What is Physical Basis of Tuning Note in Western Musical scale

  • #1

Main Question or Discussion Point

Apparently extensive music theory of European, Hellenistic and ancient world is beyond my current depth. However, it seems reasonable to expect that western musical notes have a substantive physical basis in addition to relative consistency and obvious physical parameters.

Obvious physical parameters include average range of human ear and voice, and sonic nature of materials used to build musical instruments. For instance, typical peak resonance of animal gut strings might explain common variations in tuning note for various common instruments. Less obvious physical conditions might also determine peak resonances, things like weather conditions, common building materials, room sizes and echo factors.

Less obvious physical parameters might include average human heart rate at rest or dancing, or sound resonance in bodies of listeners, in addition to obvious resonance of singers' own head and chest register. All of those factors and similar physical attributes might explain much of the background physics of music in ways that tend to limit and guide choosing favorable numbers of cycles per second (cps) for important notes.

However, even given a natural world that might give certain numbers of cps more sonic resonance, the music world still seems to go far beyond obvious resonance in imposing on itself mathematical consistency, for example, in relative tuning.

In choosing substantive tuning note for relative tuning, as well as other "raw" number definitions, a widespread assumption of the importance of mathematical consistency in western music does not seem completely explained by needs for harmony among contemporary musicians and recording and reproducing musical work.

It is a widely-held assumption that the mathematical patterns in Hellenistic music are expressions of meta-musical concepts. Music is used as a medium to express observed mathematical relation. But searching out mathematical patterns similar to relative patterns of the musical scale involves coordinating many variables at once, in order to derive relative relations, and also comparing them in an equally complex process of discerning other patterns of relative relations. Kepler and others have tried complex comparisons of relative patterns whose internal relations are similar to western musical scales. Yet the simple and obvious assumption that musical scales have a physically determined independent reference should more easily be shown by seeking only one simple independent variable.

To begin the search, it seems a reasonable method to choose the most independent variable, the tuning note used to set other notes of the scale. Two major examples are Middle C and A' over Middle C. The cps of Middle C has been discussed as an expression of binary counting, in that the usual Middle C is 260 cps, which is approximately very close to two-to-8th power, 256. However, for our purpose here it is not enough simply to describe a similar mathematical relation, like central number in a counting system. That may be in some ways similar to eight note scales, but it is still not a physical basis for 256 cps as a tuning note.

We might try to find a physical basis for Middle C as trained binary counting by musicians. For instance, maybe there is a one-second clock in the average resting heart rate, (assumed to be around 60 beats a minute,) and the computation process of tuning to Middle C is a quick approximation of counting "One, two, four, eight, sixteen, thirty-two, sixty-four, 128, 256," in one heartbeat. That assumes that the musician can be trained to count the cps of the tuning note by comparing it to the "clock" of the average resting heartbeat. That theory assumes the ability of a human ear to register sound in binary "click" unit of one sound cycle "on-off," and the ability of musician's brain to double that count eight times in one heartbeat. That is, the ear measures the speed of one click cycle, and the brain registers how many times that cycle duration is doubled during an average heartbeat.

That tuning theory for Middle C assumes an internal clock process. I am more interested in an external clock process, in addition. For instance, the observed speed of the sun through the sky, (observable directly and by shadows, etc...) may be a basis to derive tuning A'=440 cps. The length of one cycle of the 440cps sound wave seems to be approximately ground distance covered by the speed of the sun during that duration, one-440th of a second.

At this point I have to check the computations of Earth circumference at the Mediterranean latitude, Earth rotation speed, wave length of 440cps sound-wave. And if it turns out that 440hz is a wave length approximately equal to ground distance covered by observed solar movement during that time duration, we may suspect that speeds registered in other senses may be timed in a musician's brain by musical tone. The mile seems coordinated with Hellenic distance and geographical measurements so I will use it (mile = 1 longitude second.) So far I have solar speed as 900 miles per hour, divided by sixty minutes, divided by sixty seconds, equals a quarter-mile per second. 5,280 feet per mile = 1,320 feet per second. 1320 divided by 440 = 3 feet, that is, using those numbers, the sun apparently moves 3 feet per cycle of the 440cps tone A'. the length of the 440hz sound wave is 2.568 feet according to "www.mcsquared.com/wavelength.htm" applet calculator, which is labelled as based at sea level 20 degrees Celsius. Could 440cps be the result of a similar formula done by ancient mathematicians using different estimates?

It seems that 3 feet sound wavelength is approximately 365cps. (3.095 feet) Is that somehow related to 365 day year and 360 degree/60 mile per degree solar orbit?
 

Answers and Replies

  • #2
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The frequency of wind instruments (e.g., clarinet) is nearly completely independent of altitude (speed of sound independent of altitude), so it can be used at 9000' and at sea level. But there is a slight dependence on temperature and humidity. However, if a note is held too long, so that the CO2 concentration inside the instrument gets too high, the velocity of sound, and the instrument's tune, may change. See
http://en.wikipedia.org/wiki/Speed_of_sound
What exactly the western musical scale: fifth, fourth, major third, minor third, etc.
 
  • #3
Hurkyl
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...

To begin the search, it seems a reasonable method to choose the most independent variable, the tuning note used to set other notes of the scale. Two major examples are Middle C and A' over Middle C. The cps of Middle C has been discussed as an expression of binary counting, in that the usual Middle C is 260 cps, which is approximately very close to two-to-8th power, 256....
This, and everything following, is bunk, for two reasons:

(1) If you look for coincidences, you will find them; there is nothing meaningful about their existence.

(2) If you look at history, tuning varied widely: taking the figures at this site, there is nearly an entire fifth between the lowest and highest frequencies it lists as being used for the A above middle C. Any hypothesis that depends on notes having very precisely defined frequencies cannot possibly be valid.
 
  • #4
f95toli
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Also, 440 Hz is the international standard but no one is forcing you to use it. In classical music A=440 Hz is mainly used by orchestras in the UK and western Europe, elsewhere (including in the US) most orchestras have their own "standards" which can be several Hz below of above 440 Hz. Some Latin music uses -by convention- A=435 Hz

Baroch music is often performed using A=415 Hz since this is closer to what was used in those times (not to mention that e.g. a Cembalo might quite literally break due to the higher tension if you try to tune it to 440 Hz).

Finally, many guitarrists often tune their instrument down half a step of more, Jimmy Hendrix is a good example. Sometimes this is done on purpose, and sometimes it is just a the result of tuning by ear; e.g. using relative tuning with the "wrong" starting frequency.
 
  • #5
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As said earlier, the 440 Hz pitch is almost completely arbitrary, and any starting point will do. What matters is the distance between each of the 12 notes of the scale. I have studied this area for my dissertation at uni, and so I will stick my neck out and say that nobody really knows whether Just Intonation, Pythagorean Intonation or Equal Temperament etc. is the best tuning for a scale, and in at least Equal Temperaments case, why the number twelve is so special.
 
  • #6
turbo
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Guitar and some other fretted instruments are a special case, particularly in musical genres in which is is common to flat a note and bend up to it or to hit a note a bit sharp and release down to it, all restricted by the placement of the frets - fretless instruments are less restricted. Also, some guitarists are enamored of dramatic bends but are not in shape enough to make them accurately, in time and in tune, so they resort to detuning their guitars to decrease string tension to make the bends easier. I am fond of the tone of relatively heavy nickel-wound strings with wound G strings, especially with vintage-stagger Strat-type single coils. They are heavier than strings generally used by rock/blues players these days, and you have to have better hand-strength and technique to play them well. When I was hosting open-mike blues jams, local and regional guitarists would often hear me play on of those guitars through my home-made Tweed Deluxe clone and ask if they could borrow my gear during the next set. They rarely asked again because they couldn't handle the string tension.

SRV favored heavy strings, too, but he detuned significantly. It works out well, because if a guitarist plays chords in Barre permutations he or she can pretty much play in any key required. Transposition is dead-easy.

If you want to learn more than you ever thought you could use about tuning, find an old piano-tuner and ask him how he likes to tune upright pianos used in honky-tonks vs grands and baby grands used in recital halls.
 
  • #7
Hurkyl
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nobody really knows whether Just Intonation, Pythagorean Intonation or Equal Temperament etc. is the best tuning for a scale,
Best is subjective.

(I believe) just intonation always sounds best*, and I understand that is what you will hear when you listen to any symphony orchestra or similar: such instruments offer great flexibility in what pitches they can produce, and so the performer can produce whatever actual frequency 'e wants.

However, when making an instrument like the piano, such flexibility would require a bajillion keys, which is completely impractical... that's why the whole issue comes up.

*: except, of course, when the composer/performer specifically desires an unusual sound

and in at least Equal Temperaments case, why the number twelve is so special.
Almost surely because a perfect fifth is 7.01955001 / 12 of an octave.

More generally, a perfect fifth is 0.58496250072115619... octaves, and the first few best rational approximations are:

0
1
1/2 = 0.5
3/5 = 0.6
7/12 = 0.58(3)
24/41 = 0.(58536)

(the parentheses are used to denote what part gets repeated indefinitely)

Written alternatively,
2 fifths = 1.16992500... octaves
5 fifths = 2.92481250... octaves
12 fifths = 7.01955000... octaves
41 fifths = 23.9834625... octaves


I don't know if it's related, but I believe Hindu music uses 40-ish tones....
 
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  • #8
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Best is subjective.
Not necessarily.

(I believe) just intonation always sounds best*, and I understand that is what you will hear when you listen to any symphony orchestra or similar:
Hmm... I've not really heard that in anything I've read. At most, there will be a small amount of stretching of all the notes (bass becomes flatter, treble becomes sharper). I'd love a good reference if you have one available.

(I believe) just intonation always sounds best
And to my ears and many others, the reverse is actually true (apart from for building the timbre of an instrument where simple ratios are best of course). I'm not sure of your experience, but there are a few possible reasons why you may prefer Just Intonation, and the idea that our minds may be ultimately hearing different things at the very last stage of the audio/hearing process, is just one possibility. (Another one may be because certain instruments introduce acoustic distortion with ET, but not with JI, which is of course not a fault with ET, as that problem can be avoided with an ideal instrument or synthesizer).

Almost surely because a perfect fifth is 7.01955001 / 12 of an octave.
The fact that 1.5 approximately equals 2^(7/12) may be just coincidence, as they're not identical numbers. Of course as you may know, building the scale up through a series of Pythagorean fifths twelve times will result in a note almost a quarter of a semitone higher than unity. So although what you say is a bit of evidence, I would stop far short of calling it 'proof'.
 
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  • #9
chroot
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From another thread:

chroot said:
The notes were chosen so that they produce pleasing sounds when overlapped. For example, a major third interval corresponds to a frequency ratio of 5:4. All pleasing chords are based on these integer frequency ratios. (The octave is a ratio of 2:1, the perfect fifth is a ratio of 3:2, etc.)

The simplest tuning systems, like those of Pythagoras, made use of the 3:2 ratio. As it turns out, a stack of twelve 3:2 steps is almost exactly equal to a stack of seven 2:1 steps. There is a slight difference, however, and it created some very real problems. A wide variety of solutions blossomed, each a trade-off in some way. Western music eventually settled upon the equal temperament system, which fudges the ratios a bit to make everything fit mathematically, while rarely subjecting the listener to noticeable dissonances.

The point of reference for the entire system, the A above middle C at 440 Hz, is a historical choice.

- Warren
 
  • #10
Hurkyl
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Hmm... I've never heard that in anything I've read. At most, there will be a small amount of stretching of all the notes (bass becomes flatter, treble becomes sharper). I'd love a good reference if you have one available.
I think you misunderstand what I mean; the orchestral performers will adjust the frequencies of notes on the fly to produce the chords and intervals they want to hear.

If a woodwind quintet finds itself playing an A-major chord and for whatever reason the bassoonist plays an A at 112 Hz, then the french horn player will most likely match it with an A at 224 Hz, the clarinetist's E at 336 Hz, the oboist's C# at 560 Hz, and the flautist's A at 896 Hz -- no matter what frequencies the instruments are actually tuned to play those notes at.
 
  • #11
chroot
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Hurkyl's right. Chords in just intonation sound "perfect," since the wavelengths overlap exactly -- three wavelengths of one note for every two of another, for example.

A musical instrument made out of rigid material like brass cannot play just intonation in every key; only one key is made "perfect," at the expense of the others. That's why one might speak of using an A or a Bb clarinet.

As Hurkyl says, good musicians can slightly change the pitch of their instrument with their mouths, allowing them to correct for the mismatches that necessarily occur when they play in a different key than their instrument's design.

- Warren
 
  • #12
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Hi Hurkyl,
Oh right. Yes, even then, the results from studies are mixed, and even favour equal temperament, at least in more recent studies.

http://www.skytopia.com/music/theory/scale-dissertation.html#studies

Hurkyl's right. Chords in just intonation sound "perfect," since the wavelengths overlap exactly -- three wavelengths of one note for every two of another, for example.
The consonance type for partial matching (JI-like) to obtain perfect timbres is not necessarily the same consonance type as the one for the twelve intervals (general harmony/melody). After all, many ratios (even of the simpler ones) don't match many of the twelve intervals even remotely.
 
  • #13
chroot
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After all, many ratios (even of the simpler ones) don't match many of the twelve intervals even remotely.
Right, that's why only a few intervals are called "perfect."

- Warren
 
  • #14
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Right, that's why only a few intervals are called "perfect."
So Just Intonation manages to hit the fourth and fifth, but not any of the others? Isn't there just a chance that coincidence could be involved there (especially as the fifth is just an inversed fourth) ? In any case, the JI major third is said by many who prefer JI to be the 'real' major third...
 
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  • #15
The cps of Middle C has been discussed as an expression of binary counting, in that the usual Middle C is 260 cps, which is approximately very close to two-to-8th power, 256.
To echo something that Hurkyl was on about... keep in mind that 260 Hz is not a pure number, it's a measurement with units. The numerical value is meaningless. We could easily make it exactly 256, or [itex]\pi[/itex], or the price of Enron stock just by changing the definition of a second.

Although the exact value of Middle-C has varied a bit over the years, the "physics" that's driving it to remain near that value comes from the range of the human voice. If it moved too much, then the great works of music could no longer be performed by our most talented performers. I'm afraid looking for some more cosmic meaning than that, such as the speed of the sun, is really going off the rails.
 
  • #16
atyy
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So Just Intonation manages to hit the fourth and fifth, but not any of the others? Isn't there just a chance that coincidence could be involved there (especially as the fifth is just an inversed fourth) ? In any case, the JI major third is said by many who prefer JI to be the 'real' major third...
Pure numbers based on first few harmonics gets the 8 diatonic notes:

Tonic major triad:
Tonic: 1f, where f is an arbitrary frequency
Fifth: 3f
Third: 5f

Fifth major triad:
Fifth: 3f
Second: 3*3f
Seventh: 5*3f

Fourth major triad:
Fourth: f/3 (fifth below tonic)
Tonic: 3f/3
Sixth: 5f/3

The 12 comes from saying, why not set the fifth as the new tonic? Finding that it almost closes mathematically, and that musically the ear hardly notices if we fudge the maths.

But I'd be surprised if an orchestra plays just - would they tune differently for Beethoven's fifth symphony and concerto?
 
  • #17
thanks for the responses.

No overall criteria were suggested for choosing the tuning note and that note's cps;

yet it doesn't seem the result of a mysterious combination of factors, only accidentally arrived at, or random, or so many subjective inputs so varied and so complex to be practically random. The cps of the tuning notes, and which notes are used, are varied only a little, too consistent to be random. What is so memorable about those two notes, or so physically useful to drastically limit the set of tuning notes, when the entire range from 60 to 6,000 cps is randomly possible?

And why did Kepler, Pythagoras and others put such stock in a higher mathematical plan evident in the scale?

And why carefully choose the tuning notes if any cps is as memorable as any other? Who would think out the best random inputs, if only relative processing matters?

Of course musical notes could intentionally be made only a relative pattern, since scales can set ratios based on any input base note. But that doesn't mean any random input is a memorable note that can be repeated easily under different conditions.

The input note cannot be random. There have to be limits on the input note's cps to efficiently tune by. It has to make a scale mostly in hearing range, the scale has to be playable on available instruments, and we start collecting a large set of natural factors, like acoustic characteristics of environment, average expected players, listeners, etc...

Given that planning the scale inevitably involves a complex set of factors limiting or "choosing" the best cps of the tuning note, isn't it likely that a musically-minded mathematician or astronomer would try to set the ultimately best tuning cps using the best common denominator, like a universally observed natural phenomenon, or a human process universally occurring while detecting tones?

if the human brain naturally remembers a certain tone cps, well that is a natural tuning note. so why should a random set of numbers fit that set of best tuning notes? it seems most likely that the best tuning notes are as determinable as the formulas of the common chord intervals, for example, or any other relative pattern in music. Just that the tuning note is relative to some phenomena in the physical world, and thus all the other notes should be tuned to it, because it represents a good clock event. More, the best tuning notes should use the most universal criteria, and everybody should have immediate access to the phenomenon. Thus human heartbeat, or sun speed are good choices.

I can only expect to check likely criteria that the planners of Helenistic music would use. And it may not be as explicit as calculating speed of a common event. It might be that the tone is memorable because it reminds the musician of a common event, inductively, without being explicitly calculated. Or maybe best tuning notes were only determined explicitly by a wise one of ancient days, and proved sucessful and so were remembered, or were worked out in practice over generations.

Regarding range of variation, significance depends on what clocking phenomenon is suggested. A clocking event to be useful to remember a note under differing conditions should allow some variation.
Apparent speed of the sun as a clocking reminder of cps of A' is not at all locked into one cps number for all time and all latitudes. A predictable variation is sun speed itself, between 700 to 1,000 mph, aproximately. Variation might tend to prove it, if different latitudes consistently use different cps numbers for A'.

We are not looking for mysterious coincidences, rather practical ones. We are looking for the most memorable tones to use as tuning notes and it seems likely that the most memorable tones will match some fast perceptual events. Like how high can be counted between heartbeats, and the speed of the sun over the ground.

Because we are looking for a memorable number to identify a tone, the fact that cps of Middle C is close to 256 is not such a surprise since that number is a counting "milepost." The question is how that number is processed physically to recognize if a tone is close to that cps. Using a 256-based cps assumes that the ear counts by doubling, and quadrupling, etc... a sample until it matches a standard test duration. If the brain reports that the eight doublings are consistently done between standard heartbeats, while counting the cycle, positive result is registered. Eight doublings per heartbeat would be the memorable thing, and the clock phenomena is the brain circuit of doubling, with a steady hearbeat. Once the musician is trained to the importance of eight loops of that brain circuit per heartbeat, all he needs is to connect by ear to the tested tone by accurately detecting the pace of one "on-off" cycle, and input that to his own internal doubling circuit, and tell if his own internal brain circuit is going to beep eight times per heartbeat, and that is "perfect pitch" detection. Sounds easy, right?

I want to work on the sun speed theory. A lot of speculation is already done on 256 for Middle C.

Has someone reported a sun speed derivation in musical tone? Can we get a confirmation that sonic wavelengths are approximately equal to apparent ground leap of sun during same durations, and that speed of sun by that shadow speed calculation is directly proportional to note A', with its standard variations and sun speed standard variations?
 
  • #18
turbo
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Numerology doesn't enter into it at all. The frequencies that instruments are tuned to is a matter of convention. People who design and build instruments do so to optimize the sonic performance of their instruments based on these conventions. If you want to explore how the tuning of instruments affects or is affected by Western styles of music, you would be better-off studying the intervals between notes, scales, modes, etc.

BTW, people playing in small bands (all with tunable instruments) can ignore convention and tune to any frequencies that are practical, BUT once you get into a larger band or orchestra in which you have to accommodate non-tunable instruments, then you are pretty much stuck, and have to revert to the tuning convention used in the design and construction of those instruments. Ever tuned a xylophone?
 
  • #19
atyy
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No overall criteria were suggested for choosing the tuning note and that note's cps;

yet it doesn't seem the result of a mysterious combination of factors, only accidentally arrived at, or random, or so many subjective inputs so varied and so complex to be practically random. The cps of the tuning notes, and which notes are used, are varied only a little, too consistent to be random. What is so memorable about those two notes, or so physically useful to drastically limit the set of tuning notes, when the entire range from 60 to 6,000 cps is randomly possible?

And why did Kepler, Pythagoras and others put such stock in a higher mathematical plan evident in the scale?
If you want musical mathematical nonsense, think about it this way. There is no meaning to the absolute pitch because of translational invariance, a mathematical symmetry. Would you prefer this symmetry to be violated? Pythagoras etc (whose system I actually don't like) preserved the symmetry by talking about ratios, not absolute values.

The reason the above explanation is nonsense is that some things in music do depend on absolute frequency, notably (some aspects of) timbre.
 
  • #20
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MusicPhysics, please listen to what the others say - the fact that Middle C happens to fall around 256hz (256 times a second) is nothing special, since the time unit of a second itself is arbitrary. There are 86,400 seconds in a day, but there could have just as easily been 36,800 or 187,142 if history had gone slightly differently. There are 60 seconds in a minute - that's arbitrary. There are 60 minutes in an hour - again arbitrary. If base 10 had caught on sooner and/or in more cultures, I'm sure we'd be looking at say 10,000, 100,000 or even 1,000,000 'seconds' in a day.

If there is indeed any wonder and strangeness in music, then as turbo-1 said (and I said earlier), the spacings between the notes of the scale (tuning etc.) is a very rich and complex topic that deserves your attention a *lot* more.

Pure numbers based on first few harmonics gets the 8 diatonic notes:

Tonic major triad:
Tonic: 1f, where f is an arbitrary frequency
Fifth: 3f
Third: 5f

Fifth major triad:
Fifth: 3f
Second: 3*3f
Seventh: 5*3f

Fourth major triad:
Fourth: f/3 (fifth below tonic)
Tonic: 3f/3
Sixth: 5f/3
Here's a nice picture I knocked up which 'completes' the set:
http://www.skytopia.com/project/scale/ji5.html

However, f/5/5, or 5*5, (both 27 cents (1/4 of a semitone) off their nearest ET equivalent) or even simpler ratios such as 3/5 or 5/3 (16 cents off nearest ET interval) sound strange to my ears, and many others.

Perhaps even more importantly though, the notes of the scale can be defined in other ways such as 2^(n/12) which of course ia equal temperament, and for certain intervals such as the major third, 3*3*3*3 is battling JI's 5 as the 'real Mccoy'.

Apart from the elegant 2^(n/12) where the 'specialness' of the number twelve is a mystery, each of the other methods to define the intervals of the chromatic scale gets 'close' on a number of occasions, but is also flawed in some way (such as the close, but not exact intersection of the octave after 12 fifths, or how further extrapolation of intervals from 5-limit Just Intonation (such as 5/5/3) produces really unpleasant sounding intervals).
 
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  • #21
atyy
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MusicPhysics, please listen to what the others say - the fact that Middle C happens to fall around 256hz (256 times a second) is nothing special, since the time unit of a second itself is arbitrary. There are 86,400 seconds in a day, but there could have just as easily been 36,800 or 187,142 if history had gone slightly differently. There are 60 seconds in a minute - that's arbitrary. There are 60 minutes in an hour - again arbitrary. If base 10 had caught on sooner and/or in more cultures, I'm sure we'd be looking at say 10,000, 100,000 or even 1,000,000 'seconds' in a day.

If there is indeed any wonder and strangeness in music, then as turbo-1 said (and I said earlier), the spacings between the notes of the scale (tuning etc.) is a very rich and complex topic that deserves your attention a *lot* more.



Here's a nice picture I knocked up which 'completes' the set:
http://www.skytopia.com/project/scale/ji5.html

However, f/5/5, or 5*5, (both 27 cents (1/4 of a semitone) off their nearest ET equivalent) or even simpler ratios such as 3/5 or 5/3 (16 cents off nearest ET interval) sound strange to my ears, and many others.

Perhaps even more importantly though, the notes of the scale can be defined in other ways such as 2^(n/12) which of course ia equal temperament, and for certain intervals such as the major third, 3*3*3*3 is battling JI's 5 as the 'real Mccoy'.

Apart from the elegant 2^(n/12) where the 'specialness' of the number twelve is a mystery, each of the other methods to define the intervals of the chromatic scale gets 'close' on a number of occasions, but is also flawed in some way (such as the close, but not exact intersection of the octave after 12 fifths, or how further extrapolation of intervals from 5-limit Just Intonation (such as 5/5/3) produces really unpleasant sounding intervals).
Within common practice western harmony, the idea is not simple ratios. The idea is that if you choose some frequency f as a tonic, because instruments produce complex tones instead of pure tones, playing the complex tone f by itself produces the pure tones 1f,2f,3f,4f,5f,6f,7f,8f ... Usually the higher harmonics are weaker, so taking the strongest harmonics (first 5 or 6) produces the tonic (1f,2f,4f), fifth (3f,6f) and third (5f) - this is how the major triad is defined. The diatonic 8 notes (7 excluding the octave) are major triads defined on the tonic and the fifth above and below the tonic.

The 12 notes are an approximation which allows modulation so that you can use 12 sets of 8 diatonic notes, where the 12 tonics are related by fifths. People have tried keyboards with either 41 or 53 keys to the octave (see Hurkyl's calculation in an earlier post for the significance of 41), but those didn't become popular.

Essentially, ET 12 allows the circle of fifths to be used with the 8 notes of each diatonic scale.

There are instruments which produce anharmonic (ie not 1f,2f,3f,4f,...) complex tones (some gamelan instruments), and it has been postulated that the scale there is different because of the anharmonicity of single notes.
 
  • #22
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so taking the strongest harmonics (first 5 or 6) produces the tonic (1f,2f,4f), fifth (3f,6f) and third (5f) - this is how the major triad is defined.
Yes, 4:5:6 (which is 1 : 1.25 : 1.5 normalized or: Root, M3rd, P5th) uses those harmonics to define the major triad. However that is still a simple ratio is it not? Also, doesn't it seem slightly suspicious that the minor triad isn't defined very well using the harmonic series?

But getting back to the main point, I can just as easily define the major triad thusly:

2^(0/12) : 2^(4/12) : 2^(7/12) ----- which normalizes to 1 : 1.2599... : 1.4983... (equal temperament)

or even...

3^0 : 3^1 : 3^4 ------ which normalizes to 1 : 1.26562 : 1.5 (Pythagorean intonation)

Both triads differ to varying degrees from the 4:5:6 definition.

But as I've said previously, who's to say which version is necessarily the 'correct' one? It might just be the Just Intonation (/ harmonic series) defined ones (4:5:6) which is the 'compromise', especially as the studies I have posted earlier seem to prefer equal temperament. (By the way, I'll say it again, but just to clarify, the harmonic series is *great* for building the timbre of a particular note/instrument, but not necessarily for the twelve notes of harmony).

ET numbers like 1.2599 and 1.4983 may look 'ugly' compared to their JI equivalents, but on a logarithmic scale, these numbers are of course very elegant.
 
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  • #23
atyy
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Yes, 4:5:6 (which is 1 : 1.25 : 1.5 normalized or: Root, M3rd, P5th) uses those harmonics to define the major triad. However that is still a simple ratio is it not? Also, doesn't it seem slightly suspicious that the minor triad isn't defined very well using the harmonic series?

But getting back to the main point, I can just as easily define the major triad thusly:
2^(0/12) : 2^(4/12) : 2^(7/12) ----- which is 1 : 1.2599... : 1.4983...
Only the root is the same of course, the major 3rd and fifth differ very slightly to the 4:5:6 definition.

But as I've said previously, who's to say which version is necessarily the 'correct' one? It might just be the Just Intonation (/ harmonic series) defined ones (4:5:6) which is the 'compromise', especially as the studies I have posted earlier seem to prefer equal temperament. (By the way, I'll say it again, but just to clarify, the harmonic series is *great* for building the timbre of a particular note/instrument, but not necessarily for the twelve notes of harmony).
The minor triad is quite easily defined once you have the major triads on the tonic, fourth and fifth since those give you all the notes in the scale. The minor triad is the triad on eg. the sixth degree of the scale.

The problem with thinking of triadic harmony off the ET scale as primary is that there's nonthing special about the major/minor triad, or even a triad - in fact, if the ET scale is primary, one should go straight to the most chromatic Bach or Schoenberg. The ET scale is fantastic of course - no Bach, Beethoven or Schoenberg without it, but there is a special place for the simple 8 notes of the diatonic scale, rather than considering all 12 notes of the chromatic scale straightaway.
 
  • #24
atyy
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Oh, the minor third is also present in the major triad. Say the major triad on C: CEG. CE is a major third, EG is a minor third.
 
  • #25
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Oh, the minor third is also present in the major triad. Say the major triad on C: CEG. CE is a major third, EG is a minor third.
Fair enough, but then that's more like Just Intonation then (6/5 being the ratio there). I was thinking about strictly using one number from the harmonic series as the basis for the minor third, but of course, no such number exists (as no number will make 1.2 when halved repeatedly). There's no "4:5:6" equivalent for the minor triad (even 11:13:20 is way off for the 5th).


The ET scale is fantastic of course - no Bach, Beethoven or Schoenberg without it, but there is a special place for the simple 8 notes of the diatonic scale, rather than considering all 12 notes of the chromatic scale straightaway.
Okay, so the diatonic scale in equal temperament is defined by 2^(n/12) where n=0, 2, 4, 5, 7, 9, or 11. I'm not sure why those numbers are the way they are.

And JI uses relatively simple ratios to achieve many of the notes of the diatonic scale. But there are still 'rival ratios' and discrepancies which muddy the waters. For example the M3rd could equal 3*3*3*3 or 5/4, and the M6th could be 3*3*3 or 5/3. Also, 15/8 is often used for the major seventh, whilst arguably simpler ratios such as 7/5 or 11/8 don't get much of a look in. If you don't want to use higher prime numbers like 7 and 11, then the previous problem I mentioned still exists (3*3*3 vs 5/3).

I'm not saying that there isn't something to the idea of JI representing the pitches of the scale. But at the least, there are flaws, and when 'rival tunings' such as Pythagorean or ET come into play, they look just as effective at representing the notes of the scale.
 
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