John Harrison: Musicologist
Although it is widely known that John Harrison, the world-famous clockmaker, was choirmaster at Barrow Church, his detailed interest in music theory is less well known. In fact, 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 found a few decades ago in the United States Library of Congress*.
“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 of
“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 using both logarithms and the value of PI to define the ratios.
First some background. The major scale, such as that exemplified by the well known song “Do-Re-Mi” – The Sound Of Music (1965) (whose first line is Doe a Deer) uses the Tonic Sol-far notation and is sung i.e. Do, Re, Mi, Fa, So, La, Ti, Do. When sung in tune musicologists are in 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 best ratios for the frequencies of the other notes of the scale. In fact the gaps (or more correctly the ratios) between the notes are not equal and there are various schemes for defining what the individual ratios (gaps) are, note by note.
One of the earlier definitions was called the Pythagorean Scale dating back to Pythagoras of Ancient Greece. A later scale is called Just Intonation. There are many different schemes which define the notes, some using whole number fractions and 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 sceptical 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.”
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”
…and he then proceeds to define his unique definition of scale notes, based upon logarithms and PI, the ratio of a circles circumference to its diameter. John Harrison was clearly familiar with logarithms and describes their use in both tuning Church bells and in defining a musical scale.
Logarithms to the base 10, as used by Harrison, were invented by Henry Briggs (1561 to 1630) who was the first in the world to publish such tables in 1624 as “Arithmetica Logarithmica”. By coincidence Henry Briggs originated from the village of Sowerby Bridge, just 30 miles from the village of Foulby where John Harrison was born less than a century later, and one cannot help but speculate that Henry Briggs must have been famous in his local area for his Logarithmic Tables to the base ten. Indeed the fame of Henry Briggs and his Logarithms quickly spread to Europe 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 scale we widely use today.
In 1722 JS Bach published his great works the Well-Tempered Clavier Vols 1 followed by Vol2 in 1742 which were in all 24 major and minor keys. It has been said that this was a demonstration of the great advantage of equal temperament but now musical scholars are no longer so certain. It may have been an exercise in how to avoid the pitfalls of the equally tempered scale.
In John Harrison’s time there was a serious debate, and there still is, over whether this equal-temperament scale is sufficiently accurate.
It has the great advantage that tunes can be transposed into any key and yet be played on the same keyboard. But the price of such versatility is a compromise in accuracy of pitch and that accompaniment and especially chords lack purity and do not sound so smooth.
John Harrison was trying to provide an alternative to the equally tempered scale, whilst also avoiding the complicated whole number fractions of the Just or Pythagorean scales. His solution was based upon PI, an idea which he no doubt found attractive also because of the “Music of the Spheres”.
Now for the Details
In his own words from the above document he wrote “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 then proposes that an interval corresponding to two whole notes which he calls the Greater Third (on a piano is from C to E) and which he maintained corresponds to the diameter drawn around a circumference of a circle (i.e. pi radians) by saying
“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.”
That is he calculates 0.09582 as 0.30103 divided by PI. (In fact The major Third corresponding to two hole notes having an angle of on his imaginary circle of 1 radian = 57.296 degrees in John Harrison’s Scheme has an angle of 60 degrees in the modern equally tempered scale.)
Next he calculates the size of a whole note by saying “And the radius or half of which the Larger note viz 0.004791”
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”
Indeed they can and I show them all below.
Note number |
Tonic Sol-fa | Key of C | Interval =2^{(n/12)} |
Equal Temper | Musical Notation | JH Scale as Log |
JH Scale As Ratio |
0 | Do | C | 2^{0} | 1 | Unison | 0 | 1 |
1 | |||||||
2 | Re | D | 2^{(2/12)} | 1.1224 | Major Second | 0.047910 | 1.1166 |
3 | |||||||
4 | Me | E | 2^{(4/12)} | 1.2599 | Major third | 0.095821 | 1.2468 |
5 | Fa | F | 2^{(5/12)} | 1.3348 | perfect fourth | 0.126559 | 1.3383 |
6 | |||||||
7 | So | G | 2^{(7/12)} | 1.4983 | perfect fifth | 0.174470 | 1.4944 |
8 | |||||||
9 | La | A | 2^{(9/12)} | 1.6817 | Major sixth | 0.222380 | 1.6687 |
10 | |||||||
11 | Ti | B | 2^{(11/12)} | 1.8877 | major seventh | 0.270291 | 1.8633 |
12 | Do | C | 2^{(12/12)} | 2 | 0.301030 | 2 |
The formulae of each whole note of John Harrison’s are shown below
John Harrison | Formula | Value | Ratio =10^{Value} |
Cents |
Large Note | log(2)/(2*PI) | 0.04791041183 | 1.1166328 | 190.98 |
Small Note | (log(2)-5*log(2)/(2*pi))/2 | 0.03073896826 | 1.07334087 | 122.53 |
Octave | log(2) | 0.3010299957 | 2 | 1200 |
Comparison and discrepancy between John Harrison Scale and 5 Just Intonation
Scale | Do | Re | Mi | Fa | So | La | Ti | Do | SD σ: |
5 Just | 1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15:8 | 2 | |
J. H. | 1 | 1.1166 | 1.2468 | 1.3383 | 1.4944 | 1.6687 | 1.8635 | 2 | |
%diff | 0 | 0.75 | 0.2 | 0.4 | 0.4 | 0.2 | 0.6 | 0 | 0.24 |
Comparison and discrepancy between John Harrison Scale and Equally Tempered Scale
Scale | Do | Re | Mi | Fa | So | La | Ti | Do | SD σ: |
12TET | 1 | 1.1224 | 1.2599 | 1.3348 | 1.4983 | 1.6817 | 1.8877 | 2 | |
J. H. | 1 | 1.1166 | 1.2468 | 1.3383 | 1.4944 | 1.6687 | 1.8635 | 2 | |
% diff | 0 | 0.51 | 1 | 0.3 | 0.2 | 0.7 | 1.2 | 0 | 0.40 |
And for reference
Comparison and discrepancy between 5Just Scale and Equally Tempered
Scale | Do | Re | Mi | Fa | So | La | Ti | Do | SD σ: |
5 Just | 1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15:8 | 2 | |
12TET | 1 | 1.1224 | 1.2599 | 1.3348 | 1.4983 | 1.6817 | 1.8877 | 2 | |
% diff | 0 | +0.2 | -0.8 | -0.1 | +0.1 | +0.9 | -0.7 | 0 | 0.53 |
Although John Harrison’s scale compares favourably with the 5 Just Intonation, his scale does not help to facilitate key changes on a conventional keyboard, as the equally tempered scale does, and so it never gained popularity.
Nevertheless, the whole story makes the character of John Harrison, a self-taught Barrovian and expert in clockmaking, lunar longitude and now, in music theory all the more unusual and remarkable.
The sound files below contain a rising Major Scale followed by a Chord of C Major followed by a Chord of C Major7.
Article written Feb 2020 by Steve Taylor MinstP MIEEE MD www.geomatix.net