Chord and Interval Analysis

Exposing the physical guts of chords

Gene Boggs

Epistemologist-at-large

While studying...

Musical Geometry,   etc...

I decided to side-step and do some yak shaving on the size of note intervals and what acoustically makes up a chord.

This turned out to get more complicated, the closer I shaved^Wlooked.

The Problems

• There are common tables of intervals in every music science book like this list.
• But all are limited and have different slices of commonly used measures, like frequency and ratio.
• Very annoying!
• But a Table of Everything would numb your brain.
• I wanted to be able to inspect any and all possible chord combinations, but only the ones in which I was interested.

Some Visual Context

Pictures are worth at least 1000 words

The Design

• Command-line options and default settings.

• Use C major tonality. That is, C is our reference and we're in major.

• There is also "minor tonality", which swaps a couple intervals but we'll pretend it isn't important, for now.

• Build a note to ratio to named description model from the known intervals.

• Create a frequency sorted scale of names by evaluating the interval ratios.

• Compute the names, frequencies, intervals, cents and prime factorizations, etc. for each possible chord.

Use some handy CPAN bits

1. Math::Combinatorics

2. Music::Chord::Namer qw(chordname);

3. MIDI::Pitch qw(name2freq);

4. Math::Fraction; # XXX Syntax check spits warnings but does fractional approximation. Submit patch.

5. Sort::ArbBiLex;

6. Music::Scales

D#, Eb, E, Fb, E#, F?

• Please remain calm. We have to ponder "just intonation" now. :-)
• This would not be a slide if we were only concerned with equal temperament and "enharmonic equivalents."
  • C 1 (1) unison
  • ...
  • D#: 75/64 (1.17); classic augmented second
  • Eb: 6/5 (1.20); minor third
  • E: 5/4 (1.25); major third
  • Fb: 32/25 (1.28); classic diminished fourth
  • E#: 125/96 (1.30) classic augmented third
  • F: 4/3 (1.33) perfect fourth
• Check out the ratios under __DATA__ for more exciting details.

The Solution

=> Perl !

• Reveal the "guts" of chords - the frequencies of the notes and the intervals between them.

• There are over 400 known intervals! (Under __DATA__)

• Handle: Just intonation (ratio), Equal temperament (decimal), Cents, Prime factorization, Interval and Chord names.

On with the show!

• Program Settings:

 size      = 3,  # Number of notes in a chord.
 octave    = 4,  # Common octave: A=440Hz
 equalt    = 0,  # Show equal temperament.
 justi     = 0,  # Show just, ratio or natural intonation.
 intervals = 0,  # Show chord intervals.
 chords    = 0,  # Show chord names.
 rootless  = 0,  # Show 'no-root' chords.
 cents     = 0,  # Show note cents.
 freqs     = 0,  # Show note frequencies.
 primes    = 0,  # Show prime factorizations.
 tonic     = C,  # C Major tonality only for now.
 temper = 1200/log(2), # Equal temperament factor. 

Example Usage

• From the command-line:

 perl intervals --size=3 --chords --equalt --intervals C E G
 perl intervals --ch --e --i C E G  # Same as above.
 perl intervals --ch --r --j --p --i C D E
 perl intervals --j --f --i C pM3 pM7  # Pythagoras would be proud.
 perl intervals --j --ce C D D\#       # Intersting analysis...
 perl intervals --j --ce C D Eb
 perl intervals --s=8 --j --ce --f --i C D D\# Eb E E\# Fb F
 perl intervals --j --f --ce --i C 11h 7h  # Crazy! 

Real Example 1

 perl intervals --size=3 --chords --equalt --justi --intervals C E G
 C E G => {
  chords => [C],
  eq_tempered_intervals => {
   C E => 1.25992104989487,
   C G => 1.49830707687668,
   E G => 1.18920711500272
  },
  natural_intervals => {
   C E => { 5/4 => major third },
   C G => { 3/2 => perfect fifth },
   E G => { 6/5 => minor third }
  }
 } 

Real Example 2

 perl intervals --j --f --i C pM3 G
 C pM3 G => {
  natural_frequencies => {
   C   => { 1/1   => unison, perfect prime, tonic },
   G   => { 3/2   => perfect fifth },
   pM3 => { 81/64 => Pythagorean major third }
  },
  natural_intervals => {
   C G   => { 3/2   => perfect fifth },
   C pM3 => { 81/64 => Pythagorean major third },
   pM3 G => { 32/27 => Pythagorean minor third }
  }
 } 

Real Example 3

 > perl intervals --s=9 --j --f --i C D Db D\# Eb E E\# Fb F
 C Db D D# Eb E Fb E# F => {
  natural_frequencies => {
   C  => { 1/1    => unison, perfect prime, tonic },
   D  => { 9/8    => major whole tone },
   D# => { 75/64  => classic augmented second },
   Db => { 27/25  => large limma, BP small semitone (minor second), alternate Renaissance half-step },
   E  => { 5/4    => major third },
   E# => { 125/96 => classic augmented third },
   Eb => { 6/5    => minor third },
   F  => { 4/3    => perfect fourth },
   Fb => { 32/25  => classic diminished fourth }
  },
  ...
 } 

Real Example 3 (cont.)

 > perl intervals --s=9 --j --f --i C D Db D\# Eb E E\# Fb F
 C Db D D# Eb E Fb E# F => { ...,
  natural_intervals => {
   C D   => { 9/8 => major whole tone },
   C D#  => { 75/64 => classic augmented second },
   ...
   Eb E  => { 25/24 => classic chromatic semitone, minor chroma, minor half-step },
   Eb E# => { 625/576 => 1.08506944444444 },
   Eb Fb => { 16/15 => minor diatonic semitone, major half-step },
   Eb F  => { 10/9 => minor whole tone },
   E E#  => { 25/24 => classic chromatic semitone, minor chroma, minor half-step },
   E Fb  => { 128/125 => minor diesis, diesis, diminished second },
   E F   => { 16/15 => minor diatonic semitone, major half-step },
   E# F  => { 128/125 => minor diesis, diesis, diminished second },
   Fb E# => { 3125/3072 => small diesis },
   Fb F  => { 25/24 => classic chromatic semitone, minor chroma, minor half-step }
  }
 } 

Links

• At github
• This slide-show