Why do we use geometric Progression?

Question

It seems what's important in the progression of the frequencies in notes is...the difference in frequency of any two notes is not the same, but the ratio between any two notes of the same distance is the same. For example, the difference in frequency between C1 and D1 is not equal to the difference between C2 and D2… however, D1/C1 = D2/C2 In other words, D1 - C1 does not equal D2 - C2… however D1/C1 equals D2/C2. First of all, does this seem accurate?

If i understand that first part correctly... why was it laid out tat way? What is the significance of making sure the ratios are ll the same? I'm assuming it's related to scales, intervals, harmony etc... I just can't wrap my head around it. Thanks!

Answer 1

The fundamental reason we define intervals as pairs of pitches with a certain ratio (e.g., 3/2), rather than a certain fixed frequency difference (e.g, 100Hz), is that pitched instruments create overtones. In a simplified picture, if you pluck a guitar string, you will get both the fundamental frequency (let's say, 220 Hz), which corresponds the entire string oscillating back and forth. You will also get and multiples of it: twice the base frequency (440 Hz), which corresponds to two halves of the string oscillating in opposite directions, and the point in the middle staying at rest; three times the fundamental frequency (660 Hz), where the string is divided into three parts oscillating in opposite directions to their neighbors, etc. Similar things happen with the oscillating air column in wind instruments.

The absolute frequency does not matter here (within the range of the instrument) - a guitar string sounds (pretty much) the same whether you play it at 220 Hz or 180 or 270, because the frequency ratios and amplitude ratios of the overtones to the fundamental frequency are the same.

So the sound of each pitched instrument is a combination of all these frequencies, or overtones. If you combine two of these pitches, their overtones also add up - some are the same frequency, so they reinforce each other; some have frequencies in complex ratios, which tend to sound dissonant. Again, how the overtones add up is independent of the absolute frequency - you get pretty much the same sound impression if you play notes at 220 and 330 Hz, or at 300 and 450, because they're at the same ratio of 3/2, and their overtones stack up the same way.

The rest of western music theory (which notes to use together in scales, how to combine them into chords, which chords sound consonant or dissonant...) is basically a very elaborate heuristic (rules based mostly on experience) on how the combinations of overtones combine and how these combinations are perceived by the listener, mixed with habits developed during centuries of a certain musical tradition and the constraints of the instruments commonly used in that tradition.

Answer 2

It was laid out that way, because there was no choice due to the nature of sound waves, thw workings of our ears/brain combination and the intercultural agreement, that octaves are the basis on which to operate.

You somehow seem to think, that geometric is complicated and artifical, which is wrong in most of contexts related to physiology and perception:

• A salary increase of 10 $ per week has a different effect for one earning 100 $ a week as for one with 1000 $ a week
• If you want to get music noticably faster, it may not suffice to add 5 bpm, especially if you are already at 160. (This is the reason, why the increment on a mechanical metronome grows with higher settings.)
• the decibel scale for volume is misleading, since it is logarithmic. Actually you have to add energy proportional to existing level to become noticable in the same way.

Answer 3

The main reason is a physiological one. It's all in how our inner ear percieves sounds.

Play an octave. Our ear percieves them as virtually the same sound (by the reasons exposed in other answers above). Musically, nothing is added by playing an octave above the root note you're playing. Yet, in physics, an octave is represented as a proportion of frequencies (ie: double) as opposed to a fixed amount (ie: +200 Hz). This is not something we "invented", but something we discovered.

Many people seem to struggle with logarithms and geometric progressions yet this kind of progressions are quite common in physics.