Did humans invent music, or discover it?

Factual Questions
1

Dear Fans:

During an attempt to rewrite the Kodaly Method so that I could not only give the Skinnies out behind Devil’s Tower the finger, but get the Von Trapps sent to Dachow, I came to some puzzling musical arithmatacies.

To wit: If there are infinite tones between an octave (which in inself suggests “8”, no?), why and how has the moden tonal frequency structure been divided into 11 sections? (If we declare that the octave is actually the root tone, but double the frequency). 11 is a weird number. A prime number so why not 13? Why not 10? What about the current division of tones makes them “acceptable”, exposure or math?

What gives tonal steps their “finality” or “completeness”? The frequency change between half-steps is a constant, and that suggests math. Does a major scale build and come to a common sense resolution because I have been raised around music, or is something “higher” taking place, like an audible mathmatical word-problem? Human hearing with brains or simple exposure? Pachabel’s Canon in D is one big major scale, with major scales over the top, a classic, but not entirely imaginative. Possibly accepted because of mathmatical “Seasame Streetness”?

I understand that if atonality were appreciated/accepted, TOLERABLE (and therein lies the question) Phil Glass would be Bon Jovi. This isn’t about that. Atonality relies on the current music structure to validate itself.

Could the span inbetween an octave be divided into a different number of half-steps and sound appreciable? If no, is that because the math would be incorrect?

Slide trombones, slide whistles, fretless stringed instruments and Gazoos are all capable of presenting miriad tones, but are only palatable when being presented within the current musical structures. Again, Phil Glass with a Gazoo, or a bullet to the head and shit.

We have a musical structure, major and minor scales, with all the modes in between (which are really all the same scale patterns played from a different starting tonic, right?), that has some kind of mechanical “rightnesss” based on math, right?

So, did we invent music, or discover it? Darwin would problably say we copied birds or some shit. If it were there to begin with, it’s like old as math?

K-to-tha-MixoLydian

2

An Orchestra as a whole, has the ability to play in what ever tone it wishes…as long as it’s in the key of the resident piano weather. :wink:

A-440 is not a strictly enforced, all semitones are inclusive to the ensemble.

3

I don’t have much to add other than that I’ve puzzled over this myself.

Whilst studying intonation patterns in speech at university I did start to wonder about this further. Speech seems to follow a similar pattern to music. Questions end on a higher pitch, and need to be resolved, like music etc. But music seems so much more precise. Not only in pitch, but in rhythm too. I’ve often wondered why exactly 4/4, 3/4, 6/8 time signatures work so well and are so common when others are not.

I’ll be interested in any informed responses to this thread!

4

There are a number of mathematical relationships at work, and the chromatic scale we use has its origin in being able to closely approximate a large number of them. But the Western chromatic scale, and its subsets the major and minor scales, are not the only musical mode in the world. Some musical traditions have more notes in an octave, and some have fewer.

5

Sounds like a YES to me!

IN YOUR FACE MATH!!

6

Both. It was most likely discovered in the way you say, from birds or other sounds they heard or even from sounds they made. But then we invented a lot of the rest–which tones to use, the rules for what sounds good, etc. And this even changed over time.

7

There are a whole heap of questions, and a few misconceptions here.

The western (diatonic) scale has a clear history, and is a bit more complex in origin than the idea of dividing an octave into 11 separate notes. Starting here is starting at the current endpoint - the equally tempered scale. Which is a mathematical construction that is an approximation to the underlying scale tones.

In brief, a scale can constructed from the fundamental ratios: 3:2 (which) gets you a fifth) 4:3 (a fourth) 9:8 (tone) 6:5 (minor third) 5:4 (major third). The common point being using the numbers 3:4:5 in various combinations. Pythagoras did much the same, but used only two and three, with 32:27 and 81:64 being the minor and major thirds. The modern equal temperament simply uses the twelth root of two for everything.

There is clearly an underlying appealing beauty to the idea of musical tones stemming from simple ratios. That the brain finds these ratios consonant is not all that surprising, but also very deep, and not exactly understood. But the question about consonance can be rooted here. Few would argue that there is a clear critical point from which everything else stems.

But there is also a world of pain in these ratios. The problem being that the idea of key modulation is essentially impossible. The ratios work for one, and only one root note of a scale. If you try to move to use a different note in that original scale as the root note for another scale, all the intervals fall into gaps between the notes, and the entire thing is out of tune. However the desire to allow multiple keys meant that things settled down on a notation that took the seven note scale, and rattled it about so that all possible interleavings of notes, were evened out, and you ended up with a 11 note scale. This turned out to work because the ratio 3:2 (the fifth) could be used to cycle though the 11 notes (the cycle of fifths) and would get close enough (mostly) to the note you really wanted. Start on the root, multiply by 3/2, that is your next note of the cycle. Multiply that by 3/2. If the frequency is larger than an octave from the root, divide it by two to pull it back into the octave. Repeat until you have “filled” the octave. (That is the process returns a note close to the root). This process gets you your 11 note scale. The sticking point can be the definition of “filled” as it is impossible for the process to exactly return to the octave. (Unique prime factorisation theorem sees to this.) So you terminate the process when the note returned is close enough to the octave that you don’t care. If you ran the process past 11 notes it fills in the next sequence to give you a 41 note scale. Some music is scored for this scale. You can also stop at 5 notes. A heck of a lot of music is based on that scale.

However only the octaves and fifths are the right frequency, and the fifths only so for the key you started the cycle on. Everything else is slightly out of tune. Equal temperament shifts the frequencies around so that everything is equally out of tune. It is only in equal temperament that the semitones are all the same size. In all other intonations they are of different widths across the scale. By making all the semitones identical in width the instrument has no preferred key, and key modulation becomes possible with complete freedom. At the cost that some intervals are never totally happy. The fourth particularly. Over time people get used to it.

However, for instrument that don’t have a fixed set of intervals (i.e. fretless string instruments) they are free to play the notes on the consonant frequencies, and thus a good ensemble will play the correct frequency for a note in the key it is being played in, and the same note on the page could be played at 11 different frequencies depending upon the key. (More actually, as part of the skill of a good player is to tweak the tone slightly.) This can lead to conflicts, where a written note might actually be reasonably be played at two different frequencies depending upon its role - ie is it part of the melodic line, or part of a harmonic structure. A string ensemble will play a fourth perfectly. It is about the only time you will hear one.

The above is a very very brief, and off the cuff overview. PhD theses can and have been written on the subject. As Chronos points out, there is noting special about the western diatonic scale worldwide. However is is possible to meld most of them under a sufficiently rich structure. One tuning I have not mentioned is Just Intonation, which seeks to cope with many of the problems. It yields such things as keyboards with split notes. There are serious and passionate proponents of Just Intonation.

The fact that simple ratios and primes pop out is not really a surprise. The number of notes is a scale constructed with the circle of fifths will always be prime. 5, 11, 41.

Nit pick. Glass isn’t atonal. Far from it. Minimalist is the usual tag. Most of the atonalists worked in the 12 tone scale. Indeed Webern made it the cornerstone of his work, something which dominated atonalism for most of its time.

8

I’m a little confused here. Where is everyone getting the number 11 from? The chromatic scale (and, thus, circle of fifths) has 12 tones. What am I not getting? I’m not double counting the octave, either.

9

Correct, I think this is (at least for me) fuzzy counting - cycle of fifths gets 11 in addition to the octave, not including it. I need more sleep.

10

Someof this has already been touched on, but there’s a great explanation of why we use the 12-tone equal temperament scale at Twelve-Tone Musical Scale .

*The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent.

Furthermore, for the most important intervals, the fifth (3/2) and fourth (4/3), the approximations are better — within about one tenth of one percent.

Let’s compare the twelve-tone equal-tempered scale to some other equal-tempered scales.

All equal-tempered scales with 14 notes or fewer, except the twelve-tone equal-tempered scale, contain at most only three of the seven basic intervals (including the octave) within one percent.
Several equal-tempered scales with between 15 and 30 notes (notably the 19-tone and 24-tone scales) contain all seven basic intervals, but in none of these scales are the intervals more nearly pure than in the twelve-tone equal-tempered scale.
The 31-tone equal-tempered scale has all seven basic intervals to a good approximation, some with better accuracy than the twelve-tone scale, but the most important fifth (3/2) interval is less accurate than in the twelve-tone scale (218/31=1.495).
The 41-tone equal-tempered scale is the first with a better fifth (3/2) interval than the twelve-tone scale (224/41=1.5004).
The 53-tone equal-tempered scale has all seven basic intervals with a better accuracy than the twelve-tone scale (the fifth is 231/53=1.49994).*

11

The answer starts with the observation that certain frequency ratios, especially 2:1 (“octave”) and 3:2 (“perfect fifth”), are pleasing to humans. AFAIK the reason for the pleasure is not well understood but probably originates as a simple matter of compatibility or resonance in neuron excitation.

In the progression C-G-High C, which combines octave and fifth, the second interval is 4:3 (“perfect fourth”). Other pleasing intervals include 5:4 (“major third”) and 6:5 (“minor third”) which are the components of the progression C-E-G. The early Greeks beginning with Pythagoras are credited with discovering these mathematical relationships, although harmonic music had probably been discovered inadvertantly earlier.

Pythagoras developed this scheme further to have seven notes, but other schemes have been used. The black keys on the piano (bringing the total notes to 12) can vary the scheme, but are used mainly to adopt the Pythagorean 7-tone scale to other starting frequencies (“keys”).

It was Simon Stevin who invented the equal-temperament scale circa 1605. Without that, a piano would need to be retuned for every “key,” but with Stevin’s method the piano remains the same in all keys … and slightly off, with the “major third” ratio 2^(1/4) = 1.1892 instead of 1.2000.

One question I’ve asked here in similar threads, but without an unequivocal response, is:
For special concerts or recordings in a single key, why aren’t pianos or orchestras tuned to the exact Pythagorean values instead of left as “equal-temperament” … or are they?

(ETA: While I was Googling to get Stevin’s date etc., nudgenudge beat me to most of this.)

12

Meant to add - in the 12-tone equal tempered scale, it’s the major and minor third intervals, not the fourth, that are significantly “off”, of the more consonant intervals (and the thirds’ inverses, the major and minor sixths).

13

Yup, sounded wrong when I wrote it was the fourth too, I really need to post when I’m not so tired, or at least check before I post. The thirds are critical harmonically too, so this mismatch is not a trivial issue.

14

As I mentioned. Note that the “perfect fourth” will be exactly as “off” as the “perfect fifth”, due to simple arithmetic.

15

Someone had their Beats by Dr. Dre on a bit too tight during Music Appreciation in college, eh Kapowz?

On the Western side of things, certain harmonies, like a 5th (referred to as the Dominant) sounded, oh, let’s go with “proper and good in God’s eyes.”. There were a few simple intervals that were “allowed” as music evolved via Gregorian chants. Innovative church composers branched out and scales were developed that allowed (oh noes!) five acceptable notes - the pentatonic scale. At the same time, musical instrument tech was advancing, but the spaces/intervals between the notes were still being worked out. One instrument might play in one key and one key only, until Bach showed that one approach to setting intervals - a well-tempered approach (tempering being a word to describe interval distances) enabling instrument to play in different keys with being completely retuned (and which was eventually overtaken by “even” tempering, a different approach to it). My take is that working out Tempering lent itself to a fewer set of notes vs more…

So - I am surmising that a combination of the gradual expansion of “allowable” intervals and harmonies along with advances in tempering and instrument construction were both factors deeply influencing the “allowable” set of notes.

…and shit.

16

Does anyone know, do songbirds (or songwhales :)) follow a scale of some kind, versus just higher and lower pitches, but not necessarily in set ratios?

17

Perhaps this is a nitpick, but the frequency, as measured in hertz, is NOT a constant between half-steps. The RATIO is constant, but as you move up the scale, each higher half-step represents more cycles (hertz) than the last. To human ears, it sounds like the same musical interval, and perhaps that’s what you meant.

18

Perhaps the importance of small integer ratios in pleasing frequency changes has to do with the presence of small integer ratios of sine wave frequencies in the composition of single notes of real physical instruments and real physical phenomena.

19

The reason why consonant intervals are “pleasing” is far better understood than this sentence would lead to believe.

When I was a kid I used to wonder how a single speaker cone was able to produce the sound of several different instruments at the same time. As it turns out, I had it completely backwards. The real mystery is how can my brain perceive several different instruments at the same time, despite the fact that I have only two ear drums?

It turns out that identifying various sound sources from a single waveform, a task often referred to as “auditory stream integration”, is far from a trivial task. We certainly haven’t figured out all the details, and computer programs, although they’re getting better by the day, are still nowhere near as good as we are at it.

The sound information that enters the brain is coded not as a wave form, but as a spectrum. Any complex waveform (sound) can be described as the sum of simple waveforms, or “partials”. This is the basis of most signal processing and the human brain is no exception.

In order to identify sound sources, the brain makes educated guesses as to which partials belong together. Several factors that make partials more likely to “fuse” together have been identified. One such factor is that if two partials are part of the same harmonic series, then they are likely to have been produced by the same source.

When you set a taught string, or a column of air in motion, it will create a waveform that can be described by adding partials whose frequencies are at integer ratio of some fundamental frequency. If the string goes back and forth 100 times per second, there will also be smaller waves oscillating at 200, 300, 400, and 500 times per second (Hz).

Sounds that exhibit this behaviour are a minority, but since our vocal cords are one of the places where they do occur, they are extremely important to us. Hence, any other harmonic sound will trigger our brain circuitry dedicated to recognising voices.

So here be the math: the most consonant interval is the octave. It is a 2:1 pitch ratio. If I play a sound with the partials 100, 200, 300 and 400 Hz, the note an octave above will have the partials 200, 400, 600, 800. Every partial of the second note is also a partial of the first. Even if the sounds don’t completely perceptually fuse together, there will be an important part of your auditory cortex screaming “belongs together!”

The second most consonant interval is the fifth, with a ratio of 3:2. This also happens to be the ratio between the second and third partial. The ratios between successive partials of the harmonic series are 2:1, 3:2, 4:3, 5:4, 6:4, etc. These correspond musically to the octave, the fifth, the fourth, major third and minor third.

These intervals are important musically because they are important perceptually, because they arise naturally in sounds that are important for our survival.

There’s a lot more going on, but that’s the basics.

20

Fascinating. As a music-lover and psychology lit. dabbler, where would one go to read more about this?