t6799171253133723873241
p4m2n3s3d2t6637411209183323793237
p3m4n3sd3t615825915235525053225
p3m3n3s3d2t5996971481182923473221
p3m4n3s2d34415671827233129613213
p4m2s2d4t34552275 (fifth mode of limited transposition)3185
p4m2n2s2d3t24239091251225926733177
p3m3n2s2d3t24077391817225124173173
p3m3n2s2d3t23599071649222725013161
p2m4ns4d2t234313931477181322193157
p2m4n3s2d3t3118811811220329533149
p3m3n3s3d2t43711331307169927012897
p2m2n5s2d2t26218731179168326372889
p2m2n4s3d2t236511151675174526052885
p3m4n3sd3t4357951635226524452865
p2m3n4s2d2t26678691241161923812857
p2m2n5s2d2t26037291611173723492853
p3m4n3sd3t4118671587225324812841
p3m3n3s3d2t34714571579173322212837
p4m2n3s3d2t42911311443167726132769
p2m3n4s2d2t26198571427161323572761
p4m2n3s3d2t36314191581171322292757
p4m3n2s3d2t42713791421158922612737
p2m4n2s4dt285312371333135713632729
p4m2n3s4dt72512051325135517052725
p3m4n3s2d2t8218511229133124732713
p5m2n3s4d69311971323144917012709
p2m2n4s2d2t38451235 (tritone scale)2665
p3m2n4s2d2t27238131227168924092661
p3m2n4s2d2t27178431203164124692649
p4m2n3s4dt68511951385144516852645
p3m6n3d38192457 (augmented)
p3m4n3s2d2t6918111433163723932453
p2m2n4s2d2t371516252405
p2m4n2s4dt268313691381142916212389
m6s6t31365 (whole tone, second mode of limited transposition)

Modal families with 7 tones (heptatonic)

p3m4n4s4d5t50557523353215365538753985
p3m4n4s4d5t319100922073151362338593977
p3m4n3s5d4t2501114913112703339937473921
p3m3n5s4d4t2637100111832639336737313913
p2m4n4s5d4t2381111920012607335137233909
p4m4n3s3d5t249979922972447327136833889
p4m3n4s4d4t267199712732383323936673881
p3m3n5s4d4t260776119932351322336593877
p4m4n3s3d5t241599522552545317536353865
p3m4n3s5d4t2351152119892223315936273861
p4m3n4s4d4t2493114716792621288734913793
p3m4n5s3d4t263598516152365285534753785
p4m4n4s4d4t379158319692237283934673781
p5m3n3s4d4t2491142315972293275934273761
p4m4n3s5d3t2981126913411359272734113753
p5m3n4s5d3t757121313271961271134033749
p4m4n4s3d4t282997912312537266333793737
p5m3n4s5d3t701119915131957264733713733
p4m4n4s4d4t445113519552615302533553725
p5m3n2s3d5t348791122912503319332993697
p4m3n4s3d4t384797312672471268132833689
p4m4n4s3d4t275581519452425245532753685
p4m3n4s3d4t371997116572407253332513673
p4m4n3s5d3t2687140115091941239132433669 (prometheus)
p3m4n5s3d4t262388919392359301732273661
p5m3n2s3d5t346396722792531318733133641
p5m3n3s4d4t2431150719332263280131793637
p4m3n4s4d4t2367177719312231301331633629
p3m5n3s4d4t2477114318632619297933573537
p4m5n4s3d4t63195318312363296332293529
p3m5n3s4d4t2375181519052235295531653525
p4m4n4s3d4t2475159517352285284529153505
p5m4n4s4d3t949126113391703271728993497
p4m3n5s4d3t2749121116871897265328913493
p4m6n4s2d4t82794716392461252128673481
p3m5n4s4d3t2699149716231893239728593477
p4m5n4s3d4t443159118912269284329933469
p4m4n2s4d4t3471147918212283278731893441
p5m3n4s4d3t2941125914471693267727713433
p3m4n4s4d3t3747143116291881242127633429
p4m5n3s4d3t2939138314371653251727393417
p2m6n2s6d2t3136713731397149318772731 (major neapolitan)3413
p3m5n4s4d3t2885124513351875271529853405
p5m4n3s3d4t292593512552515267533053385
p3m4n4s4d3t3861123914911869266727933381
p4m3n5s4d3t2733120717691867265129813373
p5m4n3s3d4t274391918492419250732573301
p4m5n4s2d4t287192316512483 (double harmonic)2509 (hungarian minor)28733289
p4m5n3s4d3t285513951485 (aolian #4, romani scale)18452475 (minor neapolitan)27453285
p4m6n4s2d4t82388318432459248929693277
p5m3n4s4d3t2727148317211837241127893253
p5m4n4s4d3t695146517651835239529653245
p4m4n4s3d4t2439176318192267292929573181
p3m3n6s3d3t3877124316911747266928932921
p3m3n6s3d3t3731162717391753241328612917
p4m4n5s3d3t2875143516451715248527652905
p4m4n4s5d2t21371 (altered)1389 (half diminished)146117071749 (acoustic, lydian dominant)2733 (heptatonia seconda, ascending melodic minor, jazz minor)2901 (lydian augmented)
p4m4n5s3d3t28591459 (phrygian dominant, spanish romani)16431741 (ukranian dorian, romanian scale, altered dorian)2477 (harmonic minor)27772869
p6m3n4s5d2t1387 (locrian)1451 (phrygian)1453 (aeolian, natural minor)1709 (dorian)1717 (mixolydian)2741 (major, ionian)2773 (lydian)

Modal families with 8 tones (octatonic)

p4m5n5s6d6t25091151262333593727391140034049
p4m5n6s5d6t26391017236732313663387939874041
p4m5n5s6d6t23832033223931673631386339794037
p5m5n5s5d6t25071599230128473471378339394017
p5m5n5s6d5t210131277134327193407375139234009
p5m4n6s6d5t27651215202526553375373539154005
p5m6n5s4d6t28311011246325533279368738913993
p5m5n5s6d5t27031529202123993247367138833989
p5m5n5s5d6t24472019227130573183363938673981
p5m5n4s5d6t35031823229929593197352738113953
p5m4n6s5d5t310051275169526852895349537953945
p4m5n6s5d5t37631631200924292863347937873941
p5m5n5s5d5t310031439166125492767343137633929
p4m6n4s7d4t313751405152520052735341537553925
p4m5n6s5d5t38931247200326713049338337393917
p6m5n4s4d6t3927999251125473303332136993897
p5m5n5s5d5t38631523199724792809328736913893
p5m4n6s5d5t37351785199524153045325536753885
p5m5n4s5d6t34791991228730433191356936433869
p6m4n4s5d6t34951935229530153195355536453825
p5m5n5s5d5t39891271187126832983338935393817
p6m5n5s5d5t27591839197724272967326135313813
p5m5n6s4d5t39871659174325412877291935073801
p6m5n5s6d4t214031517171119732749290334993797
p5m6n6s4d5t28911647197124932871303334833789
p6m5n4s5d5t39831487185325392791331734433769
p7m4n5s6d4t214551515172519652775280534353765 (bebop dominant)
p6m5n5s6d4t213911469178119632743302934193757
p6m5n5s5d5t29571263195926793027338735613741
p6m4n4s4d6t49752535 (fourth mode of limited transposition)33153705
p6m5n4s5d5t39431511194925192803330734493701
p5m5n6s4d5t38791779194724872937302132913693
p5m5n5s5d5t37511913194324233019325935573677
p5m7n5s4d5t29551655189525252875299534853545
p4m7n4s6d4t313991501187919092747298734213541
p5m7n5s4d5t28871847190724912971300132933533
p6m6n5s4d5t29511767185125232931297333093513
p5m5n6s5d4t314991723175119012797290929233509
p6m5n6s5d4t214671719177318992781290729973501
p4m6n4s6d4t41495188527953445 (sixth mode of limited transposition)
p5m5n6s5d4t314631757177118832779293329893437
p4m4n8s4d4t41755 (octatonic, second mode of limited transposition)2925

Modal families with 9 tones

p6m6n6s7d8t351123033199364738713983403940674081
p6m6n7s7d7t3102112792687339137433919400740514073
p6m6n7s7d7t376720412431326336793887399140434069
p6m7n7s6d7t3101916632557287934873791394340194057
p6m7n6s8d6t3140715332037275134233759392740114053
p6m7n7s6d7t389520352495306532953695389539954045
p7m7n6s6d7t3101518552555297533253535381539554025
p7m6n7s7d6t3153117272029281329113503379939474021
p7m6n7s7d6t3147117892027278330613439376739314013
p7m7n6s6d7t395920232527305933113577370338993997
p7m6n6s6d7t4100719512551302333233559370938273961
p6m7n6s7d6t4152718872013281129913453354338193957
p6m6n8s6d6t4175917872011292729413053351138033949
p6m7n6s7d6t4150319172007279930513447357337713933
p7m6n6s6d7t499119992543304733193571370738333901
p8m6n6s7d6t3151919671981280730313451356337733829
p7m7n7s6d6t3178319031979293929993037351735473821
p7m7n7s6d6t3177519151975293530053035351535653805
p6m9n6s6d6t3191130033549 (third mode of limited transposition)

Modal families with 10 tones

p8m8n8s8d9t41023255933273711390339994047407140834089
p8m8n8s9d8t41535204528153455377539354015405540754085
p8m8n9s8d8t41791204329433069351938073951402340594077
p8m9n8s8d8t41919203930073067355135813823395940274061
p9m8n8s8d8t41983203130393063356735793831383739634029
p8m8n8s8d8t520153055 (seventh mode of limited transposition)357538353965

Modal families with 11 tones

p10m10n10s10d10t520473071358338393967403140634079408740914093

Modal families with 12 tones

p12m12n12s12d12t64095 (chromatic 12-tone)