Music Sequencing with Dynamic Transposition

The purpose of this software is to read music scores in Dynamic Transposition notation (PDF) and convert them into WAV files. This simplifies music sequencing and uses a more harmonious musical scale.

Contents

Method

In just intonation scale frequencies of musical tones are given by rational fractions of the base frequency. For example, if the base frequency of note A in 4-th octave is freq(A4)=440Hz then frequencies of related harmonic intervals, corresponding to C#5 and E5 in octave 5 will be: 5/4*440=550 and 3/2*440=660 respectively.

Corresponding minor chord frequencies will be: 440 (A4), 440*6/5=528 (C5), and 660 (E5). Thus, the frequencies for A4, C5, C#5, E5 are:

440 A4
528 C5
550 C#5
660 E5

This is in contrast to the standard Chromatic scale where the corresponding frequencies are approximately:

440.00 A4
523.25 C5
554.37 C#5
659.26 E5

From this comparison it is clear that the scale based on rational fractions produces musical intervals with better resonating frequencies. This makes music more harmonious when plaid in such scale. We will refer to the scale based on rational fractions (i.e. fractions of integers) as rational scale.

Acoustic instruments can be tuned in rational scale but then they can only play in one octave, i.e. in a single key. Going to a different octave will change the intervals, so the same chords will sound differently. This is the reason why rational scale is not commonly used.

That’s where the method of dynamic transposition comes to the rescue. This technique makes it possible to play in rational scale in different keys and different octaves using computer generated tones. All one needs to do to switch from one key to another is to shift the frequency of all notes by a corresponding rational fraction. For example, the frequencies of the major chord above in key A and then in key D will be:

Key A: 440, 550, 660
Key D: 6/5*440, 6/5*550, 6/5*660

To play in key D we multiplied all three frequencies of key A by 6/5 which corresponds to the frequency shift from A4 to D5. This is akin to transposition which can be used on electronic instruments. Here transposition is done on-the-fly, switching from one key to another while playing the composition, thus the name dynamic. Also, the shift in frequency is done by rational numbers which produces more harmonious intervals than in conventional Chromatic scale.

Implementation

A music score in rational scale for several instruments can be written in a simple text file. Two notations can be used: explicit and implicit. Explicit format uses numbers and implicit - letters. For example, explicit format will look like this:

5:3 1:1:2 5:4:2 3:2:2

which instructs three instruments to play a chord of three notes at frequencies:

instrument 1 (1:1:1) plays at frequency: base_freq*(5/3)*(1/1*2^2)
instrument 2 (5:4:2) plays at frequency: base_freq*(5/3)*(5/4*2^2)
instrument 3 (3:2:2) plays at frequency: base_freq*(5/3)*(3/2*2^2)

where “base_freq” is the base frequency which is the same for the whole composition and is selected at the beginning (e.g. 440Hz).
The fraction given at the beginning of each row (5:3 in our example) is used to multiply the base frequency of the composition and make it base frequency of that row. In this example:

base_freq_row = base_freq * 5/4

Essentially the base frequency was transposed to become the frequency of that row and all notes on that row will use it to set their frequencies. Thus, the fraction in the first column determines dynamic transposition since it changes the base frequency on-the-fly during the composition.

All other numbers in the row determine frequencies of the notes plaid by respective instruments. Notation “a:b:c” defines the factor that the base frequency of the row is multiplied by, that is

note_freq = base_freq_row * a/b * 2^c

where

a is the numerator, 
b - the denominator, and 
c - the octave (notation 2^c means raising 2 to power c).

We can also substitute letters for fractions for convenience. For example, that line can also be written in implicit format as

A C2 E2 G2

where number 2 on the right of each letter represents the octave and the fractions are assigned to letters as:

A=5:3
C=1:1
E=5:4
G=3:2

The letters can be arbitrary, but in this case we chose them to correspond to key C. So, the above line (A C2 E2 G2) corresponds to chord C-E-G transposed to key A, which will result in chord A2-C#3-E3 actually plaid. This way one can play all keys using only the notes in key C which is another advantage of this method.

For example, if we want to repeat the sequence of main triad notes in four different keys: C, Am, F, G (i.e. three major and one minor - Am) we can do it as this:

C C2 E2 G2
A C2 Eb2 G2
F C2 E2 G2
G C2 E2 G2

where the first letter in each row corresponds to the frequency shift for the entire row (i.e. the key plaid). Note that we did not need to change the notes C,E,G for any of major keys, and only introduced Eb=6/5 (i.e. E-flat) for a minor key Am. The resulting sequence is much simpler than it would be in a conventional music score.

The complete composition may look like this:

; Definitions:
Am = 5:6
C  = 1:1
E  = 5:4
Eb = 6:5
G  = 3:2
; Score:
C C2 E2 G2
Am C2 Eb2 G2
F C2 E2 G2
G C2 E2 G2

Thid will play chords in keys: C, Am, F, and G. Note that fraction 5:6 was labeled as Am to signify that this is a minor chord. This was done mainly for clarity and would work with any other label. The semicolon (;) is used for comments.

Time duration for each row is referred to as tick and is set at the beginning of the score file which is called preamble. Each tick can be further subdivided into smaller time intervals referred to as pics. The number of picks in a tick is also defined in the preamble.

Several ticks comprise a beat, and several beats comprise a measure where corresponding numbers (i.e. ticks-per-beat and beats-pre-measure) are also set in the preamble. These parameters are used when one wants to output only specific measures from a composition. This can be done by supplying clip parameters for the DT compiler (run DT compiler with –help option to see more information).

To make note duration longer than one tick one should put ‘+’ in the next line corresponding to the position of respective instrument. For example, in this chord:

A B2 C2 D2
A +  -  -

instrument 1 plays note B2 for the duration of two ticks (two consequtive lines) and the rest of instruments play respective notes (C2,D2) for one tick only. The dash symbol ‘-’ is used to indicate pause for respective instrument.

To make notes with a duration less than one tick (i.e. several picks) one can use this notation:

A -     C2,2
A B2.1  D2

In this case note B2.1 starts one pick before the tick corresponding to the line where this note is located and note C2,2 starts two picks before the tick corresponding to the next line (i.e. before D2).

There are also special lines to set the volume (amplitude) and panning for each note. These lines begin with special tags: amp for amplitude and pan for pan. Sample score files supplied with this package provide examples of their usage. In the examples file-extention dtt is used for explicit and dtm for implicit file formats.

The source codes included in this package are written in GO language. They produce the DT-executable that compiles dtm or dtt files into MIDI or SCound SCO files. The supplimentary Unix shell scripts are used to run the DT-compier and convert the MIDI/SCO file to WAV or MP3 file.

Files

The files in this package include the source files and build scripts for dynamic transposition (DT) compiler as well as examples of score files and compiled compositions.

Usage

Building DT compiler

Editing DTM/DTT files

To edit DTM/DTT files on Linux you can use vim editor with syntax highlighting set to asn68k. In the score part use tabs instad of spaces to separate columns for respective instruments (see manual in man directory).

Using DT compiler

The DT compiler executable by itself converts dynamic transposition scores into MIDI or SCound files. To produce actual audio files, such as WAV or MP3 one can use different tools, such as Fluidsynth, or FFmpeg on Linux. To simplify this we provide two Bash scripts: make_midi.sh to create a MIDI file and then produce a WAV file with fluidsynth and make_sco.sh to create a CSound score file and then compile it into a WAV file with CSound.

Examples below show how to compile dynamic transposition formats, DTM (using letters) or DTT (using numbers) into WAV or MP3 files using MIDI or CSound isntruments on Linux.

Compiling to MIDI

To compile a DTM or DTT file to MIDI file a GoMIDI go-library is used by the DT compiler. The produced MIDI file can then be converted to WAV file with FluidSynth on Linux. The make_midi.sh script provided in this package will run the DT compiler and convert MIDI to WAV. For example this command

sh make_midi.sh canon.dtm

will convert canon.dtm file into canon.mid file and then call fluidsynth to produce canon.wav file. One can also play the MIDI file directly with fluidsynth.

To include tracks with percussion instruments into MIDI file one can use use example drums.dtt provided in this package which can be compiled with make_drums.sh script as:

sh make_drums.sh drums.dtt

which will produce the drums.mid file and convert it to drums.wav by calling fluidsynth engine.

To convert WAV file to MP3 on Linux one can use the FFmpeg utility.

Compiling to CSound

To compile a DTM or DTT file to CSound sco-file one can use make_sco.sh file provided with this package. For example, to compile the Ave Maria example score one can use this command:

sh make_sco.sh avamaria.dtm

This will produce a CSound score file avemaria.sco and then call csound engine to convert it to avemaria.wav file using CSound orchestra file avemaria.orc and a header file avemaria-head.sco file which are included in this package.

Compilation options

Sometimes it can be useful to compile only certain measures from the score. For that one can use special compiler options. For example, to compile a midi file with two measures starting with the third measure, one can invoke the compiler with this command:

dtc -m 3 -n 2 score.dtm output.mid

or the respective Bash script as:

sh make_midi.sh score.dtm 3 2

If the first measure starts after a few introductory beats which one wants to skip one can indicate this with the -s option, like this:

sh make_midi.sh score.dtm 3 2 1

This will skip one introductory beat. That is, measures will be counted from the second beat in the compisition.

References