David Ryan

David Ryan

Musical frequencies in Just Intonation are comprised of rational numbers. The structure of rational numbers is determined by prime factorisations. Just Intonation frequencies can be split into two components. The larger component uses only integer powers of the first two primes, 2 and 3. The smaller component decomposes into a series of microtonal adjustments, one for each prime number 5 and above present in the original frequency. The larger 3-limit component can be notated using scientific pitch notation modified to use Pythagorean tuning. The microtonal adjustments can be notated using rational commas which are built up from prime commas. This gives a notation system for the whole of free-JI, called Rational Comma Notation. RCN is compact since all microtonal adjustments can be represented by a single notational unit based on a rational number. RCN has different versions depending on the choice of algorithm to assign a prime comma to each prime number. Two existing algorithms SAG and KG2 are found in the literature. A novel algorithm DR is developed based on discussion of mathematical and musical criteria for algorithm design. Results for DR are presented for primes below 1400. Some observations are made about these results and their applications, including shorthand notation and pitch class lattices. Results for DR are compared with those for SAG and KG2. Translation is possible between any two free-JI notations and any two versions of RCN since they all represent the same underlying set of rational numbers.

David Ryan

Musical chords, harmonies or melodies in Just Intonation have note frequencies which are described by a base frequency multiplied by rational numbers. For any local section, these notes can be converted to some base frequency multiplied by whole positive numbers. The structure of the chord can be analysed mathematically by finding functions which are unchanged upon chord transposition. These functions are are denoted invariant, and are important for understanding the structure of harmony. Each chord described by whole numbers has a greatest common divisor, GCD, and a lowest common multiple, LCM. The ratio of these is denoted Complexity which is a positive whole number. The set of divisors of Complexity give a subset of a p limit tone lattice and have both a natural ordering and a multiplicative structure. The position and orientation of the original chord, on the ordered set or on the lattice, give rise to many other invariant functions including measures for otonality and utonality. Other invariant functions can be constructed from: ratios between note pairs, prime projections, weighted chords which incorporate loudness. Given a set of conditions described by invariant functions, algorithms can be developed to find all scales or chords meeting those conditions, allowing the classification of consonant harmonies up to specified limits.

David Ryan

A notation system was previously presented which can notate any rational frequency in free Just Intonation. Transposition of music is carried out by multiplying each member of a set of frequencies by a single frequency. Transposition of JI notations up by a fixed amount requires multiplication to be defined for any two notations. Transposition down requires inversion to be defined for any notation, which allows division to also be defined for any two notations. Each notation splits into four components which in decreasing size order are octave, diatonic scale note, sharps or flats, rational comma adjustment. Multiplication can be defined for each of the four notation components. Since rational number multiplication is commutative, this leads to a definition of multiplication for frequencies and thus notations. Examples of notation inversion and multiplication are given. Examples of transposing melodies are given. These are checked for accuracy using the rational numbers which each notation represents. Calculation shortcuts are considered which make notation operations quicker to carry out by hand. A question regarding whether rational commas should be extended from 5-rough rational numbers to all rational numbers is considered which would greatly simplify notation multiplication. This approach is rejected since it leads to confusion about octave number. The four component notation system is recommended instead. Extensions to computer notation systems and stave representations are briefly mentioned.