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W. Unruh

Temperaments are ways of tuning the notes on a fixed note instrument (eg, piano, organ, lute, guitar,...) in order to accomplish certain ends in terms of the various harmonic and dissonant intervals. The necessity for temperaments arises because of the conflict between the desire to have certain intervals (especially the perfect fifth, the perfect fourth and the major thirds) be as harmonious as possible while still allowing one to play the instrument in as many different keys (ie starting the scale on as many different notes) as possible.

In the following I will examine what makes certain musical intervals harmonious (discovered already by the Pythagorean school in the 5th century BC) and how this conflicts with the greatest possible flexibility in the playing of fixed tuned instruments. The standard temperament in use today is the equal temperament, in which all semi-tones are the same size, and has sometimes been regarded as the worst of all possible temperaments.

{\titlerm Pythagorean Intervals}

While at the beginning of this course I defined a series of musical intervals by the ration of frequencies, such that an octave (a ratio of 2) is made of of 12 "equal" semi-tones, with each semi-tone up being a ratio of $2^{1/12}\approx
1.059$ higher in frequency than the one below. As a part of this definition of the pitches used to make music, certain intervals were defined. In particular in the so called major scale, the sequence of intervals TTSTTTS (T=tone, S=semi-tone) were used to define the various intervals of the musical scale. The note a tone above the base of the scale (the Do, in the Do-Re-Me nomenclature) is called the major second, the one two tones above the major third, the one two tones plus a semi-tone (or 5 semi-tones) is the perfect fourth, the one 7 semi-tones above is the perfect fifth, etc. But this "explanation" or rather description of the scale gives one very little idea of why in the world musicians would have hit on this slightly weird division of the pitches.

In order to get some feeling for why the scale was organized in this way, we will make use of some physical features of any sounds, especially continuous sounds, which make up the experience we call a pitch. A note with a definite pitch tends to be a sound with a definite repetition in time, a definite period. However such notes are almost never pure sinusoids- they have a more or less complex shape when one views the pressure distribution in the note as a function of time. We have learned that such a complex distribution of pressure can always be represented by a sum of pure sinusoidal sounds, a fundamental, which has the same fundamental period as the sound, and its various harmonics (frequencies which are integer multiples of that fundamental frequency).

One of the most astonishing feature of human hearing is that that huge set of disparate frequencies ( and we know that those different frequencies excite different parts of the basilar membrane, and send signals down different nerves to the brain- almost as if the ear were doing a Fourier analysis of the sound, and reporting to the brain about each of the separate sinusoidal frequencies that make up the complex sound) are experienced by us as a single pitch. Those separate components are amalgamated in the brain as a single pitch, with the different amplitudes changing the "tone-colour" of the note, but not the pitch or the unity of the experience of that sound. Ie, it seems that the brain tried to amalgamate frequencies which are integer multiples of each other into a single sound experience.

{\stitlerm Octave}

This gives a clue as to why the octave therefor would be regarded as somehow fundamental. Two notes, and octave apart, share harmonics. Each harmonic of the higher note is also a harmonic (an even harmonic) of the lower note. One would therefor expect that two notes an octave apart played at the same time would tend to be experienced not as two separate notes, but rather as a single experience, as coming close to being a single note at the pitch of the lower.

Now, this amalgamation is not perfect. Both the inevitable variations in pitch, amplitude, etc of the two separate sources, and the ability of the hearing to remember differences and maintain them even when the overlap of harmonics would tend to blend the experiences, means that we do not entirely blend the two notes. We hear them as separate notes, but also feel them as being unified, of being harmonious. Two singers or two instruments playing an octave apart tend to be regarded as harmonious.

Note that it does not take much of a shift in the pitch of one of the notes to destroy that feeling of harmony. In the example where I shifted the frequency of all of the odd modes of a guitar string by putting a tiny piece of tape at the center of the string, the change in pitch of the odd numbered modes was less than a quarter of a semi-tone. But this was enough to present a very strong impression of two different pitches to the mind, as contrasted with the situation with no tape, where the odd modes and even modes formed a harmonic series. In the latter case, it was almost impossible to hear the sound as two pitches. It was a single pitch, even though nothing had changed in the amplitudes of the various modes (which now formed a harmonic series).

This leads to the hypothesis that when notes share harmonics, the ear will tend to amalgamate the notes, and make the experience of the sound to be much more of a unity, than if they do not share harmonics.

{\stitlerm Pythagorean Fifth}

The second case where harmonics are shared is when the second note is a factor of three higher in frequency than the bottom note, which one would also expect to be a very harmonious note. If we take that note down an octave, the frequency ratio of the higher note to the lower is now 3 to 2 ( ie 1.5 times the lower frequency) and these two notes again share a lot of harmonics. Every second harmonic of the top note is now the same as every third harmonic of the lower note. This ratio is the ratio called a perfect fifth, and is again considered a very harmonious combination of frequencies- not as harmonious as the the octave, but still harmonious. Again, looking at music theory throughout the ages, and looking at music from a wide diversity of cultures, two notes separated by this ratio are considered harmonious.

This fact of the relationship of frequencies (or actually in their case of the lengths of strings required to produce the notes) as whole number ratios for precisely the musical pitches considered to be harmonious was a discovery made by the Pythagoreans in ancient Greece (4th or 5th century BC) and was the key which showed them that the physical world was governed by mathematics.

This perfect fifth defined by the ratio of frequencies of 3 to 2 (ie 3/2) is called the Pythagorean fifth, and was considered throughout the middle ages to be the most harmonious interval outside the octave. Throughout history the fifth has formed the most consistent intervals in melodies and music.

{\stitlerm Pythagorean Fourth}

Once one has the octave and the fifth, another interval comes into being, that which is called the perfect fourth. If we consider the progression from a note, to the one a fifth above it, and then to an octave above the original, we have a new interval between the fifth to the octave. This is NOT another fifth. Instead going from 3/2 of the original frequency to twice the original frequency means that you have gone up a ration of 4/3 in going from the fifth to the octave, a new ratio, and again a ratio of whole numbers. Again, two notes a fourth apart share harmonics- not as many as with a Pythagorean fifth, but now the every fourth harmonic of the lower note is shared with every third harmonic of the higher note. Again we would predict that two notes a fourth apart would again be harmonious, not as harmonious as the fifth (not as many harmonics are shared) but harmonious nevertheless. And again, within the music theory, the interval of the fourth is again considered a harmonious interval.

{\stitlerm Pythagorean Tone}

Our range of notes is now become rich. If we consider a note, a fourth above that note, a fifth above that lowest note, again there is an extra interval between that fourth and that fifth, a ratio which we have not seen yet. This interval is called a second, and is a whole tone. Its frequency ratio is given by the ratio of 3/2 (the fifth) to 4/3 (the fourth) and is the ratio of 9/8. This ratio of 9/8 is called the tone. It is not particularly harmonious. Two notes a Pythagorean tone apart do share a few harmonics, but not very many. (every ninth harmonic of the lower is shared with every eighth of the one a tone above).

How do we fit all this together? In our original investigation we found that six tones fit together to make an octave. Do six Pythagorean tones fit together to make an octave? If we go up six tones- ie go up in frequency 9/8 six times, or $\left( {9\over 8}\right)^6 \approx 2.027$, we find that this is not exactly an octave, it is slightly larger by about 1.3% or about a quarter of a semi-tone. This difference between six Pythagorean tones and an octave is called the Pythagorean comma, and its existence was recognized already by the Pythagoreans. Just as and octave is not evenly divided by two prefect fifths (they differ by a tone), six tones do not make up a octave (they differ by this Pythagorean comma). One solution, used in some forms of music around the world is to introduce a new subdivision equal to the comma, a micro-tone. But Western music has not in general gone this route.

{\stitlerm Pythagorean Major Third and Semi-tone}

However there is another interval. If we go up by two tones, (a major third) we get $\left( {9\over 8}\right)^2= 81/64$. We note that this interval shares almost no harmonics ( the 81st harmonic or the 64th harmonic of a note take it outside the range of hearing for most notes). This interval, the major third, was considered a dissonance, not a harmony in the Pythagorean system used in the west until about the 14th century.

We note that again, two tones do not add up to a perfect fourth. There is again something left over. This amount left over, this difference between a Pythagorean major third and the perfect fourth, is called a semi-tone. Under the Pythagorean system, this semi-tone is the ratio between the perfect fourth, 4/3, and the Pythagorean major third, 81/64, and is thus a frequency ratio of 256/243$\approx $ 1.045. Note that this is significantly smaller than the semi-tone we used of about 1.059. Note also that two of these Pythagorean semi-tones do not make a Pythagorean tone.

In the Pythagorean system, there are different semi-tones, some smaller, some larger, depending on exactly where they occur in the music.

While the Pythagorean system is a rational system, it has problems. We note that since semi-tones differ in size, that two semi-tones do not make a tone, that six tones, or twelve semi-tones do not make an octave, one could easily run into trouble. How does one tune one's instrument, especially if the instrument has fixed tones like the a lute, a viola da Gamba (the family of bowed string instruments which has frets like a guitar) a harpsichord or an organ. In particular what happens if one wants to play starting on a different note than the one the instrument was designed for (lets say because the singer's range of singling does not encompass the same range as the octave the instrument is tuned for)? does one re-tune the whole the instrument each time one changes key? (how does one re-tune the organ, which would require shortening or lengthening the organ pipes?).

{\titlerm Just Third}

The problem became much worse in the late middle ages, when the third became regarded as a harmony. Now the Pythagorean third, a ration of 81/64=1.2656 was not harmonious, but very close to that ratio is the ration of 5/4=1.25. Two notes a ratio of 5/4 apart again share quite a few harmonics ( every fifth harmonic of the lower is the same as every fourth harmonic of the higher note), and our theory that the ear hears shared harmonics as harmonious would again predict that this interval would again be harmonious (not as much as the perfect fourth or perfect fifth, but certainly much more so than the tone). Again the music theory bears this out, because around the fourteenth century the major third came be be regarded as harmonious, at the same time as the concern over the deviation from the Pythagorean system became acute. This third, called the just major third, does not fall into the Pythagorean system at all. And it also makes the problems more difficult. The semi-tone between the major third and the perfect Pythagorean fourth is now the ration of 4/3 over 5/4 or 16/15$\approx $ 1.0667 is much larger than the semi-tone we discussed (1.059) and is much much larger than the Pythagorean semi-tone. This drive to bring the third into harmony (instead of disharmony- remember that the major third is one of the principle intervals used for the various men's voices in Barber-shop quartet singing), simply compounded the problem. Semi-tones were proliferating like mad. The fact that the next ratio or 6/5 is very near what was called the minor third brought in still another semi-tone, the ratio between the major and minor third or 25/24$\approx $ 1.041 which is even a smaller ratio than the Pythagorean semi-tone of 1.045.

At the same time as this was happening, music was getting richer. Musicians began to want to do what is called modulation, in which the piece of music would start in one scale, and part way through switch to another scale (ie have some other note, usually the perfect fifth of the first scale, be the base or Do of the new scale. Instead of boringly (as we would now say) just twiddling around with one scale, and one set of pitches and harmonies, one could add excitement by now using another set of notes and pitch relations as well. But things now were becoming intolerable. Two notes which in the first scale bore some relation ( say between the just third and the prefect fifth) now had a new relation ( that between the major sixth and the octave) of the new scale. If one tuned ones keyboard so all the relations in the original octave sounded good, the relations in this new scale didn't. And one could not simply re-tune in the middle of a piece.

{\titlerm Temperaments}

So, the problem was, what do you give up. You could tune the instrument so that melodies or harmonies played on one scale sounded good, but on some nearby scale they would sound weird or even horrible. Or did you tune so that things sounded somewhat bad on any scale. Throughout the 15-19the century this battle raged, producing a huge variety of tuning systems ( temperaments as they are called). Each tried to strike a different balance between having some of the keys (ie scales based on different base notes) sound good (ie the important intervals having whole number ratios) while other would sound weird.

Even by the 17th century it was known (published by Mersenne, a French musician and mathematician) that one could define a semi-tone so that exactly 12 fit into the octave. This would make the perfect fifth slightly sharp over the Pythagorean, the fourth slightly flat, and and the major third noticeably sharp in comparison with the just third, sharp enough to really bother musicians.

Bach's Well Tempered Claver was a series of pieces written for a harpsichord to show the range possible with a particular temperament which his friend devised ( called Well Tempering) This was not equal temperament. The purpose of these pieces was not, as many music teachers say, to show that each key sounded the same, but rather that each key in this well tempered system sounded different. This temperament tried to preserve the thirds in some of the keys, and the fifths in most, but as a result the thirds and fifths in some of the keys were decidedly out. Pieces written in the former keys tended to sound harmonious, giving a feeling of peace and harmony, while the pieces in the latter keys sounded weirder.

One of the musical tools available to the composers in the 17th to 19th centuries was precisely that the various keys, especially on keyboard instruments, sounded different. You got different moods depending on which keys you used. going to a different key was not simply a matter of starting higher or lower in pitch, the piece actually sounded different. A piece written in C felt different from one written say in C# ( a semi-tone higher).

Around the end of the 19th century, possibly driven by the mass manufacturing of musical instruments, and the unwillingness of people to re-tune their instruments (a constant feature of life before then), the equal temperament came into vogue, and has remained in vogue ever since. This temperament is the one in which all semi-tones are the same, a frequency ratio of about 1.059. In this temperament all keys are the same (except for the absolute pitch of the notes). There is no difference (except pitch) of a piece played in C or C# or F or B. The whole point of the Bach Well Tempered Clavier is lost. The additional emotional impact carried by the different relations between the notes in the different keys which composers relied on in the late 18th and early 19th centuries for music on keyboard instruments is lost. Thirds, again as in the early Middle Ages, have become almost dissonances, they are so far from the harmonious just thirds. Fifths are slightly less harmonious.

Of course, the development of the piano (as we shall see later) hid some of this loss. The different modes of the piano strings are not, as they are for the less stiff brass keys of a harpsichord or the forte-piano (the piano of Beethoven's time), in a nice harmonic relation. A piano note is naturally a harsher, and more dissonant note, all by itself. The frequencies of the modes are not harmonics of the fundamental (they almost are and the lowest modes have frequencies that are very close to being harmonics) and the ear, or rather the brain more or less presents them to us as a single pitch, but also notices that the relations are not harmonic and ``complains" by presenting the experience to us as harsher and more dissonant than for those other instruments. To some extent the harshness of the ``out of tune" fifths and thirds is hidden in the natural harshness of each individual note of the piano itself.

(Note that this is not to say the piano is worse or better- one may want that additional harshness, and may regard it as a feature of the music, rather than a detraction, but however one regards it, this difference is there).

In order to hear the various intervales in the various temperaments, see this web page

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Bill Unruh 2003-02-12

Copyright W G Unruh