Bryden Allen's Website

The Lost Chord

I learnt about the difference between the natural scale and the equi-tempered scale when I was at school doing physics. We even went far enough to calculate the error in the equi-tempered scale, which corresponds the ratio 81:80. But even my Dad, who was a physicist and taught me to play the piano, was very vague on this subject. Currently very few people understand this subject at all. I have just looked up the subject on Wikipedia and it is not covered at all.
            This topic is subject to fashion. Bach in the 17th century introduced the equi-tempered scale because it allowed some instruments (like pianos and guitars) to modulate to different keys more easily (previous to this the most people played and sung in the natural scale). And most people followed Bach’s lead. Then Helmholtz (2 century’s later) revived the natural scale and children were encouraged to re-engage in the natural scale again (using the doh, ray, me, fah, soh, lah, tee, doh system). But this idea has now mostly been forgotten and the equi-tempered scale now reigns supreme (often called diatonic or equal spaced scale).
            I am afraid that, if you don’t understand the physics behind the natural scale, then this my “lost chord” won’t mean very much. But I shall persevere.


The following diagram shows the difference between the equi-tempered scale and the natural scale.


diagram showing difference between equi-tempered and natural scales



The notes in the equi-tempered are marked by upward arrows and the notes of the natural scale are marked by downward arrows. The notes of the equi-tempered scale come from 12 equally spaced semi-tones over the octave - whereas the notes from the natural scale come from the precise ratios that the human ear can recognise.
            You can see that the notes ‘ray’, ‘fay’ and ‘soh’ are almost exactly agree with each other. But the notes ‘me’, ‘lah’ and ‘tee’ differ by the ‘comma’ (the ratio 81/80). This error in the equi-distant scale is not very large and it can easily be ignored. But a slow tune played or sung in the natural scale will give the listener a better feeling of perfection. This is partly the reason why harmony-singing groups should sing unaccompanied.
            Now you can see that the natural scale accurately includes all the intervals associated with the ratios 2, 3/2, 4/3, 5/4, 5/3, 6/5 and 9/8). Thus this scale could be naturally extended to include the notes associated with the intervals 7/6, 7/5 and 7/4. These three notes then form a minor chord (i.e. they are in the ratios 6/5 and 5/4). So these notes can be could be regarded as a “lost chord” because they are chord and are all related to the key note. Thus it would be interesting to see if the human ear could recognize these intervals and so we could have music in the natural scale of a more sophisticated form. However you can also see that these notes are nowhere near any equi-tempered notes and so there is no possibility that such music could be played on equi-tempered instruments like the piano, guitar or organ.

So this is my “Lost Chord”. I don’t think this idea will ever be used in practise. But it is an interesting idea.

{The “lost chord” term comes from the poem called “A Lost Chord” by Adelaide Proctor. But in this poem the chord is played on the organ. Thus there can be no connection between this idea and the well-known reference.}



You might now also like to look back at:

either my "Home Page" (which introduces this whole website and lists all my webpages),

or  “Bryden’s Inventions(which introduces this major set of webpages).


My next normal webpage is More Efficient Road Systems.


Updated on 17/11//2016.


































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