This page is about finding every possible mode of limited transposition (MOLT) in arbitrary equal temperatements. It also contains a list of every mode of limited transposition for some equal temperaments. Prime numbered tone scales are not included because they have no MOLTs; rather, they are the basis for MOLTs in composite numbered tone scales. Through interpolation of these prime tone scales all modes of limited transposition in all tone scales are derived. For a more detailed explanation, see here. For information about the history and other properties about these scales, see the wikipedia article on them.
Because many of the scales listed cannot be expressed in the 12 tone scale, the lists numbers the scales from $0$ to $n-1$ for $n$-tone equal tempered scales. When it is possible to represent the scales as notes of the 12 tone scale, the lists start with C. When it is possible to translate the scale to the 24 scale this is done, again starting with C.
Note: my use of "scale" means simply a set of pitch classes which includes at least the pitch class of the key. The term "interpolation" is used as in Nicolas Slonimsky's Thesaurus of Scales and Melodic Patterns, and is described below.
MOLT list:The concept of a mode of limited transposition (MOLT) is usually defined in two ways; a scale with a limited number of transpositions before the scale's notes repeat themselves, or a scale which has less modes than notes in the scale. These two explanations describe the same set of scales. Although some of the scales were used before him, Messiaen defined MOLTs originally, and his definition also did not allow MOLTs which were truncations of other MOLTs. Although he also listed all the possible MOLTs, his list contradicted this rule; only two MOLTs (both of which were included in his list– his 3rd and 7th MOLTs) are not truncations of other MOLTs. For the sake of simplicity, we consider truncations to be MOLTs too.
Finding MOLTs for the 12 tone equal tempered scale is somewhat simple. First, we take the prime factors of 12, because 12 is the number of tones in our scale. These prime factors are 2 and 3. We then divide up our scale by those intervals and get the tritone {0, 6} and the augmented chord {0, 4, 8}. We call these prime scales.
Any interpolation of either of these prime scales is a MOLT. If we continue to interpolate both of these scales we will find every MOLT; that is to say, there are no MOLTs which cannot be discovered through interpolation of these notes.
Interpolation is a process for deriving new scales from another, smaller scale. To interpolate, start with a scale, e.g. {0, 4, 8}, and add a note either below or above each note. An interpolation might be {0, 1, 4, 5, 8, 9} (bolded notes are those which are part of the original scale); we might call this an interpolation of one semitone. An interpolation of one whole tone would be {0, 2, 4, 6, 8, 10}; this is the whole tone scale. We can interpolate a scale more than once. However, when we interpolate the second time, we do not interpolate with the already interpolated notes. For example, if we were to interpolate the tritone, {0, 6}, twice, we would first interpolate it by the first interval (e.g., a semitone, the result being {0, 1, 6, 7}), and then by the second (e.g., a minor third, the final result being {0, 1, 3, 6, 7, 9}). Thus, the initial intervals in the first scale form a tree of related scales through interpolation.
A disadvantage of using this method is that there will be duplicate scales derived from both prime factors. For example, we can find the whole tone scale by interpolating the augmented chord once by a whole tone, or by interpolating the tritone twice by a whole tone and a major third. MOLTs which can be found by either prime scale have a number of notes which is divisible by both prime factors of 12.
This process works equally well for all equal tempered scales. For example, finding the MOLTs of the 10 tone scale involves first prime factorizing 10 (2 and 5, which in turn yield the prime scales {0, 5}, the tritone, and {0, 2, 4, 6, 8}). After this, we interpolate until we find all the scales (there are 17 in all, if we include the PSSes and the "chromatic" 10 tone scale). Because both 12 and 10 have 2 as a prime factor, the scales share a PSS, the tritone.
The relationship between PSSes and equal tempered scales is also interesting. The 3 tone scale, for example, is merely the 3 PSS in the 12 tone scale, i.e. the augmented chord. Similarly, the 6 tone scale is a subset of the 12 tone scale. In fact, the 6 tone system is simply the whole tone scale. Because of this, we can write 6 tone scales in terms of notes from the 12 tone scale. Thus, the 6 tone equal tempered scale is a "sub-system" of our 12 tone scale. Naturally, all MOLTs of the 6 tone scale remain MOLTs in the 12 tone scale.
True MOLTs– that is, the MOLTs which are not truncations of other MOLTs– are the chromatic scales with a single PSS removed. One might also think of this as "uninterpolating" a PSS. For example, Messiaen's 7th MOLT is the chromatic scale with a tritone removed, and his 3rd is the chromatic scale with an augmented chord removed.
Scales from any $n$–tone equal temperament scale (TET) can be represented as a binary number with $n$ digits (including leading zero digits). For example, the C major scale in 12-TET:
1 0 1 0 1 0 1 1 0 1 0 1 = 101010110101b
B A♯ A G♯ G F♯ F E D♯ D C♯ C
which is $2741$ in base 10. (This method of representing scales is key-agnostic and the key of C was chosen for readability.) Notice how the least significant bit represents the lowest note in the scale. Every scale is represented as the odd numbers as we do not considers sets of notes which do not include the first note to be scales (the least significant bit of odd numbers in base 2 is always 1).
Finding all the MOLTs first involves finding the unique prime factors of the TET, then using the factors $p$ to generate a scale which equally divides the TET into $p$ notes (called the prime scale). For each of these prime factors $p$ of TET $t$, the corresponding prime scale is $\sum_{n=0}^{p-1} 2^{t/pn}$. For example, the prime factors of 12 are 2 and 3, which means its prime scales are $\sum_{n=0}^1 2^{12/2n} = 65$ (the tritone) and $\sum_{n=0}^2 2^{12/3n} = 273$ (the augmented chord). This sum is of a geometric series and can be rewritten: $$\sum_{n=0}^{p-1} 2^{t/pn} = \frac{2^t-1}{2^{t/p}-1}$$ These prime scales are MOLTs, and overlapping transpositions of these prime scales are MOLTs. The formula $S = p(2n+1)$ while $S$ is less than $2^tet$ represents all the MOLTs from prime scale $p$.
The number of MOLTs for a TET $t$ generated by its prime factor $p$ is given from the formula $2^{t/p-1}$. The total number of MOLTs in a TET is the sum of these for each prime factor of that TET. Note that this counts duplicate scales found by each prime twice or more. For example, the whole tone scale 1365 is both $65(20 + 1)$ and $273(4 + 1)$ (i.e. interpolation of the tritone by a whole tone and a major third and interpolation of the augmented chord once by a whole tone). Currently I don't see any way to skip these duplicates without manual checking.
Because scales are numbers, we can manipulate scales using arithmetic operations. For example, the union of scales $S1$ and $S2$ is given with $S1 + S2$ where neither scale has any bits both equal to 1 (i.e. no base 2 carries occur; in reality this is a bitwise OR).
As mentioned above, the sum of unique scales is the combination of those scales. $2^n$ for any non-negative integer always creates scales which are unique from one another: $$2^0 = 001b$$ $$2^1 = 010b$$ $$2^2 = 100b$$ etc. This example also demonstrates that $2^n+S$ where $S$ does not already contain the $n$th note means "$S$ with the $n$th note." $2^n$ itself is just $n$th note.
To see what is meant by $2^{t/pn}$, we look at the augmented chord (where $t$ is 12 and $p$ is 3): $$2^{12/3(0)} = 1 = 000000000001b = C$$ $$2^{12/3(1)} = 16 = 000000010000b = E$$ $$2^{12/3(2)} = 256 = 000100000000b = G♯$$ The difference of the digit position of 1 between each note is 4 (that is, $p/t$). Prime scales of prime $p$ are scales $p$ notes that are equally distributed.
As stated above, the sum can be rewritten as:
$$\frac{2^t-1}{2^{t/p}-1}$$The numerator, $2^t-1$, is the chromatic scale in TET $t$. $2^n$ is the $n$th note, regardless of TET, and is only a scale when it is the 0th note (because scales must include the first note); $2^n-1$ means "every note up to but not including the $n$th note". It is a scale containing the first $t/p$ notes of its TET.
Division means the inverse of interpolation. So this equation states that the prime scale from prime factor $p$ of TET $t$, when interpolated by every note up to but not including the $t/p$th note (i.e. $2^{t/p}-1$), produces the chromatic scale (i.e. $2^t-1$).
Multiplication of prime scale $P$ by another scale $S$ means interpolating $P$ by each note in $S$. For example, the whole tone scale is $65\times21$. $65$ is the tritone $1000001b$, and $21$ is $10101b$, which is C D E in the key of C. Through multiplication the pattern C D E is transposed to each note contained in the tritone, yielding C D E (i.e. $21\times1$, where $1$ is the first note contained in $65$) plus F♯ G♯ A♯ ($21\times64$, $64$ being the second note contained in the tritone).
Essentially, this formula says that every possible scale (i.e. $2n + 1$) between $1$ and the first non-one interval of a prime scale multiplied by that prime scale is a MOLT.
The Haskell code below is more concise than the old code I wrote in Scheme, due to the new method of representing scales as numbers; it is also more efficient. Most of the time spent running the program is probably in translating the numbers into readable scales for printing. The old Scheme program was used to produce the lists on this page.
primeFactors :: Integer -> [Integer] primeFactors x = [pf | pf <- filterPrimes [2..x-1], x `mod` pf == 0] filterPrimes :: [Integer] -> [Integer] filterPrimes [] = [] filterPrimes (a:d) = a : filterPrimes (filter ((/=0).(`mod`a)) d) primeScale :: Integer -> Integer -> Integer primeScale ntet prime = (2 ^ ntet - 1) `div` (2 ^ (ntet `div` prime) - 1) findMolts :: Integer -> Integer -> [Integer] findMolts ntet ps = takeWhile (< 2^ntet) [ps, ps*3..] molts :: Integer -> [Integer] molts ntet = concatMap (findMolts ntet . primeScale ntet) (primeFactors ntet) -- showing scales twelveTones :: [String] twelveTones = ["C", "C#", "D", "D#", "E", "F", "F#", "G", "G#", "A", "A#", "B"] bits :: Integer -> [Integer] bits n = takeWhile (>0) (iterate (`div` 2) n) showScale :: [a] -> Integer -> [a] showScale t s = [k | (k, n) <- zip t (bits s), n `mod` 2 == 1] printMolts :: [Integer] -> Integer -> IO () printMolts scale ntet = mapM_ (print . showScale scale) (molts ntet)