On the Principle of Relativity of Music

Yeo Doglas (Singapore, Independent Researcher)

Published in musi.philica.com

Abstract
This article attempts to apply the Principle of Relativity to music, and tries to explain two phenomena (perfect pitch, and characteristics of keys) with it.

On the Principle of Relativity of Music (PRM)

Contents:
(i) Principle of Relativity of Music (PRM)
(ii) Applications of PRM
(iii) Potential Flaws of PRM
(iv) References

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(i) I will first quote the physical Principle of Relativity by Galileo (1564-1642): "Without external reference, it is not possible to distinguish between a stationary body and a body moving at a constant velocity".

We will now attempt to investigate whether the Principle of Relativity applies to music as well, in particular to the pitch of musical chords.

Now, the stationary body referred to by Galileo could be replaced by a "stationary chord" in this case, ie. a chord that does not "move".

A body moving at a constant velocity could be replaced by a chord "moving" at a fixed amount for all the notes in the chord, such that the chord is shifted but its nature (major, minor, etc) remains unchanged. An example would be the movement of the chord CEG to FAC. In this case, all the notes in the chord have moved by a constant amount—5 semitones. The major nature of the chord also remains unchanged.

Hence, we can formulate the Principle of Relativity of Music:"Without external reference, it is not possible to distinguish between a stationary chord, and a chord moving at a constant velocity". In effect, we have just assumed that a musical chord can be considered as a physical body, an assumption which may or may not be justified.

A valid theory must be able to make predictions that coincide with real phenomena. So let us explore the predictions that the Principle of Relativity of Music (PRM) would make.

Assuming that the PRM is true, it would mean that it is not possible to distinguish between the two cases:

Case 1) a C major chord, followed by another C major chord (a stationary chord)
&
Case 2) a C major chord, followed by a D major chord (a chord "moving" at constant velocity)

However, commen sense and experience tells us that it is very possible to distinguish between the two cases mentioned above. In fact, it is nearly impossible not to tell the difference, unless one is tone deaf! This seems to show that the PRM leads to a reductio ad absurdum.

However, a closer inspection reveals that we have left out a key phrase of the PRM — "Without external reference". Thus, perhaps the PRM is not beyond salvage.

In particular, if we define our short-term musical memory (STMM) as external reference, then the PRM begins to make more sense. Using the above example of case 1 and case 2, we can distinguish between them as we have been using our short-term musical memory (STMM) of the first C major chord, to compare with the second D major chord that we hear, and that is counted as external reference.

Short-term musical memory (STMM), by definition, is only for a short period of time. Hence, to eliminate the possibility of external reference to STMM, let us consider the following cases:

Case 3) a C major chord, followed by another C major chord after a time interval of one day.
&
Case 4) a C major chord, followed by a D major chord after one day.

(Note: one day can be replaced by one week/year if necessary)

Now, most people (including myself) cannot distinguish between Case 3 and Case 4, hence showing that the PRM is valid if we consider our STMM as external reference.

However, defining STMM as external reference seems to be somewhat arbitrary. This will seem less so if we consider the fact that when analysing a musical chord, anything not belonging to that chord can be considered as external, and that includes our musical memory of the previous chord.

Lastly, as of this sentence, PRM seems to work for all chords, and the words "constant velocity" seem to be redundant. However, this is not so. The main importance of "constant velocity" is to ensure that the nature of the chord remains unchanged. (ie. a major chord remains a major chord, a minor chord remains minor, etc.)

If the nature of the chords were to be changed, we would be able to distinguish between them, even without referring to our STMM. This is done by referring to our long-term non-musical memory (LTNM). (eg. verbal memory of the word "major", etc.)

Why is short-term musical memory (STMM) considered external reference, but not long-term non-musical memory (LTNM)?

First of all, I will make an assumption that there is no such thing as long-term musical memory, regarding pitch. I do not refer to the memory of ntoes (such as memorising a Beethoven Sonata, which is possible), but to the memory of the exact pitch or frequency of the notes, which is highly impossible. If it were possible, there would be great arguments over whether the orchestra is tuned to A=440 Hz or A=442 Hz, since one or the other will not match the long-term musical memory of the audience. Experience tells us that, most people cannot tell the difference between the two.

Long-term non-musical memory is not considered as external reference, because in this case, we are only concerned with the pitch of the notes, which the above paragraph shows that it should be "stored" in the short-term musical memory.
Hence, a person upon hearing a C major chord followed by a D minor chord (after a time interval of a day) will be able to tell the difference by reference to his/her long-term non-musical memory (LTNM) of the nature of the chord, and that is not considered as external reference.

Hence, we have shown to some extent, that each part of the PRM appears to be valid.
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(ii) Applications of PRM

If we accept that PRM is true, we can use it to explain several phenomena:

1) Firstly, PRM seems to be conflicting with the fact that some individuals have "perfect pitch". If we assume that PRM is true, then perfect pitch must be due to some form of external reference, to short-term musical memory or otherwise.

I can think of 3 factors that may help perfect pitch:
a) Length of short-term musical memory (STMM)
An individual with a longer STMM may be distinguish between pitches more easily, as they can "store" musical memories of previous chords for a longer time for reference.

b) Refresh rate of STMM
A person who is constantly in an environment with perfectly tuned music (eg. a musician) will have his/her STMM constantly refreshed. This partially explains why the majority of people with perfect pitch (eg. Mozart) happen to be musicians.

c) Reference to resonance.
A person may also use the phenomena of resonance (of his body) to determine the exact pitch of a note.

2) Secondly, PRM can refute the old theory that different keys have different characteristics, for example in the English translation of Helmholtz's Tonempfindungen:

C major - Pure, certain, decisive; expressive of innocence, powerful resolve, manly earnestness and deep religious feeling.
Db major - Fulleness of tone, sonority and euphony
E major - Joy, magnificence
E minor - Grief, mournfulness, restlessness
F major - Peace, joy, light, passing regret, religious sentiment
F minor - Harrowing, melancholy
F# major - Brilliant, very clear
Gb major - Softness, richness

By PRM, the above statements have no validity, and at the very most only exist in the minds of the composer. This is in part proven by the fact that Plato associated C major with "sorrow, weakness and self-indulgence, while Helmholtz associates it with brightness and strength", an obvious contradiction.

(iii) Flaws of PRM

1) Consider the case that a chord "moves" at such a high "velocity" that it enters the ultrasonic region. PRM would not work, since the human ear cannot hear notes above 20 000 Hz. A theory must be able to work for all cases, so PRM's failure in this case may be significant, although this case is rather extreme. It could be fixed though, by adding the phrase "within defined limits" to PRM.

2) The human memory cannot be just categorised into two discrete categories— short-term and long-term, as in some cases there may be overlaps. Hence, the explanation regarding STMM and LTNM may be a case of oversimplification.

(iv) References

1) "Science & Music", Sir James Jean
2) "Einstein", Joseph Schwartz and Michael McGuinness