newsham@wiliki.eng.hawaii.edu (Timothy Newsham) (07/10/90)
i am interested in finding out how our present (western) keyboards
evolved and the theory behind them. i know that the keyboard started
out with the white keys and then keys where added (like Bb to avoid
using tritones) but how did we originally arive at the 7 note scale
that the modes are based on? i am interested at the foundations in
the overtone series in particular. i have also read about a theory
that the minor triad is generated by undertones but i fail to find
out how when i look through the undertone series. anyone have any
information on this or any good references for me to check out?
oh, and i am also interested in the scales that evolved seperately
(ie. indian classical) and how those came about.
thanks in advanced...
-Tim Newsham
newsham@wiliki.eng.hawaii.edujthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) (07/10/90)
The major scale is indeed based on harmonics. In the key of C (what else?) :
Note Decimal frequency fraction
C 1 1
D 1.125 9/8
E 1.25 5/4
F 1.333333... 4/3
G 1.5 3/2
A 1.666666... 5/3
B 1.875 15/8
C 2 2
This is of course the true tempered scale.
Johan Thornton
EE UBC
------- _/__/ -----------------------------------------------------
_| ___| E l e c t r i c a l | Johan Thornton, Esq.
| | |_/ E n g i n E E r i n g |-------------------------
|/| __| U n i v e r s i t y | jthornto@fs1.ee.ubc.ca
|-| |/__ o f B r i t i s h |-------------------------
| |_____| C o l u m b i a | This space for rent
---- |__|/_| ------------------------------------------------------ROGER@pucc.Princeton.EDU (Roger Lustig) (07/10/90)
In article <8547@uhccux.uhcc.Hawaii.Edu>, newsham@wiliki.eng.hawaii.edu (Timothy Newsham) writes: >i am interested in finding out how our present (western) keyboards >evolved and the theory behind them. i know that the keyboard started >out with the white keys and then keys where added (like Bb to avoid >using tritones) but how did we originally arive at the 7 note scale >that the modes are based on? Well, keyboards have had the 7 + 5 arrangement for a LONG time. Paintings from the 14thC make this clear. By that time, there was music with all 12 pitch-classes; and the theoretical existence of these goes back a long way before that. >i am interested at the foundations in >the overtone series in particular. Actually, there is less to this than meets the eye, as Tallulah Bankhead once said. All the damn theories of division of the octave, or of the whole tone, or whatever, do not explain the music being written when the theories were made. Music theory goes back to ancient Greece, and has been misunderstood and done badly ever since. This is not to say that the theories aren't interesting; just that overtones wson't get you very far in ANY attempt to explain music. >i have also read about a theory >that the minor triad is generated by undertones but i fail to find >out how when i look through the undertone series. anyone have any >information on this or any good references for me to check out? There's a good reason for this: the minor triad has nothing to do with undertones. Nor has it anything to do with 10:12:15 or whatever. True, that sounds like a minor triad, just the way 4:5:6 sounds like a major triad. But to derive a theory from this involves eventually confronting the nasty issue of pieces beginning in C minor that *ought* to resolve to A flat eventually (the fundamental...) Better to think of minor as major with a screwy third. >oh, and i am also interested in the scales that evolved seperately >(ie. indian classical) and how those came about. A good article on Mode is the one in the New Grove Encyclopedia. It's by Harry Powers, is alas only about a third of what he wrote and is still 75 pages, and is amazing in its scope. All kinds of world musics are covered, as is the growth of modal theory and practice in the West. In the 15 years since he wrote it, Powers has written more, as have others; it's a BIG topic. Roger Lustig (ROGER@PUCC.BITNET roger@pucc.princeton.edu) Disclaimer: I thought it was a costume party!
maverick@fir.berkeley.edu (Vance Maverick) (07/11/90)
In article <1307@fs1.ee.ubc.ca>, jthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) writes: > The major scale is indeed based on harmonics. In the key of C (what else?) : > [scale omitted] > This is of course the true tempered scale. This is actually the "Ptolemaic" or "Just" scale. It has problems if you want to play anything but I/IV/V; the supertonic triad, for example, has the "wolf fifth" 40/27 (ratio 1.48 instead of 1.5). "Tempering" it would be to tweak some of the intervals so the worst case doesn't sound so bad. The idea that the scale and the chords are based on the overtone series dates (I think) from Rameau -- there's not much evidence we can bring to bear, since people were using scales before they knew much about harmony, and also before they wrote much about either.
jthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) (07/11/90)
Sure, people were using scales long before they figured out what harmony
means. The reason they used them was that it sounded good. A major chord,
say C-E-G has frequency ratio 4:5:6. The three superimposed sounds "beat"
in a very short interval. Inharmonic tones have a longer (and unrelated)
beat interval, so it sounds unpleasant.
------- _/__/ -----------------------------------------------------
_| ___| E l e c t r i c a l | Johan Thornton, Esq.
| | |_/ E n g i n E E r i n g |-------------------------
|/| __| U n i v e r s i t y | jthornto@fs1.ee.ubc.ca
|-| |/__ o f B r i t i s h |-------------------------
| |_____| C o l u m b i a | This space for rent
---- |__|/_| ------------------------------------------------------ROGER@pucc.Princeton.EDU (Roger Lustig) (07/11/90)
In article <1307@fs1.ee.ubc.ca>, jthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) writes: >The major scale is indeed based on harmonics. In the key of C (what else?) : > Note Decimal frequency fraction > C 1 1 > D 1.125 9/8 > E 1.25 5/4 > F 1.333333... 4/3 > G 1.5 3/2 > A 1.666666... 5/3 > B 1.875 15/8 > C 2 2 >This is of course the true tempered scale. This is, in fact, one tuning that sounds like what we think of as the major scale. It is not, however, what the major scale is 'based' on in any historical or practical sense. Nowadays, in any music that uses more than one pitch at once, the above is utterly useless because the interval d:f is nasty. (27:32; try it!) Reconciling the harmonic series with the practice of harmony is well-nigh impossible; check out things like Rameau's _Generation Harmonique_ for an example that trips over itself in all kinds of ways. So, since intervals and their physical ratios have been known for much longer than there's been a major scale as such in use, and since the various diatonic modes were all considered to be pretty much equal in value, I can't see how the major scale is 'based' on the overtone series. Most music using the major scale requires great deviations from pure intonation to sound at all nice. Roger Roger Lustig (ROGER@PUCC.BITNET roger@pucc.princeton.edu) Disclaimer: I thought it was a costume party!
maverick@fir.berkeley.edu (Vance Maverick) (07/11/90)
In article <1312@fs1.ee.ubc.ca>, jthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) writes: > Sure, people were using scales long before they figured out what harmony > means. The reason they used them was that it sounded good. A major chord, > say C-E-G has frequency ratio 4:5:6. The three superimposed sounds "beat" > in a very short interval. Inharmonic tones have a longer (and unrelated) > beat interval, so it sounds unpleasant. Again, you're basing a claim about preharmonic music on a harmonic claim (Helmholtz's this time, instead of Rameau's). Why should people who never isolated the sound of the first, third and fifth degrees together have worried about its beating?
alves@castor.usc.edu (William Alves) (07/11/90)
In article <8547@uhccux.uhcc.Hawaii.Edu> newsham@wiliki.eng.hawaii.edu (Timothy Newsham) writes: >i am interested in finding out how our present (western) keyboards >evolved and the theory behind them. i know that the keyboard started >out with the white keys and then keys where added (like Bb to avoid >using tritones) but how did we originally arive at the 7 note scale >that the modes are based on? This is not at all an easy question, and yet there are some intriguing cross-cultural similarities. For example, systems with 5, 7, or 12 notes per octave are relatively common in different cultures. Many have noted that 5+7=12, and Barbour has pointed out that the most common microtonal systems he has studied include 19, 31, and 50 notes per octave, forming a "Fibonacci-like" sequence. Some have argued that tuning systems based on successive tuning of 3/2's (fifths) come closest to completing the oc- tave at those numbers. I don't think that is necessarily true, and Horn- bostel's theory of tuning by "overblown-fifths" (slightly sharp successive fifths supposedly created by overblowing a flute) has been blown way out of context. Clearly there is no unified field theory of scales. Why the 7/12 system in the West? There's no one good answer that I'm aware of. >i am interested at the foundations in >the overtone series in particular. i have also read about a theory >that the minor triad is generated by undertones but i fail to find >out how when i look through the undertone series. anyone have any >information on this or any good references for me to check out? For some reason, the fact that the minor triad does not appear in successive natural harmonics has been very troubling to some theorists seeking to prove the god-given naturalness of the Western scale system. I think that the theory you refer to may be that of Zarlino, who tried to argue that if the succession major third->minor third occurs naturally then one need only go the other way to get the minor third->major third relationship of the minor triad. Partch called the former "otonalities" and the latter "utonalities." In fact, neither has anything to do with "undertones." Hinde- mith went through a convoluted system of "proving" that the 12-tone sys- tem was the most natural extension of the harmonic series. In fact, he just went the long way around to derive a 5-limit just intonation scale, something that's been around at least since the early 16th century. He then proceeds to throw out the tuning system in favor of equal temperament be- cause it's "close enough". All of this (with the possible exception of Partch) is theory after the fact. I don't think there is One True Scale that music then adheres to. Rather, I think that whether a composer or a culture decides to go with 12-tone equal temperament, or 43-note 13-limit just intonation (Partch), or 6-tone quasi- equal temperament (Thailand), or a whole multitude of heptatonic and penta- tonic systems (Indonesia) has to do with the music and the instruments first. In most cases, it is impossible to say what criteria were used or exactly how they developed historically. Also keep in mind that often theory does not agree with practice. Bill Alves
newsham@wiliki.eng.hawaii.edu (Timothy Newsham) (07/11/90)
i fail to see how you arrive at a couple of those values as overtones.
how does a value of 2/3 fit in? division by a power of 2 is allowed
because it represents changing the octave, but where do we get a 2/3?
the only possible explanation i can think of is perhaps it is in the
overtone series of one of the other prominant notes?
-TimROGER@pucc.Princeton.EDU (Roger Lustig) (07/12/90)
In article <8561@uhccux.uhcc.Hawaii.Edu>, newsham@wiliki.eng.hawaii.edu (Timothy Newsham) writes: >i fail to see how you arrive at a couple of those values as overtones. >how does a value of 2/3 fit in? division by a power of 2 is allowed >because it represents changing the octave, but where do we get a 2/3? >the only possible explanation i can think of is perhaps it is in the >overtone series of one of the other prominant notes? Well, you've just put your finger on one of the reasons why it's a bogus explanation for the major scale. 2/3 is in fact a perfectly pleasing-sounding ratio, but it makes the numbers imply funny things, and it causes the wolf (27/32 or 27/40). If you take common denominators and make everything whole notes, you have made a scale that goes 24, 27, 30, 32, 36, 40, 45, 48. Now, note that the tonic is 24, which suggests that the fundamental underlying the whole works must be -- yup, the subdominant!!! That is, in a C scale, the underlying fundamental that generates all the notes is an F. See the problem? Now, with Lydian mode, you don't have this problem, because you have an F instead of an F. But in Lydian mode, we tend to 'correct' the sharp 4th down to a perfect 4th because we don't like a tritone with the tonic, so from experience, we know that there's nothing 'prior' about the Lydian. In our music, the major scale is obviously prior to others. So much for overtone theory... Roger Lustig (ROGER@PUCC.BITNET roger@pucc.princeton.edu) Disclaimer: I thought it was a costume party!
torkil@psivax.UUCP (Torkil Hammer) (07/12/90)
# of context. Clearly there is no unified field theory of scales. Why the # 7/12 system in the West? There's no one good answer that I'm aware of. I don't think that there is an aggreed-upon theory among scholars, but it seems to me that one reason is the absense of gongs and the presence of sustained-tone music in the early Western music. Early Western music was done with a capella singing, bowed and blown instruments. The instruments were tunable, but had no fingering, so they could not retune during a performance. Sustained-tone music sounds sweetest if it is harmonic. As we have seen, the diatonic (7/12) scale supports harmonies nicely, and the almost identical intervals of 2,2,1,2.. encourages modulation to different keys while staying in tune with the instruments. In contrast, gongs produce disharmonic overtones and discourage harmony music. The same is true to a lesser degree for bells and drums, which are not native to early Western music, either. Torkil Hammer
smoliar@vaxa.isi.edu (Stephen Smoliar) (07/13/90)
In article <11271@pucc.Princeton.EDU> ROGER@pucc.Princeton.EDU writes: > >Better to think of minor as major with a screwy third. > This is probably the best summary I have ever read of the "Combinations" chapter of Heinrich Schenker's HARMONY! Of course, he then goes on to argue that you can play the same game with ANY step of the major scale. This is how he then explains all the modes. I'm not sure anyone has ever bought into that generalization. ========================================================================= USPS: Stephen Smoliar USC Information Sciences Institute 4676 Admiralty Way Suite 1001 Marina del Rey, California 90292-6695 Internet: smoliar@vaxa.isi.edu "It's only words . . . unless they're true."--David Mamet
ROGER@pucc.Princeton.EDU (Roger Lustig) (07/13/90)
In article <14260@venera.isi.edu>, smoliar@vaxa.isi.edu (Stephen Smoliar) writes: >In article <11271@pucc.Princeton.EDU> ROGER@pucc.Princeton.EDU writes: >>Better to think of minor as major with a screwy third. >This is probably the best summary I have ever read of the "Combinations" >chapter of Heinrich Schenker's HARMONY! Of course, he then goes on to argue >that you can play the same game with ANY step of the major scale. This is how >he then explains all the modes. I'm not sure anyone has ever bought into that >generalization. How about Ed Cone? (Probably got it off an episode of Benny Hill or something...) Seriously, it's not a bad idea if you're talking about the use of mode in the common-practice period. Major IS prior to the other modes; note how pieces in minor have a funny habit of ending in major -- and NEVER the other way around. When you get down to it, you can, with little trouble, substitute the equivalent major harmonies for the chord on any scale degree of the minor. And minor is an odd case to begin with, as it comes with two notes that alternate to begin with -- and a set of rules for choosing one or the other. And it's major-key practice that determines those rules: when the harmony demands it, you step out of the minor scale and use the major-key note -- or if the melody demands it. 'Melodic' minor is a majorized minor. And once you start in with the most obvious chromatic alteration of all -- V of IV -- there goes your third! Roger Lustig (ROGER@PUCC.BITNET roger@pucc.princeton.edu) Disclaimer: I thought it was a costume party!
smoliar@vaxa.isi.edu (Stephen Smoliar) (07/14/90)
In article <10737@chaph.usc.edu> alves@castor.usc.edu (William Alves) writes: >I don't think there is One True Scale that music then adheres to. Rather, I >think that whether a composer or a culture decides to go with 12-tone equal >temperament, or 43-note 13-limit just intonation (Partch), or 6-tone quasi- >equal temperament (Thailand), or a whole multitude of heptatonic and penta- >tonic systems (Indonesia) has to do with the music and the instruments first. >In most cases, it is impossible to say what criteria were used or exactly how >they developed historically. Also keep in mind that often theory does not >agree with practice. > It is important to remember that, when it comes to trying to reconcile equal temperament with the harmonic series, you will always have to contend with mathematics. Since all integer roots of two are irrational, you are never going to get any which provide ratios which will line up with those of the harmonic series. What makes matters worse is that there will always be a trade-off. That is, as you improve your approximation to 3/2, you lose on 5/4 . . . unless you choose to go for some extremely fine division of the scale, in which case you are no longer talking about anything which can be resolved as an equally tempered gamut. ========================================================================= USPS: Stephen Smoliar USC Information Sciences Institute 4676 Admiralty Way Suite 1001 Marina del Rey, California 90292-6695 Internet: smoliar@vaxa.isi.edu "It's only words . . . unless they're true."--David Mamet