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Chip Booth
Jul-21-2005, 12:27pm
I have asked this question to a number of learned people and surfed endlessly but I have never found an answer as to where the major scale comes from. Is it arbitrary, is there some mathematical formula?

I teach music and I tell my students that this is a like a geometry proof, something must be "given", and that is the major scale. If you accept that the scale exists everything else makes sense...

Chip

glauber
Jul-21-2005, 12:38pm
Do a Google for Pythagorean tuning, just intonation, etc. It's a mix of what sounds pleasant to the ear, combinations of notes that don't generate secondary beats, and the search for finding simple mathematical formulas for these. Pythagoras gets most of the blame, but the mesopotamians already had something similar going, before him. The whole thing is rooted in the mystical belief that the whole reality is connected, and simple mathematical formulas underlie everything. There was a very strong connection between music and astronomy, for example. Search for "music of the spheres".

Chip Booth
Jul-21-2005, 12:49pm
Thanks, glauber, I'll do some looking. I do know about the Music of the Spheres, even played in a group named that at one point http://www.mandolincafe.net/iB_html/non-cgi/emoticons/biggrin.gif

Chip

arbarnhart
Jul-21-2005, 7:57pm
A text I have contributes it largely to Pythagoras, but has an interesting little footnote that says it is generally accepted that "Pythagoras" is not just one person. Hmmm...

OdnamNool
Jul-22-2005, 12:56am
Facinating stuff, isn't it? #I really am not very learn-ed when it comes to history.

My guess would have been that the major scale was derived from the Gregorian chants of the Medieval times. #Especially, the "Ionian" mode which is like a C major scale. #But... that Pythagorus cat was much earlier, like a B.C. dude, right?

Lefty&French
Jul-22-2005, 4:55am
Facinating stuff, isn't it? #I really am not very learn-ed when it comes to history.
And mathematics, too ? http://www.mandolincafe.net/iB_html/non-cgi/emoticons/biggrin.gif

Pythagoras was a Greek Mathematician born in 569 B.C. who studied math, music, and astronomy.

OdnamNool
Jul-22-2005, 6:14am
Correct. In fact, I don't know much about nuthin. My father is a HUGE Pythagoras fan. He has enlightened me throughout my up-bringing... My dad knows everything about math, and everything else...actually... Yes. I'm bragging... http://www.mandolincafe.net/iB_html/non-cgi/emoticons/smile.gif

OdnamNool
Jul-22-2005, 6:46am
Oh, Lefty&French!

I just read my reply, and I hope it was not mis-interperated! What I meant was simply that, yes, it's correct that I am un-learn-ed in history... and math too! But, yeah... I know a little about Pythagoras. That damn theorum was drilled into me! http://www.mandolincafe.net/iB_html/non-cgi/emoticons/biggrin.gif

glauber
Jul-22-2005, 6:57am
One interesting thing is that the relations between notes were agreed on much, much earlier than the specific pitches. For a long time, pretty much each locality had one or more tuning standards. Not much of a problem for string players, since they can always retune, but wind players who travelled had to carry multiple instruments to accomodate. A=440Hz was officially established in 1939. Before that, tuning had varied by as much as a whole tone down or up, generally starting lower and gradually drifting up. Even today there is a tendency to make instruments in A=444Hz, for example, in Japan.

Flowerpot
Jul-22-2005, 9:29am
(Nerd alert)

My own theory: the natural vibrational modes of vibrating objects have harmonics. Everything from stringed instruments to bells to wind instruments will produce a series of harmonics, given by multiples of the fundamental (or lowest note). Ok given that:

Harmonics which fall of multiples of 2 sound like the same note to our ears. 2x, 4x, 8x the fundamental sound like higher replicas of the original. So far, not so interesting to the ear.

However, other harmonics sound like different notes. Take a guitar or mando string, pluck a harmonic where it sub-divives the string length into 3 segments, and you get 3x the open note. This sounds like a new note.

Now let's figure where these naturally occuring notes fall. Let's tune a string to 100Hz to make the math easy. So 2nd harmonic = 200 Hz, 3rd = 300 Hz, etc. Now find what closest sub-octave of these harmonics would be: 200 Hz reduces to 100 Hz, so like I said, 2x, 4x, 8x harmonics do not make new notes, but fall back on the original tone. How about 3x? It's equivalent sub-octave is 150Hz... hmmm, that's a different note than 100Hz, cause it's not exactly double. 5x makes a note with an equivalent sub-octave at 125Hz. What happens when you put together 100Hz, 125Hz, and 150Hz?

drum roll...

A major triad! A 'nice' sounding chord... all produced by one vibrating body.

Now there is a problem, though... if you want an instrument which can play in different keys, you can't use those frequencies exactly. For instance, if you re-tune our string to 125 Hz, the "triad" would fall on 125, 156.25, and 187.5 Hz... note that the 156.25 is not the same pitch as 150, so a single instrument could not make the notes neccessary for both keys without re-tuning it. What we need is an approximation, something that will be close enough to the natural pitches in any key... the "even tempered scale".

So take an octave, divide it into 12 equal parts; equal, that is, in the sence that the ratio of one frequency to the next is constant all the way through. If I apply this to my 100Hz starting point, the pitches I get are:

100 Hz
105.95
112.25
118.92
125.99
133.48
141.42
149.83
158.74
168.18
178.18
188.77
200

The major 3rd and 5th fall at 125.99 and 149.83. Pretty close to 125 and 150, right? The 5th is almost dead on, but the 3rd is just off enough to be noticable. Instruments without frets can adjust to make it sound less even-tempered and more "natural".

This doesn't answer the question about the major scale, just the major triad... but extending the math to the 7th and 9th harmonics gives me equivalent notes (relative to our fake 100Hz fundamental) of 175, and 112.5 Hz, close to the even-tempered flat7 and major2 (ninth). So I reckon the major chord with flat 7, predominant in bluegrass, are indeed "ancient tones", notes which were heard by cave men banging on sticks and blowing on flutes, occuring naturally in a series of harmonics.

glauber
Jul-22-2005, 9:31am
Flowerpot, yes, this is basically it. I believe the early experiments Pythagoras' people did were with vibrating strings. This makes sense because you can see the vibration nodes there, unlike what happens, for example, with a flute.

One more thing: the idea of "consonant" versus "dissonant" intervals have to do with how the vibration nodes of the 2 strings superpose on each other. If they match, then you don't have any of the low frequency beats that we hear as dissonance. You can see this, incidentally, when you tune 2 mandolin strings to unison, by ear. When they're close, you start hearing a beat, then as they get more in tune, there are fewer beats and then suddenly none, and the sound is much fuller - you have achieved consonance.

One of the weird cruelties of the physical world is that you can't have a perfect tuning system and still allow for playing in more than one key, so our modern tuning systems are approximations to the mathematical ideal.

Lefty&French
Jul-22-2005, 9:32am
I just read my reply, and I hope it was not mis-interperated! #
OdnamNool, you weren't! http://www.mandolincafe.net/iB_html/non-cgi/emoticons/smile.gif
But I think you were correct with your Gregorian and medieval times idea. Their chants carry on the greek idea of music (and maths!) during these very troubled times.
My two eurocents,

otterly2k
Jul-22-2005, 10:08am
Not that I disagree with anything in particular that has been said here... but I do want to observe that the major scale as we know it is not the basis for all musics in the world.

Western music has developed around a certain set of ideas about notes and how to divide up tones and times. But there are other systems that divide things up differently. What sounds "right" or "consonant" varies among cultures. (e.g. Balkan music uses a lot of intervals that contain nodes/beats...and that is what they're after!) Some systems (e.g. Indian music) use what we would call quarter tones-- variations in pitch that most of us can barely distinguish in a scale that are as meaningful in their contexts as the difference between a C and a D in ours. Different scales as the bases, etc.

So, just a suggestion that while these tones may be ancient tones, there are others out there as well.

glauber
Jul-22-2005, 10:38am
There are. I bet they have interesting stories too. The original question was how did we arrive at the specific choices that are used in Western music.

otterly2k
Jul-22-2005, 10:53am
true.
meant no harm.

Chip Booth
Jul-22-2005, 12:47pm
Yes, my original question was indeed about western music. I know that eastern music uses all sorts of other tones, any info anyone has is welcome to share, it's all interesting.

Interesting explantion Flowerpot, I buy into most of that. The only thing I see missing is that you never seem to have calcuated a half step. I guess you can extrapolate a half step based on the third-to-fifth interval compared to the whole step intervals you calculated.

I have not yet had time to do my research about Pythagorus yet. It's an interseting coincidence that I use the geometry proof example with my students if indeed the origin lies with fellow(s?) responsible for the theorum.

Chip

glauber
Jul-22-2005, 1:21pm
It's an interseting coincidence that I use the geometry proof example with my students if indeed the origin lies with fellow(s?) responsible for the theorum.
Not really a coincidence. Those guys thought these things were related (http://en.wikipedia.org/wiki/Pythagoras#Scientific_contributions).

sunburst
Jul-22-2005, 1:26pm
I've always sort of thought Flowerpot's explanation was a likely one for the origin of the chromatic scale, but I haven't been able to make the leap from there to the major scale. Maybe it is just as simple as... "it sounded good".

PaulD
Jul-22-2005, 3:41pm
The only thing I see missing is that you never seem to have calcuated a half step
Chip; maybe I'm misunderstanding your statement (wouldn't be the first time), but Flowerpot's explanation calculates half steps rather than whole steps. Applying the pattern for a given scale determines the placement of whole steps (i.e. Major: 100 Hz, 112.25, 125.99, 133.48, 149.83, 168.18, 188.77, 200; Natural Minor: 100 Hz, 112.25, 118.92, 133.48, 149.83, 168.18, 178.18, 200).

Paul

wharfrat
Jul-22-2005, 4:31pm
Notes in the major scale occur in nature as harmonics of a tone. If you pluck an open guitar string the first harmonic is the octave, the next is the fifth, then the major third, minor third... so on.

Pythagoras was just the guy who figured out the ratios for this. Over the last hundred years, these intervals have changed to the even tempered scale, which is different than the well tempered, and harmonic intervals. http://www.mandolincafe.net/iB_html/non-cgi/emoticons/biggrin.gif

Chip Booth
Jul-22-2005, 6:14pm
PaulD, Flowerpot did calculate half steps but that was after the initial "discovery" of the major triad in natural harmonics.

quote:
"So take an octave, divide it into 12 equal parts; equal, that is, in the sence that the ratio of one frequency to the next is constant all the way through"

That is a leap that seems to come with the foreknowledge of the 12 tone chromatic scale rather than "finding" them in the natural harmonics.

Or am I completely confused and missing the point? Wouldn't be the first time. Nonetheless I am learning a lot and getting much closer to the answer I am looking for, for which I am thankful for all the input. I expect that, as John suggested, at some point someone said "that sounds good" and it stuck, but it seems there is a strong precedent in nature.

Chip

billkilpatrick
Jul-23-2005, 5:26am
"it sounded good" gets my vote.

there's an interesting treatise on 5c. oud from the 13th cent. which stipulates that the tuning should be in straight 4ths but it doesn't mention pitch. in trying to imagine a group of medieval players tuning up for a performance i think it was simply a case of the best player going "hmmmmm" and everyone else twisting their pegs accordingly.

that "hmmmm" note is the key to the whole subject, i think. i sing "my-dog-has-fleas" to tune my charango and oud and manage to get it right more often than not.

does the human voice have a primal key or pitch? would something a child might say in any part of the world - something like "nyah-nyah, nyah, nyah-nyah" - be closer to one key than any other, in all cases?

assuming that the major scale is the original, i think it evolved from sounds the human voice makes, not vice versa.

interesting thread - bill

PaulD
Jul-23-2005, 1:59pm
Chip, it does look like I misunderstood your statement. Not that I know what I'm talking about here, but what others have said about the general scales/intervals/pitches being "discovered" first and then "scientists" (mathemeticians)working out ways of describing those intervals with formulae to transpose them seems logical. (pardon the run-on sentence, but this keyboard sucks and I'm not rewriting it! http://www.mandolincafe.net/iB_html/non-cgi/emoticons/smile.gif ) How we divide an octave is not standard between cultures, as Otterly and others have pointed out, so how we settled on 12 half-notes and an 8 note scale makes for an interesting discussion.

JimD
Jul-23-2005, 3:50pm
Ah, one of my favorite topics -- tuning systems and scales.

To add my $.02 to the already interesting thread:

Bill brings up an interesting point about children chanting "nyah - nyah" #Leonard Bernstein once pointed out that this chant (down a m3, up a P4, down a M2, down a m3) is so ubiquitous around the world that it may be the "Ur-song" (first or original song). It is a subset of the major pentatonic scale which, in turn is a subset of the diatonic major scale.

Many old Gregorian chants seem to be pentatonic with 2 common passing tones. If these 7 tones are taken together you get the diatonic pitch set (major scale and modes). Later music tended more toward a 7 note scale and, in some styles and pieces, use of the chromatic passing tones. So by this way of thinking, we have grown, in a few hundred years, from a system of 5 + 2 tones to a system of 7 + 5.

Regarding equal tempered tuning: This isn't a western invention either. It was created by a Chinese theorist, Chu Tsui-Yü in 1584. It wasn't in common use in Europe until Chopin's time. That's right -- after Beethoven, Mozart, Bach etc.

The earliest European use of equal temperament for fretted instruments seems to be Vincenzo Galilei (around 1600) who describes a method for calculating fret placement for lutes that is a pretty good approximation of 12 equal.

For anyone interested in tuning, there is a fascinating 2-part article by Bradley Lehman in the Feb. and May 2005 issues of Early Music. He describes how Bach encoded the tuning system of the Well-tempered Clavier into a design that he drew on the title page of the manuscript.

For PaulD:

Níl mórán Gaeilge agam.

Cad é seo: #"Is fearr Gaeilge briste na bearla cliste!" ?

Pardon my Béarla. Is it:

"Broken Irish is better than clever English." # ?

I'm just learning -- this is the best I could do with a smattering of grammer and a dictionary.

PaulD
Jul-23-2005, 6:02pm
Jim; That's correct... and I couldn't have gotten it except we found that line as part of a poem (IIRC) when we were looking for a name for our band. We settled on Gaeilge Briste because we felt Broken Irish described the music we played most accurately! http://www.mandolincafe.net/iB_html/non-cgi/emoticons/biggrin.gif

Thanks for the addition to this thread... the history lesson here is fascinating!

CraigF
Jul-23-2005, 6:42pm
To expound on Berstein's "nyah-nyah..." thing, these 3 tones are the first three non-octave harmonics (removing duplicate octaves). i.e. G E (Bb). The Bb is actually a microtone not in the chromatic scale, but it is the actual interval used in the nyah-nyah.

Also, the 12 step chromatic scale comes from the circle of fifths. The circle uses fifths because it is the dominant. It is the dominant because it is the first non-octave harmonic, thus the strongest.

billkilpatrick
Jul-24-2005, 5:20pm
is it possible that an international recitation of "nyah-nyah" might have been influenced by english language or american style television? i don't doubt that leonard bernstein heard it - it was certainly a subtle and very intelligent observation - but how could he or anyone else be sure where it came from?

it's not sung but the word "ok" is - i imagine - spoken almost everywhere on earth.

Richard Polf
Jul-26-2005, 11:19am
Bernstein took this idea from the theory of the Hungarian composer Zoltan Koldaly (1882-1967), who used it as a basis for a system of musical education of children that bears his name. Along with those of the German composer, Karl Orff (1895-1982), the most influential of the 20th Century.

billkilpatrick
Jul-26-2005, 11:39am
i knew orff was very influential as a teacher but the only thing i know about koldaly is his name.

do you know if either of them - or anyone else, for that matter - have speculated on the possibility of an "original tuning" - a primary relationship of notes which correspond to the basic sounds of human speech when sung?

relux
Jul-27-2005, 12:18am
Here is a detailed discussion about the major scale: The Diatonic Scales (http://http://www.andymilne.dial.pipex.com/Diatonic.shtml)



The diatonic scale is a very important scale. Out of all the possible seven note scales it has the highest number of consonant intervals, and the greatest number of major and minor triads. The diatonic scale has six major or minor triads, while all of the remaining prime scales (the harmonic minor, the harmonic major, the melodic and the double harmonic) have just four major or minor triads.


Etc...

Lee
Jul-28-2005, 4:32pm
Whenever the diatonic scale is discussed I always find it most illustrative to go to the piano for some hands-on explanation. The two theoretically perfect intervals for this discussion are the 2/1 Octave and the 3/2 Fifth. Beginning down low towards the left hand side of the keyboard find C, then play the fifth up from there, it being the G. Then play the next fifth up from there, it being the D. Then play the next fifth up from there, it being the A. Then keep on going in this fashion until you once again land on a C. (A fifth is seven half-steps, don't forget to count the black notes.) You'll find the sequence of notes looks like this:
C,G,D,A,E,B,F#,C#,G#,D#,A#,F,C (Piano tuners refer to all the black notes as sharps, there being no such thing as "flats" in our jargon) What we've done is to use the two ratios of the fifth and the octave to "discover" all twelve notes of the diatonic scale; more or less.
Note that we spanned seven octaves before landing on another C. Note that we spanned twelve fifths before landing on another C. Mathematically speaking seven spanned octaves are described as 2/1 to the seventh power, which equals 128. The twelve spanned fifths are described as 3/2 to the twelfth power, which equals a monstrous looking fraction 531441/4097, which equals 129.71467. The problem is; 129.71467 doesn't equal 128. The last step of the fifth's sequence takes us just a tad further than the C attained by doing the seven octaves. The difference between the 128 and 129.71467 is called "Pythagoras' comma" in honor of the discoverer. To compensate for this discrepency the piano tuner narrows all the fifths perceptably and widens the octaves almost imperceptably. This is what's called tempering the scale. The reknown "equal temperament" being when each half step is 1/12th of the octave, or the twelfth root of 2/1.
This same mathematical phenomenon can be seen with the major third, which is a 5/4 ratio. Start at C and play the third up from there, it being the F. Then play the next third up from there, it being the A#, then play the third up from there, it being the C one octave from whence we began. Mathematically the sequence of three contiguous major thirds which formed the octave is described as 5/4's to the third power, which equals 1-61/64's. This is just a tad short of the perfect 2/1 octave. Hence the piano tuner widens the thirds during the tempering process.
To see what's happening with the tempering process look at the fifth formed by the C-G, and beginning at that G look at the G-upperC interval which is a major third. If we keep both C's stable and "adjust" the G note slightly flat, we're simultaneously narrowing that C-G fifth while widening the G-C major third. What a lucky break, eh?!
It's really amazing how close the twelve note scale comes to adhering so closely to the perfect "just" intervals, yet not quite.

Chip Booth
Jul-28-2005, 5:42pm
Been gone for some days, thanks so much for all the fascinating information I have come back to. Now I just have to read it. But I think in the quick scan I just did that the circle of fifths comes closest to what I consider a logical explanation of where the twelve half steps come from. How obvious that was...

Chip