John Harrison

It is widely known that John Harrison, the world-famous clock-maker,  was choirmaster at Barrow-Upon-Humber, Church of Holy Trinity in North Lincolnshire, UK.  However his detailed interest in music went far beyond this relatively humble role. 

 

He was particularly interested in the mathematical definition of the notes comprising a musical scale, a subject known as musicology. He describes his contribution to the field in two of his works, one of which was only recently found in the United States Library of Congress. Its lengthy title encompasses three of his interests; time, longitude and music or as he put it,

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“A Description concerning such Mechanism as will afford a nice, or true Mensuration of Time; together with Some Account of the Attempts for the Discovery  of the Longitude by the Moon; and also An Account of the Discovery of  the Scale of Musick”. by John Harrison, London 1775.  Pages 67 to 108 offer

“A true and full account of the foundation of Musick, or, as, principally therein, of the Existence of the Natural Notes of Melody” of circa 1776.

 

In these manuscripts he describes his own unique mathematical method (or as we would now say ‘algorithms’) for determining the notes of the musical scale; this method used logarithms and the value of PI to define the musical intervals between the notes.

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Click below to hear the musical scales - see if you can tell the difference.

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The sound players contain a rising Major Scale followed by a Chord of C Major followed by a Chord of C Major7.

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Background

First some background. The major scale, using the Tonic Sol-far notation is sung "Do,  Re, Mi, Fa, So, La, Ti, Do". (The scale is exemplified by the well known song “Do-Re-Mi” – The Sound Of Music (1965), whose first line is Doe a Deer).  When sung in tune by a reasonable good singer musicologists are in general agreement that the last top Do is exactly twice the frequency of the first bottom Do. However there is no such universal agreement over the exact intervals for the frequencies of the other notes of the scale. In fact the intervals (or more correctly the ratios) between the notes are not equal and there are various schemes for defining what are the best ratios (gaps) are, note by note.

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One of the earlier definitions was called the Pythagorean Scale dating back to Pythagoras of Ancient Greece which uses whole numbers 2,3,4. A later scale which is based on the Pythagorean, but extends it to include the whole number 5, is called Just Intonation 5. There are many different schemes which define the notes, some using whole number fractions and some more recently using logarithms.

It seems John Harrison had studied music theory in some depth, for he clearly knew of the definition of the notes in the Pythagorean Scale and of the scales defined by Just intonation. In one of these systems a Tone Major differs from a Tone Minor by a fraction called a Comma which is determined by the ratio of 81/80. John Harrison was skeptical of the use of such fractions in musical melody and he wrote

 

“I have found …. that the natural Notes of Melody are certainly free from all the inconsistent nonsense arising from the Imagination of there being a Tone Major of 9 to 8, a Tone Minor of 10 to 9, and a nonsensical Comma (as being their difference) of 81 to 80.”

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and again

 

“whereas on the contrary, melody ought certainly to have been, nay must certainly be the Motive, for even most of the Country Plow-men and Milk-maids can naturally sing a Tune “

 

He goes on to write

 

“for easiness or perspicuity, instead of working with the Ratios themselves take their Logarithms as in manner below”

 

John Harrison then proceeded to create his unique definition of scale notes, based upon logarithms and PI, the ratio of a circles circumference to its diameter. John Harrison was clearly very familiar with logarithms and describes their use in his method of tuning Church bells and in defining a musical scale.

 

Logarithms in Music

Natural logarithms (i.e. logarithms to the base e) were invented by Napier in Scotland and published in 1614, these were changed to the more useful form, to the base 10, Henry Briggs some ten years later in 1624 as “Arithmetica Logarithmica”.  By coincidence Henry Briggs originated from the village of Sowerby Bridge, which was just 30 miles from the village of Foulby where John Harrison was born around 70 years after Brigg's publication. One cannot help but speculate that Henry Briggs must have been famous in his local area for his Logarithmic Tables, and that John Harrison familiarized himself with their use, becoming very competent in their theory. This contrasts with the idea that John Harrison was a simple carpenter. The fame of Henry Briggs and his logarithms quickly spread to Europe, their application rapidly spread to the definition of musical scales and soon they were used by Johann Faulhaber in Germany to create a musical scale of equal temperament, in around 1631. This was the first printed solution of an equally tempered scale and is the basis of the scale widely used today. By 1688, just 5 years before John Harrison was born, the Italian instrument maker Bartolomeo Cristofori (1655-1731) of Padua is credited with the invention of the modern musical keyboard with its familiar arrangement of black and  white keys. Without logarithms the piano keyboard could not have been invented.

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The reason logarithms and musical scales are so intertwined is because notes, like logarithms mix the idea of addition and mutiplication. In the equally tempered scale, each successive key plays a note whose frequency is that of the previous key multiplied by a fixed ratio. By the time you have played 12 successive notes in the scale you should have mutiplied the original note by the ratio 2. Their are 8 whole notes in a scale but 12 half notes (semitones), because some whole notes are actually semitones. This can be represented  as travelling clockwise around a circle, with the notes being equally spaced stopping points on the circle. Since an octave (Do to top Do) represents a doubling of the frequency  the circumference length should be proportional to the logarithm of 2. For the equally tempered scale the circle is divided into 12 equal portions, one for each semitone. The scale of 8 notes in terms of intervals is "tone, tone, semitone, tone, tone, tone, semitone" in the equally tempered scale and forms the basis for the representation of white and black notes on a piano.

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John Harrison knew this theory well. He states

“Let the octave, as above be represented by the logarithm of 2 and let that same number be also esteemed as the circumference of a circle viz. 0,30103”

 

He was trying to find a better scale than either the equally tempered or the Just or Pythagorean scales, an alternative which would avoid the complicated decimal fractions of the former, and also avoid whole number fractions of the latter. Instead it would replace them with the mathematical value of PI, an idea which he no doubt found attractive because of its association with the circel and, no doubt,  the“Music of the Spheres”.

 

He proposed that the interval corresponding to two whole notes (which he calls the Greater Third  now called a major third e.g. on a piano is from C to E) would be based upon PI and would have the value of the ratio of the diameter to the circumference of the circle by saying

 

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“Then [as I am shew or verify before I have done] the diameter will  be the greater 3rd Viz. ,09582.” and “As 3.1416 is to 1; so is ,30103 to ,09582.”

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That is he calculates 0.09582 as 0.30103 divided by PI. Thus his Greater Third subtends an angle of one radian i.e. 57.296 degrees, whereas in a modern equally tempered scale the interval now known as the Major third corresponds to two hole notes and has an angle of 60 degrees. Next he calculates the size of a whole note by saying

 

“And the radius or half of which the Larger note viz 0.004791”

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And finally he says there are five whole notes and two half notes in a musical scale  

 

“And as thence, Five times the larger note subtracted from the octave will leave ,06149, so the half of which must be the lesser note [since five of the larger notes, and two of the lesser notes exactly compleat or make up the octave] Hence the lesser note is  ,03074. And from these by addition and subtraction, all the others may be found” 

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Although its accuracy compares favourably with the 5 Just Intonation, it does not assist in making key changes which is the unique advantage offered by the equally tempered scale. Consequently although accurate, John Harrison’s scale never became popular. 

 

Nevertheless, the whole story fascinating story makes  the character of John Harrison all the more unusual and remarkable. His self taught expertise, as well as including clockmaking, astronomy, lunar motion and marine navigation we now must  add  music theory.

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