Harmonic Interference

I always wondered what was meant by the full tempered clavichord. Without going into a discussion of J. S. Bach and history on the matter, it never seemed to be relevant because I played the trombone. Pitch and tuning were never a problem since the ear could be relied upon for a great sounding musical chord. The slide permitted any vibration frequency, or pitch, that was needed to sound good. Actually, playing with other brass players with valves on their instruments, they could even "lip it" at the mouthpiece to play the correct pitch according their ear.

So some years ago, I started to look into the differences between the 12-tone "tempered scale", and something called the "harmonic scale". On the horn, there are a number of tones that can be played in any slide position by "buzzing" one's lips into a mouthpiece. There is the "pedal tone", which is the fundamental frequency and the first harmonic. Then, the next higher tone is just twice that frequency, being the second harmonic, and is an "octave" above the pedal tone. All of the tones are a function of the length of the air column, temperature and pressure, etc. The third harmonic is the 5th tone in the major scale. Next is the 4th harmonic which is two octaves above the 1st harmonic. The next higher frequencies, in terms of scale intervals, are the major 3rd, another 5th, a 7th (this is the flatted 7th interval of the major scale), and then the 8th harmonic which is just an octave above the 4th harmonic. After that, it gets more complicated and you can almost play a complete scale using the higher harmonics. You will see this on the next page.

It turns out that these natural vibrations or tones at harmonic frequencies can be used to construct the "harmonic scale" beginning on a certain note. Chords using these notes, and within the sense of the key based on the "tonic" pitch . . . that is the first tone of the harmonic scale, sound rich and beautiful to the ear. Conversely, the same chords played using a tempered scale can sound a little "tinny". All pianos and organs use the tempered scale. Bach found out that if he tuned his keyboard device harmonically, it was great, except when he changed key. So he tuned the 12 tones by applying a constant frequency ratio for each half step. Setting that ratio to the 12th root of 2, the frequency of each key on the piano increases by 1.059463 times the previous key. If you multiply that number by 12, you get exactly 2 and this is consistent going from the first note, and then crossing 5 black piano keys and 7 white piano keys to the next white key and a frequency exactly twice that of the starting key.

So, the ear is the thing that seems to sort all of this out. But what about the physics of the situation. Having Excel is a wonderful way to study this . . . so I did! it's based on the Bb harmonic and tempered scales.

The following screen shot shows part of a spreadsheet, and a chart. There is also a macro sheet that helps to quickly change conditions when operating this workspace in the Microsoft Excel application.

This plots the sinusoidal pressure of 4 sound waves, each of a different frequency, emanating from the ends of 4 trombones, or any other group of musical instruments. The chord is a Bb7th, so it's a combination of the 1st, 3rd, 5th and flatted 7th tone of the major scale. The green line shows how the combination of the pressures from each tone results in the additive pressure wave that is a characteristic of the chord (bold green line). Notice that this chord "footprint" is very symmetrical and returns to form the next major peak at about a frequency of 59 cps, or about 0.017 seconds after leaving the source for the first full pattern.

In this little control panel, you can see some the choices, but this will be explained later.

Now, for comparison, we can select the same Bb7th chord but this time we will generate the sinusoidal pressure "footprint" using the notes from the Tempered Scale. Notice that it's very similar to the Harmonic chord pattern, but there are some differences. First, the pitches are slightly different. Second, the "hills" are lop sided and the whole pattern is asymmetrical. Third, the next peak pressure is generated with individual tone pressures that are not lined up, thus causing a slightly different shape. This is a visual description of a Bb 7th chord that would really hurt the ears compared to a harmonically produced chord by a group of musicians. The ear seems to naturally adjust the frequencies for harmonic compatibility. Trombones (use primarily harmonic pitches) and string instruments (use primarily fundamental pitches) are perfect for this since the length of the vibrating string or air column is used to produce the 12 tone scale. Other instruments that depend on specific configurations of valves or finger holes require more subtle and complicated adjustments, but can be done effectively by skilled musicians.

I'll give you the scale information next, a couple of other examples, and the file with operational buttons that you can download to explore other chord combinations using Excel on your own computer. Also, I'll provide a little instruction how to operate the file.

Harmonic Interference
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