In 2021, Michael Coughlin made a concerted effort to organize some of his thoughts about melody and music. He had wanted for years to write a series of essays about stringed instruments throughout history, how they were tuned, how their notes harmonized, and much more. He decided to tackle just one tiny aspect of this set of thoughts.

Michael felt his main challenge was how to explain this complex topic to a person who was new to music. He struggled to find ways to make this easy to understand. He kept putting the project aside and then returning to it.

Here is the essay from 2021. Michael was a brilliant, talented person, and it shows in the variety of interests he held.

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What makes a beautiful musical melody? With few exceptions, each note is consonant with the note before it and after it . After the invention of the lyre and harp, when two notes would be played together to tune the instrument, it was discovered that consonance is mathematically related to the length of strings and the tension they are stressed with. In recent times this has been expressed in terms of the frequencies of notes. Two notes are consonant when the ratios of their frequencies are small whole numbers. A ratio of two to one is the most consonant and is named the octave. Next is three to two, named a fifth, and four to three, named a fourth. As the numbers in the ratio get larger the notes are less consonant. If notes are not tuned precisely then they will not be ratios of small whole numbers — they will be ratios of large numbers so will be dissonant instead of consonant.. With numbers as large as five and seven things start to change from consonance to dissonance.

The problem of tuning a lyre or harp was solved early on by tuning pairs of strings in ratios of 3/2, 4/3, and 2/1. A collection of seven notes tuned this way was found to be particularly useful. After a thousand or two years this musical scale of seven notes was named the Pythagorean tuning. Unfortunately this does not exactly match what a singer wants to sing. But it comes close. The harpist can change the notes that are out of tune for each song.

Multiply the fraction 3/2 by itself four times. Also add 4/3

4/3, 1, 3/2, 9/4, 27/8, 81/16, 243/32

divide by 2 to make the fractions be between 1 and 2

rearrange them in order of size, put 2 at the end

  1   9/8   81/64   4/3   3/2   27/16   243/128   2
cents 294   408     498   702    906      1110  

Or start the Pythagorean scale at the high note and tune down
1/1, 8/9, 27/32, 3/4, 2/3, 16/27, 9/16, 1/2

Of the many musical scales that have been used to improve on the Pythagorean, one called the just diatonic scale has had the most influence. It uses intervals with small whole numbers. First written about around 130 AD by Claudius Ptolemy of Alexandria. it was also investigated by prominent musical theorists such as Gioseffo Zarlino and Hermann von Helmholtz. It is not favored by composers for orchestras since there are still small changes needed to keep everything in tune with a large number of instruments and key changes.

A comparison between a Pythagorean scale and the justly tuned diatonic scale.

Pythagorean 1 9/8 32/27 4/3 3/2 27/16 16/9 2/1
major scale 1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
minor scale 1 9/8 6/5 4/3 3/2 8/5 9/5 2/1

How does this work out for frequencies? Musical notes are designated by letters. Let the first or lowest note in the scale be ” C “. The notes in a scale would be C D E F G A B. So the sixth would be ” A “. The pitch frequency of the note A was defined to be 440 Hz by an international conference in 1939. The pitch of the sixth note in the above Pythagorean scale is 27/16 times the first, so C would be 16/27 x 440 or 260.74 Hz.

Classical music has a requirement that any melody can be played at a different pitch by starting on any note. No problem for singers. For musical instruments five extra notes (flats or sharps) needed to be added to the original seven to attempt this. But it is still not possible to be perfectly in tune with the Pythagorean scale or the just diatonic scale. Many attempts were made to create such a well tempered scale and about two hundred years ago it was decided to use a compromise named the equally tempered scale. Easy to do with a guitar; much harder with a piano. The pitch of each note was given by multiplying by a factor of the twelveth root of two (1.059463). So if the note A is 440 Hz then A# is 440 x 1.059463 = 466.16 and B is 440 x 1.059463 x 1.059463 = 493.88 We lose the idea of using the ratio of small whole numbers but this compromise comes very close to sounding like a justly tuned diatonic scale and that scale can be played starting on any note.

Let us return our attention to the ratio of small whole numbers. The singing voice and certain other musical instruments (violin, trombone, Hawaiian guitar) do not have the limitations of the harp family. The singing voice can change the pitch of notes by any ratio. It is not limited to using the pitch of only a few harp strings. Around 1960 Warren Creel and Paul Boomsliter experimentally measured the pitch of familiar melodies produced by expert musicians and found they chose rational intervals but in combinations that did not match any conventional scales or temperaments. The instruments they used were a monochord with a six foot long string tuned with a steel rod like a Hawaiian guitar and a reed organ with a selection of keys tuned to confirm the measurements discovered with the monochord.

Start with a series of fractions 3/2, 4/3, 5/4, 6/5, 5/3 … these are intervals from the just diatonic scale. Add a few more fractions that don’t seem to be used as much 7/6, 8/5, 7/4. What Creel and Boomsliter discovered was that these intervals were only part of what was used to play music. They needed to be combined in other patterns. Start with one note in a series. Multiply by each ratio to have a selection of pitches to choose from. But then multiply all by another ratio to have a different selection. This is repeated several times. Unlike conventional musical theory that has notes related to a tonic pitch, Creel and Boomsliter found that what could be thought of as the tonic pitch changed as the melody progressed.

The following chart illustrates this. The vertical scale is frequency ratios in cents and fractions. Three generations of pitches are plotted horizontally. The left hand axis is marked every hundred cents and the corresponding Do Ray Mi Fa Sol La Ti Do syllables are indicated. The horizontal green lines mark the pitch of piano keys or guitar frets.

The first generation of ratios is 1/1, 7/6, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, and 2/1. A collection of these notes is plotted as horizontal lines with their pitch values shown in cents. Next on this chart the second generation of ratios is 3/2 or 3/4 times the first generation. A third generation would be 3/2 or 3/4 times the second generation or 9/4 or 9/8 times the first. This is the same idea of producing the Pythagorean scale applied to more than one note at a time.

Music exists in major, minor and modal forms. Creel and Boomsliter discovered that using different ratios to change to the second generation made the melody change musical modes. To quote Warren Creel (1976) “Our experiments in Albany indicate that music in a major mode uses 3/2 of the first generation as reference note for the second. Music in a minor mode uses 6/5 of the first generation as reference for the second. Music in a blue mode uses 5/4 of the first as reference for the second.” Warren also states that several more generations are needed using the ratio of 3/2.

Creel and Boomsliter theory is a very radical departure from older theories. As a melody progresses each change of pitch is a ratio of small whole numbers, so it is a consonant interval with none of the out of tune beating of equal temperament. But notes that are several steps apart can have pitch ratios that are made of large integers. They need not come close to the notes of the equally tempered scale. Perhaps even more important such melodies can not be written as ordinary sheet music.

Composers of “modern” microtonal music frequently use equal tempered scales with any steps besides twelve. These are doomed to be out of tune.

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Fractions with small integers and corresponding cent values

cents = 3986.313714 * log10(frequency ratio)

1/1 = 0, 7/6 = 266.87, 6/5 = 315.64, 5/4 = 386.31, 4/3 = 498.04, 7/5 = 582.51, 3/2 = 701.95, 8/5 = 813.69,
5/3 = 884.36, 7/4 = 968.83, and 2/1 = 1200.
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A quote from the Catgut Acoustical Society —
“Warren Creel and Paul Boomsliter worked together closely in the field of auditory perception and organization, studying pitch relations and durational effects in tonal perception, sensation, rhythm and cadence in language which they studied through music and poetry with related psychoacoustic studies, including animal experimentation. Starting in the early 1960s, Warren worked as a research associate in the Department of Surgery, Albany Medical College”

Paul C. Boomsliter and Warren Creel, “Extended Reference an Unrecognized Dynamic in Melody”, Journal of Music Theory, Vol. 7, No. 2 (1963), pp. 2-22″;

Warren Creel, “Musical Intonation For Fiddlers”, The Catgut Acoustical Society Newsletter, Number 26, November 1, 1976

Michael Coughlin Obituary – Boston Writing Group