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Natural Science Forum / Physics / Acoustics / June 2004Why are there 7 discrete notes? A possibly stupid question about sound...
I'm a biologist, with no background in music and limited physics, so excuse me if this is a stupid question. Why are there 7 discrete notes? The pitch we hear is a function of frequency, which is a continuous, not discrete quantity. So, at first I thought it was kind of like light and colors - we make up arbitrary names for sections of the spectrum, but in fact there are an infinity of colors (or however many the human eye can distinguish within the EM spectrum). So, maybe each note is just an arbitrary part of the frequency spectrum which people have agreed to divide into 7 sections. But one thing bothers me: why does it "wrap around"? That is, on a piano keyboard, after you've gone A,B,C,D,E,F,G, the next one is A again, one octave higher, but in some sense, A again. How does this work? And, why does it wrap around like that? If there is a real sense in which the sounds after "G" are "A" again, does this mean that there is something to the "7 notes" beyond just convention? Is there any discreteness in the sound spectrum which is real (a real feature of the physics as opposed to arbitrary convention)? I am interested in special numbers which come up in various areas of math and science. Is the "7" here real, in the sense that the value of Pi is "real" and not simply arbitrary human convention? Probably not, but I would appreciate an explanation of where I've gotten confused. How do the 7 notes and the octaves relate to the continuous spectrum of air wave frequency? Thanks in advance for any info.  SignatureMike Levin mlevin77@comcast.net > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? Hello? What about the 12 tone scale? The human ear is capable of discerning more than seven notes in an octave. And what is sacred about octaves? Bob Kolker Well, I used to be a biologist (geneticist) too. But somewhere back in the 60's I (and a few others) started playing the guitar. There's nothing special about our scale; I've heard the ear can fairly readily distinguish intervals only about one-fourth the size of our current semitones. Now an octave has real meaning in that going up an octave the frequency precisely doubles. An interesting fact I can up with (likely not original) is that the circle of fifths (which means -- illiterate musicians! -- up FOUR whole notes, 7 semitones) works because there are 12 (now) equally spaced semitones and 7 (and 5, going down) are relatively prime to twelve. The others (2, 3, 4, 6, 8, 9, 10) divide 12 or have common factors with 12. HTH John SignatureJohn T Lowry, PhD Flight Physics 5217 Old Spicewood Springs Rd, #312 Austin, Texas 78731 (512) 231-9391 jlowry100@earthlink.net > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. On 5/15/04 2:55 PM, in article BItpc.1514$H_3.113@newsread1.news.pas.earthlink.net, "John T Lowry" <jlowry100@earthlink.net> wrote: > Well, I used to be a biologist (geneticist) too. But somewhere back in > the 60's I (and a few others) started playing the guitar. There's > nothing special about our scale; I've heard the ear can fairly readily > distinguish intervals only about one-fourth the size of our current > semitones. Now an octave has real meaning in that going up an octave the > frequency precisely doubles. Ah! That answers my question exactly. The number of notes in an octave is indeed arbitrary, and the octaves wrap around because the ear detects doublings of frequency as similar notes. Makes sense. Thanks!! SignatureMike Levin mlevin77@comcast.net >On 5/15/04 2:55 PM, in article >BItpc.1514$H_3.113@newsread1.news.pas.earthlink.net, "John T Lowry" [quoted text clipped - 10 lines] >indeed arbitrary, and the octaves wrap around because the ear detects >doublings of frequency as similar notes. Makes sense. Thanks!! The notes in the scale aren't really arbitrary. They are based on harmonic relationships that make many of the overtones coincident. In some cultures, this results in a very limited scale, with just five notes. Play C, D, E, G, A on a piano to get the idea. Western music has simply extended the scale to fill in all the blanks, so to speak. d -- http://www.pearce.uk.com > >On 5/15/04 2:55 PM, in article > >BItpc.1514$H_3.113@newsread1.news.pas.earthlink.net, "John T Lowry" [quoted text clipped - 16 lines] > notes. Play C, D, E, G, A on a piano to get the idea. Western music > has simply extended the scale to fill in all the blanks, so to speak. That is, of course, equivalent to playing on the black notes only, starting at F# Franz > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 19 lines] > > -- Try: http://www.google.com/search?q=history+music+scales > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. The simple explanation goes back to the Pythagoreans... Start from C, take a quint (frequency of C times 3/2) and go to G. This C and G sound great when played together, or one after the other. Now take another quint from G and go to D, then to A, then to E, then to B, then to F, and finally back to C - You now have 4 times the frquency of your original C. C x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 = 17.09 C That is slightly over 16 C, i.o.w. you now span 4 octaves. Now collapse them all together in one octave by taking half of the frequencies of all the ones that didn't end up in the first octave. That is your scale. Not quite - It would be nice if (3/2)^7 were 16. Now, do we have some other way of producing a power of 2 by taking a power of 3/2? Yes.... try (3/2)^12 = 129.75 =~ 128. Does 12 ring a bell? http://www.musemath.com/flash/contents.swf http://www.musemath.com/flash/math.swf hth Dirk Vdm > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? In European music, there are 12 logarithmically equally spaced notes to the octave, not 7. In some Asian music, there are up to 24 notes to the octave. A violin can play a complete continuum. > The pitch > we hear is a function of frequency, which is a continuous, not discrete [quoted text clipped - 10 lines] > "G" are "A" again, does this mean that there is something to the "7 notes" > beyond just convention? THe "7 white notes" is a cultural convention. The relationship between a note and its octave is what matters as far as your question goes. Take a pure sinewave note and one an octave higher. Play them together. The second one now simply sounds the second harmonic of the first. In the process, it loses its indentity as far as the listener is concerned. It simply sounds as if it has modified the timbre of the first note. [snip] Franz Michael Levin (or somebody else of the same name) wrote in message <BCCBD9DE.17384%mlevin77@comcast.net> thusly: > I'm a biologist, with no background in music and limited physics, so > excuse me if this is a stupid question. Why are there 7 discrete notes? [quoted text clipped - 18 lines] > octaves relate to the continuous spectrum of air wave frequency? Thanks in > advance for any info. May I refer the interested reader to the article http://www.deniseswanson.com/stybr/stybr-mm.htm by contemporary American composer David Stybr, which sets out the rationale behind the diatonic and chromatic scales, simply but systematically. SignaturePaul Townsend I put it down there, and when I went back to it, there it was GONE! Interchange the alphabetic elements to reply > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? ... There's also sharp and flat, and "atonal" music has been around for centuries. There's no accounting for harmonious taste. [Old Man] > ... > Mike Levin To help quantify this thread, can someone make a short list of note frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, etc.... An octave or so around middle C, or A would really help; state the pitch criterion ("piano"?, concert pitch, or what have you)... Angelo Campanella > To help quantify this thread, can someone make a short list of note > frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, > etc.... An octave or so around middle C, or A would really help; state > the pitch criterion ("piano"?, concert pitch, or what have you)... Try this one: http://www.phy.mtu.edu/~suits/Physicsofmusic.html and specially http://www.phy.mtu.edu/~suits/scales.html Dirk Vdm > To help quantify this thread, can someone make a short list of note > frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, > etc.... An octave or so around middle C, or A would really help; state > the pitch criterion ("piano"?, concert pitch, or what have you)... They are all over the web. For instance: http://www.mjorch.com/hertz.html Start with the A above middle C as 440 Hz. exactly. Compute from there. > To help quantify this thread, can someone make a short list of note > frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, > etc.... An octave or so around middle C, or A would really help; state > the pitch criterion ("piano"?, concert pitch, or what have you)... A just above middle C is 440 Hz Any other note n semitones above A then has a frequency 440 * 2^(n/12) Franz Of course, if you play the bagpipes, then the formula is probably a bit off. Michael > > To help quantify this thread, can someone make a short list of note > > frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, [quoted text clipped - 6 lines] > > Franz Seems to me middle C at 440 Hertz would do it; you can figure the rest. I recall reading, in Helmholtz's Sensations of Tone that, when he wrote it (couple of centuries ago) scales of European bands differed in their definition of say middle C by more than a note! SignatureJohn T Lowry 5217 Old Spicewood Springs Rd, #312 Austin, Texas 78731 (512) 231-9391 jlowry100@earthlink.net > To help quantify this thread, can someone make a short list of note > frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, > etc.... An octave or so around middle C, or A would really help; state > the pitch criterion ("piano"?, concert pitch, or what have you)... > > Angelo Campanella Oops! I see it's the A above middle C, not middle C itself, that's 440 Hertz! Sorry. SignatureJohn T Lowry 5217 Old Spicewood Springs Rd, #312 Austin, Texas 78731 (512) 231-9391 jlowry100@earthlink.net > To help quantify this thread, can someone make a short list of note > frequencies in Hz A-G, do re mi....do, sharp and flat perturbations, > etc.... An octave or so around middle C, or A would really help; state > the pitch criterion ("piano"?, concert pitch, or what have you)... > > Angelo Campanella >I'm a biologist, with no background in music and limited physics, so excuse >me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] >7 notes and the octaves relate to the continuous spectrum of air wave >frequency? Thanks in advance for any info. There's an essay on the topic by Isaac Asimov. This is a paste from www.asimovonline.com - Music to My Ears Subject: musical scale First Published In: Oct-67, The Magazine of Fantasy and Science Fiction Collection(s): * 1968 Science, Numbers, and I It explains the matter very nicely. -- john > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 8 lines] > octave higher, but in some sense, A again. How does this work? And, why does > it wrap around like that? Each higher A is a vibration twice as fast as the next lower A. Its called an octave. >If there is a real sense in which the sounds after > "G" are "A" again, does this mean that there is something to the "7 notes" [quoted text clipped - 6 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. Most humans don't use the 7 note scale. Five might be the most popular. Jazz uses (at least) 12 notes per octave. Ancient Thailand used 33 or so. > > I'm a biologist, with no background in music and limited physics, so excuse > > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 25 lines] > Most humans don't use the 7 note scale. Five might be the most > popular. But so terribly boooooooring. Dirk Vdm > > I'm a biologist, with no background in music and limited physics, so excuse > > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 8 lines] > > octave higher, but in some sense, A again. How does this work? And, why does > > it wrap around like that? My home made mathematical explanation to this is as follows: We probably want a logarithmic scale, such that the frequency ratio between two adjacent notes is the same all over the frequency range. So, the frequency of a tone, starting at the with number 0 at 110Hz ("randomly" selected :-) ) would be 110 * 2^(n/x). n is the note number. Now we have to select the value of x. If x=1 then we would get one tone per octave. If x=2 we would get 2 tones per octave etc. So which value for x should we choose? It turns out that frequencies with an integer ratio such as 2/3 or 4/5 sounds pleasing together. This is particularly true for tones consisting of an harmonic series. So if we try a few different value for x, it turns out that x=12 gives a lot of frequency ratios which are near to integer ratios. 2^(7/12)~2/3, 2^(5/12)~3/4, 2^(4/12)~4/5 and 2^(3/12)~5/6. So x=12 is a good choice if we want to play chords that sounds nice. There are other values for x that also fit this rather nicely, 18 is one, but it is not as good as 12. So, if you count the keys on a piano, including the black ones, you will find that there are 12 of them per octave. <snip> > My home made mathematical explanation to this is as follows: We > probably want a logarithmic scale, such that the frequency ratio [quoted text clipped - 16 lines] > So, if you count the keys on a piano, including the black ones, you > will find that there are 12 of them per octave. No, no, and no. That's a selfconsistent theory that does not account for the data. The value of x was not chosen to form either intervals or a scale ( a distinction which seems to have escaped you.) Octaves and simple intervals are givens in Western culture (and octaves apparently in all cultures. But not intervals nor scales.) Western culture attempted to fit simple intervals together into a diatonic scale. Other cultures have other scales of various other numbers of notes, some much more and some much less than the Western seven diatonic and twelve chromatic notes. When scales were produced from simple intervals there was always a bit left over. After a few centuries trying different ways to apportion that bit left over, it was decided that apportioning it equally across the scale would be efficient. That required your mathematical equation and it required x = 12. There was no trying different values of x nor any choice involved. > <snip> > > [quoted text clipped - 39 lines] > equation and it required x = 12. There was no trying different values > of x nor any choice involved. Well, I admit that the theory is kind of home made and not historically correct and I did not mean that some matematician sat down and figured it all out. But I think the math stuff is a nice way of understanding why 12 semitones per octave works, in that the value x=12 produces intervals that are close to integer ratios. But then, that is of course only my opinion. >> > My home made mathematical explanation to this is as follows: We >> > probably want a logarithmic scale, such that the frequency ratio [quoted text clipped - 45 lines] > > But then, that is of course only my opinion. Check out what weapon designers are doing with ionised laser's. It has something to do with pulsating the laser at a certain frequency so as to be able to send an electric charge along it. Great for disabling people and damaging computer components! :-P You're both right, and you're both wrong. Yes, it does have to do with math, and yes, different cultures came up with different scales, but the development of the seven-note diatonic scale began with pythagoras. According to legend, Pythagoras discovered the foundations of music by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. Specifically, he noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note! Pythagoras rushed into the blacksmith to discover why, and he found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were (12:9 = 4:3 = musical fourth) and (12:8 = 3:2 = musical fifth). Interestingly, if you invert the ratio of hammer B (making it 3:4 = 12:8) it becomes the ratio of hammer C, and vise-verse, thus, the musical fourth can be described as an inverted fifth, and vise verse. The space between B and C is a ratio of 9:8, which is equal to musical whole tone, or whole step interval. The ratio 2:1 produces perfect consonance, that is, for each cycle of the lower frequency, there are exactly two cycles of the frequency one octave higher. The ratio 3:2 produces a very similar kind of consonance. It's all about symetry. The development of western music is very much like biological evolution. A variety of musicians, scholars, and methematicians over the course of many centuries stumbled accross the natural laws that make music sound good. It isn't purely cultural -- that is, there are mathematical reasons that those specific notes sound good, but the seven-tone diatonic scale didn't simply pop into being on the discovery of any one person. The modern scales you hear today in popular music came from the development of the equal-tempered scale. Before equal and well-tempered scales came about, different keys sounded more in tune -- as long as you played only in that key. That's because they were tuned for perfect consonance in the 5ths and 3rds for that specific scale. The problem was, the different scales sounded out of tune with each other, meaning that when you modulate keys, the key you modulate to sounds out-of-tune. The well-tempered clavier was an attempt to confront this problem. J.S. Bach championed it with "the well tempered clavier" -- A selection of fugues and preludes that is still very popular today. Equal temperement didn't become a standard until the late 1800's. The idea was to make every scale equally out of tune. They did this by dividing the octave into an equal number of cents (cents being a measure of pitch variation). This division was optimized to make 5ths and 3rds sound particularly good with the octaves for every scale. It wasn't an arbitrary choice. It had a very specific purpose. This is just a rough overview. If you want all the details (including how you can mathematically discover the whole 7-tone diatonic scale), I refer you to these excellent sources: James, Jamie _ The Music of the Spheres: Music, Science, and the Natural Order of the Universe _. New York, Grove Press, 1993 "Pythagorean hammers" _ Harvard Dictionary of Music _. Second Edition, Massachusets: Harvard University Press Signature~ <http://www.dilvie.com/> I will continue another branch of the thread, to which replies were given about a month ago, but to which I didn't notice until later. It is amended to this branch because this one is still new enough to allow direct replies. From: crynwulf (lyttlec@earthlink.net) Subject: Re: Why are there 7 discrete notes? A possibly stupid question about sound... Newsgroups: alt.sci.physics.acoustics, alt.sci.physics, sci.physics, comp.music Date: 2004-05-16 21:16:34 PST >Gulliver wrote: >> ... >> An 'octive' on a piano consists of 7 (not eight - count 'em ABCDEFG(7) >> A (the first note in the next 'octive' makes eight)) white keys and 5 >> black keys (12 notes total). >> ... > >You forgot about the "H" note. (Yes, there was an H but it got replaced by >G#/Aflat on the tempered scale). The notes were based on ratios of small >integers from Greek philosophy. There were notes for going from C1 ... C2 >and another set going from C2 ... C1. The tempered scale replsce these with >one set that doesn't quiet match either. For example, it replaces >B1(rising) and B1(lowering) with a sort of not too bad sounding median tone >called B. I've been told that American Indians find piano music terrible >because they hear the hammers and all the notes are a bit off. On the other >hand, their music is so different from ours, we can't even hear it. From: Eberhard Sengpiel (esengpiel@t-online.de) Subject: Re: Why are there 7 discrete notes? A possibly stupid question about sound... Newsgroups: alt.sci.physics.acoustics Date: 2004-05-17 00:04:13 PST >No, the englisch note B is still called in Germany >an H and the englisch B flat is called a B. [quoted text clipped - 4 lines] >and sound studio techniques >http://www.sengpielaudio.com So, on a modern German scale the notes are ABHCDEFG and on the proposed theoretical one it was ABCDEFGH. Were there any pianos or harpsichords in the history of their manufacture, that had 8 white keys and 4 black keys, or even other configurations, rather than the modern 7 - 5 configuration? First of all it's not a stupid question at all - it 's a very complex one. But the answer is there aren't 7 discrete notes. The western tuning system has 12 chromatic steps, other tuning systems range from 5 to several hundred. However, there is substantial literature on why certain degress of pitch discrimination are universal - it has to do with the feature of the mammalian auditory system called the critical band. The critical band is a filter function of the ear which describes how close in frequency two tones are before they can no longer be resolved into discrete pitches. In order to avoid a very lengthy message here, I would recommend you look at Fletcher's experiment, in which listeners were asked to detect a pure tone embedded in background noise - the tone was kept constant in frequency but varied in amplitude, and the bandwidt of th enoise was varied. This allowed the generation of a working definition of critical band as the bandwidth of noise that alone contributes to the masking of a pure tone embedded within that noise. In other words, the stretches of the basilar membrane in the inner ear act as a series of bandpass filters - if two sounds are too close in pitch or frequency, they will begin to sound "rough" (think of D and D#) because they are imposing on each others filter edges. Thios also explains why some intervals are universal in all tunings - they are separated by more than a minimal fraction of the critical band, and often in complex sounds, their harmonics or partials will overlap. Hope this helps. Seth Horowitz, PhD CTO, NeuroPop As a function of the width of the critical band > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. An 'octive' on a piano consists of 7 (not eight - count 'em ABCDEFG(7) A (the first note in the next 'octive' makes eight)) white keys and 5 black keys (12 notes total). A note that is one 'octive' higher in pitch is exactly twice the frequency of the same note one octive lower. That might be why the notes might seem to sound similar in some ways when they may be an octive higher or lower than a specific note. I am not sure if notes are officially set at base 12 logarithmic increments of frequency or not, or wheather they are set at specific frequencies based upon some arbitrary criteria about 'sounding best' by some official music organization. This scale seems to have evolved out of the middle ages, however the 'half-step' (base 24) and 'quarter-step' (base 48) is referred to from classical times in ancient Greece and Rome, and other scales such as the 'pentatonic' are known from other cultures. There is nothing that might preclude, however, that a scale might not exist or be used in other increments besides base 12. It can be done on any nonexclusively tonically stepped instruments. (Like trombones, string instruments without clefs(sp.?), synthesizers, etc.) >> I'm a biologist, with no background in music and limited physics, so >> excuse me if this is a stupid question. Why are there 7 discrete notes? [quoted text clipped - 22 lines] > A (the first note in the next 'octive' makes eight)) white keys and 5 > black keys (12 notes total). You forgot about the "H" note. (Yes, there was an H but it got replaced by G#/Aflat on the tempered scale). The notes were based on ratios of small integers from Greek philosophy. There were notes for going from C1 ... C2 and another set going from C2 ... C1. The tempered scale replsce these with one set that doesn't quiet match either. For example, it replaces B1(rising) and B1(lowering) with a sort of not too bad sounding median tone called B. I've been told that American Indians find piano music terrible because they hear the hammers and all the notes are a bit off. On the other hand, their music is so different from ours, we can't even hear it. > A note that is one 'octive' higher in pitch is exactly twice the > frequency of the same note one octive lower. That might be why the [quoted text clipped - 15 lines] > on any nonexclusively tonically stepped instruments. (Like trombones, > string instruments without clefs(sp.?), synthesizers, etc.) SignatureRuss Lyttle Not Powered by ActiveX http://home.earthlink.net/~lyttlec/philosophy/logos.html > Russ Lyttle wrote: > You forgot about the "H" note. (Yes, there was > an H but it got replaced by G#/Aflat on the > tempered scale). No, the englisch note B is still called in Germany an H and the englisch B flat is called a B. The G#/Aflat was never the note H. Eberhard Sengpiel German forum for microphone recordings and sound studio techniques http://www.sengpielaudio.com Just a couple more thoughts. That assumption of an analogy to light, with a continuous spectrum, is flawed. Real musical notes are never one frequency, they are a set of frequencies related sufficiently that the brain assigns one pitch. That A 440 you mentioned can be one frequency on a synthesizer, but any real instrument plays the 440 and a series of higher notes, probably at least five. (Also, though light is continuous across the spectrum, color is determined by 3 receptors in the retina.) It is true that when the ratios of frequency are in simple whole numbers the ear assigns some qualties to them. That description of just "tuning" posted above was almost correct and has been known since the early greeks. The most perfect consonance is the octave, and there is still no explanation for why. Even to nonmusicians A 220 and A 440 both sound like A, and nobody knows why. So there is no answer for your question about wrap around. However your assumption of a major scale always fitting inside the octave is only partially correct. All cultures seem to have the octave but not all have 8 diatonic steps in it, nor 12 chromatic steps. Indian musicians use 22 or soemthing like that. Interestingly enough those with perfect pitch in other cultures have it only for the notes that correspond to the Western chromatic scale. Perhaps there is something physiological involved. The error in the description of just and equal temperament is understanding why. You can't make a scale with just intervals. When you add up all the simple integer ratios you DON'T wrap around, you have some left over. Temperament systems are just different ways to apportion that error. Equal temperament puts it evenly across every interval. Just assigns more to some than others. There are many other ways to go about it, all have advantages and disadvantages. Equal is in style now as you noted and key independence is one of the advantages. > It is true that when the ratios of frequency are in simple whole > numbers the ear assigns some qualties to them. That description of [quoted text clipped - 3 lines] > A 440 both sound like A, and nobody knows why. So there is no answer > for your question about wrap around. That is fascinating. You are saying the 2:1 ratio defining the octave not merely a cultural artifact but is related to the way we process sound in our brain. That is very deep . That is the kind of thing that neuroscience should bust its gut (so to speak) to find an explanation for. If we can find a physical/physiological explanation for things like this we can reduce mental processes to brain processes in detail. Bob Kolker The reason that an octave is processed as the same "pitch" has been explained neurologically - it has to do with overlap of processing of harmonics by hair cells in the middle ear. Pitches made of complex tones (rather than sine waves) share the most harmonics and hence the most neurosensory overlap in any two note interval. This is also the basis of the "octave errors" that occur in pitch identification. Seth Horowitz, PhD CTO, NeuroPop > The reason that an octave is processed as the same "pitch" has been > explained neurologically - it has to do with overlap of processing of [quoted text clipped - 5 lines] > Seth Horowitz, PhD > CTO, NeuroPop You could very well be right, my knowledge being a bit dated with respect to the octave mystery. I would point out though that the "off by a fifth" error is common but octave errors are not. Also, octaves played as sine waves still sound like the same note. While most real musical notes have considerable harmonic content, at some volume levels many do not. For example, flutes played softly are pretty close to sine waves; trumpets and trombones in the high register have little harmonic content, almost none when played softly. And curiously, the brain will put together a perceived pitch even if only a few of the harmonics are present, as long as they are in the right ratio. One classic example is the brass player's "pedal" tone, which generally has little or no fundamental content, yet is heard perfectly well. I need to correct an earlier error of mine. I stated dogmatically that there is no way to put pure (simple integer ratio) intervals together into a scale. That is correct for the western diatonic scale. But it seems perfectly conceivable to define a scale with fewer intervals, maybe a pentatonic one, where all the intervals would sound pure and you would get wrap around. Then, you wouldn't need a temperament system. > But it seems perfectly conceivable to define a scale with > fewer intervals, maybe a pentatonic one, where all the intervals would > sound pure and you would get wrap around. Then, you wouldn't need a > temperament system. If by pure you mean integer ratios in the major chords, try it. :-) Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > > But it seems perfectly conceivable to define a scale with > > fewer intervals, maybe a pentatonic one, where all the intervals would [quoted text clipped - 4 lines] > it. :-) > Bob Your personal attacks have become all too boring. Accordingly, I was just wondering if there might be any chance, at any future time, in any forum that you might have something intelligent and/or technically substantive to say. > > > But it seems perfectly conceivable to define a scale with > > > fewer intervals, maybe a pentatonic one, where all the intervals would [quoted text clipped - 9 lines] > any forum that you might have something intelligent and/or technically > substantive to say. Instead of talking nonsense, you might try what Bob suggested. You will soon find out that you cannot achieve a "wrap". Franz >> The reason that an octave is processed as the same "pitch" has been >> explained neurologically - it has to do with overlap of processing of [quoted text clipped - 21 lines] >only a few of the harmonics are present, as long as they are in the >right ratio. Right ratio? Are you sure? Have you ever heard a radio talk show's voice, which you know really well, when he is on tape, and if the tape is running 0.001% fast, you can detect it immediately. The ratio is there, but still, the error is audible. >One classic example is the brass player's "pedal" tone, >which generally has little or no fundamental content, yet is heard [quoted text clipped - 7 lines] >sound pure and you would get wrap around. Then, you wouldn't need a >temperament system. Mr. Dual Space (If you have something to say, write an equation. If you have nothing to say, write an essay). > >> The reason that an octave is processed as the same "pitch" has been > >> explained neurologically - it has to do with overlap of processing of [quoted text clipped - 27 lines] > can detect it immediately. The ratio is there, but still, the error is > audible. You missed the point. Reread. [snip] Franz >And curiously, the brain will put together a perceived pitch even if >only a few of the harmonics are present, as long as they are in the >right ratio. One classic example is the brass player's "pedal" tone, >which generally has little or no fundamental content, yet is heard >perfectly well. When I was in school a teacher took a sawtooth oscillator (which contains even and odd harmonics) and put it through a very steep filter, essentially removing the fundamental entirely. The perceived "pitch" was still that of the (removed) fundamental. It's as if our ear/brain system latches onto the "repeat rate" of the waveform to determine pitch. To make this clearer, imagine a sawtooth wave with a fundamental at 110 Hz. This would contain harmonics at 220, 330, 440, 550, 660 and so on, up to the bandwidth of the system. If we filter out the 110 Hz, you might think that the "pitch" we sense would be that of a 220 Hz note. But we seem to sense that the waveshape still only "repeats" 110 times per second. It's kind of a "highest common factor" problem. The various sine waves that make up the waveform are only "in phase" 110 times per second. If we could manage to remove all the "odd" harmonics (the 330, 550, 770 etc.) then the waveform would repeat 220 times per second and we would hear the pitch as A 220. I think. :) Greg G. In alt.sci.physics.acoustics Greg G <gdguarino@verizon.net> wrote: > But we seem to sense that the waveshape still only "repeats" 110 > times per second. Nope. If you put 3 tones together, at 1400Hz, 1600Hz, and 1800Hz, you perceive a pitch of 200Hz. But if you put 1500Hz, 1700Hz and 1900Hz together, the envelope is periodic with a frequency of 200Hz, the waveform is periodic with a frequency of 100Hz, and the perceived pitch is ambiguous, with 2 pitches perceived at about 185Hz and 215Hz. The pitch percept us somewhat vague, though. Think in terms of an auto-correlation in this case. There are illustration in many places on the web, one being http://www.isr.umd.edu/CAAR/posters/ARO97.pdf (2.9 Megs...) Look at page 5, if you have the patience.I am sure there are better places. Didier SignatureDidier A Depireux ddepi001@umaryland.edu didier@isr.umd.edu 20 Penn Str - S218E http://neurobiology.umaryland.edu/depireux.htm Anatomy and Neurobiology Phone: 410-706-1272 (lab) University of Maryland -1273 (off) Baltimore MD 21201 USA Fax: 1-410-706-2512 >In alt.sci.physics.acoustics Greg G <gdguarino@verizon.net> wrote: > [quoted text clipped - 7 lines] >with 2 pitches perceived at about 185Hz and 215Hz. The pitch percept us >somewhat vague, though. Hmm. I can't claim to have tried anything as odd as that, meaning odd harmonics only, starting with such high ones, and presumably without any "natural" amplitude distribution. We did try raising the high-pass filter to remove more and more of the lower harmonics in addidtion to the fundamental. The sense of pitch remained the same even though the "tone" got very thin indeed. Greg > In alt.sci.physics.acoustics Greg G <gdguarino@verizon.net> wrote: > [quoted text clipped - 7 lines] > with 2 pitches perceived at about 185Hz and 215Hz. The pitch percept us > somewhat vague, though. This reminds me of the ring modulator, a device which from two sounds produces sum and difference tones. So 300 and 500 yield 100 and 800. Usually the input is a musical instrument and a constant tone. The result is non-Pythagorean but may still sound musical. Try "Hymn Of The Seventh Galaxy" where these unearthly scales are used to great effect. > This reminds me of the ring modulator, a device which from two sounds > produces sum and difference tones. So 300 and 500 yield 100 and 800. I would expect 200 and 800. Thank you for this hint. While I asked for something similar on June 1, I am disappointed that both your posting and that of mine were largely overlooked and not understood. What about autocorrelation, I found out that most likely it is not an additional operation after cochlear frequency analysis but it includes both this analysis and a second neural analysis together. So it is pretty similar to cepstral analysis. Chen-Gia Tsai provided some intriguing examples. Can you point me to other ones? Kind regards, Eckard > Nope. If you put 3 tones together, at 1400Hz, 1600Hz, and 1800Hz, you > perceive a pitch of 200Hz. But if you put 1500Hz, 1700Hz and 1900Hz [quoted text clipped - 9 lines] > > Didier > >And curiously, the brain will put together a perceived pitch even if > >only a few of the harmonics are present, as long as they are in the [quoted text clipped - 23 lines] > > I think. :) It is possible that the restoration of the fundamental occurs by virtue of non-linearities in the auditory system. That would cause the generation of sum and difference frequencies in you ears. In your example, the difference between any two neighbouring overtones is in fact the missing fundamental. Franz >>>And curiously, the brain will put together a perceived pitch even >>>if only a few of the harmonics are present, as long as they are in the >>>right ratio. One classic example is the brass player's "pedal" tone, >>>which generally has little or no fundamental content, yet is heard >>>perfectly well. For a weird tonal experience, go to http://asa.aip.org/sound.html and play "Risset's Continuous Scale" (bottom of page). It's a seamles wrap-around. Angelo Campanella > http://asa.aip.org/sound.html > > and play "Risset's Continuous Scale" (bottom of page). > > It's a seamles wrap-around. The whale songs were more interesting. If you listen to them enough you can get to like them. Has anyone tried to represent whale song in some kind of comprehensible scale? Bob Kolker > I need to correct an earlier error of mine. I stated dogmatically > that there is no way to put pure (simple integer ratio) intervals [quoted text clipped - 3 lines] > sound pure and you would get wrap around. Then, you wouldn't need a > temperament system. No. It is mathematically impossible, if wrap around is required for each note. If you think otherwise, let us see the sequence. Franz > > I need to correct an earlier error of mine. I stated dogmatically > > that there is no way to put pure (simple integer ratio) intervals [quoted text clipped - 10 lines] > If you think otherwise, let us see the sequence. > Franz Franz, You clearly can't get all the scale notes with any kind of integer ratio rule and still get wraparound, you are correct. However you can get most of them (see Arthur Benade, "Fundamentals of Musical Acoustics). If you defined a restricted scale that only had the notes you can get, you could make the intervals pure. I didn't stop to check if the pentatonic scale, with only five notes, might possibly work. To me that would not be a useful solution but it would be enough to make me admit I'm wrong about no possible way to create a scale. The reason I get involved in these temperament discussions is that somebody always compares the modern, "compromised" equal temperament scale to some older "correct" scale, with all sorts of mystical descriptions being added. The point is there are no correct scales nor can there be, there are only different types of compromises. For another perspective do a google on "lucy tuning," you'll see what I mean. > > > I need to correct an earlier error of mine. I stated dogmatically > > > that there is no way to put pure (simple integer ratio) intervals [quoted text clipped - 21 lines] > stop to check if the pentatonic scale, with only five notes, might > possibly work. I did. It does not work. There is no possible scale for which it would work, not even one as simple as C G Try it and you will see. > To me that would not be a useful solution but it would > be enough to make me admit I'm wrong about no possible way to create a [quoted text clipped - 7 lines] > another perspective do a google on "lucy tuning," you'll see what I > mean. Franz > > "Franz Heymann" <notfranz.heymann@btopenworld.com> wrote in message > news:<c8o9i3$7fd$2@sparta.btinternet.com>... [quoted text clipped - 35 lines] > simple as C G > Try it and you will see. What do you mean by "integer ratio rule"? If you mean frequencies in ratios of whole integers, obviously the smallest possible interval is the octave. No, that can't be it. If you mean rational ratios, then surely there are idefinitely many rational ways to factor 2 into numbers 1 < x < 2; e.g., 3/2, 4/3. Perhaps you mean that there is no such number x such that x^n = 2 for any n > 1. In other words, no rational scale with equal intervals? Is that true? > > > "Franz Heymann" <notfranz.heymann@btopenworld.com> wrote in message > > news:<c8o9i3$7fd$2@sparta.btinternet.com>... [quoted text clipped - 43 lines] > If you mean rational ratios, then surely there are idefinitely many > rational ways to factor 2 into numbers 1 < x < 2; e.g., 3/2, 4/3. Strictly speaking Tim should answer this, because he said it. However the meaning is that it has been attempted since the time of the Greeks to construct a scale in which each note is related to at least one other note by a rational factor in which the numerator and denominator are small numbers. > Perhaps you mean that there is no such number x such that x^n = 2 for > any n > 1. In other words, no rational scale with equal intervals? > > Is that true? There is no such scale in which any of the notes may be taken as a starting point for a scale, such that all these possible scales have the same frequency relationships between the notes. Franz > There is no such scale in which any of the notes may be taken as a > starting point for a scale, such that all these possible scales have > the same frequency relationships between the notes. And the equal tempered 12 tone scale just serendipitously provides a fair approximation to a relatively large set. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > > There is no such scale in which any of the notes may be taken as a > > starting point for a scale, such that all these possible scales have > > the same frequency relationships between the notes. > > And the equal tempered 12 tone scale just serendipitously > provides a fair approximation to a relatively large set. Quite. It is a interesting exercise to construct equally tempered scales with an arbitrary number of intervals per octave. I have done that for 7 <= N <= 17. N = 12 wins hands down. Franz > > > There is no such scale in which any of the notes may be taken as a > > > starting point for a scale, such that all these possible scales [quoted text clipped - 11 lines] > > Franz I think blues needs at least N=24. There a lot of string bending. 48 is not uncommon in blues. I've also seen N=48 used in voice recordings. <snip> > What do you mean by "integer ratio rule"? > > If you mean frequencies in ratios of whole integers, obviously the > smallest possible interval is the octave. No, that can't be it. For example 3/2 is a ratio of "whole integers" - a redundancy as ALL integers are "whole numbers". > If you mean rational ratios, then surely there are idefinitely many Do you mean "indefinitely" or "definitely"? > rational ways to factor 2 into numbers 1 < x < 2; e.g., 3/2, 4/3. THere are only two factors of 2, and they are 1 and 2. Factors are integers. Having done a lot of work with continued fractions, I can assure you that any positive real number can be approximated to any required degree of accuracy (epsilon, to a mathematician) by a ratio of two integers. Pi is well approximated by 22/7 if you are an engineer, and by 355/113 if you are a scientist. The key to musical "harmony" is that two notes have frequencies that are in a ratio of whole numbers to each other so that they will share some harmonics. For example, the frequency of concert A is 440 Hz. Its (first, second, and third) harmonics (respectively) will be the frequencies: 440 * 2 = 880 Hz 440 * 3 = 1320 Hz 440 * 4 = 1760 Hz This will harmonize with notes that have frequencies equal to the harmonic frequency divided by *some other* integer, such as 880 % 3 = 293.333 Hz 880 % 4 = 220 Hz 1320 % 2 = 660 Hz 1320 % 4 = 330 Hz 1760 % 2 = 880 Hz 1760 % 3 = 586.666 Hz All of these notes will harmonize with concert A. > Perhaps you mean that there is no such number x such that x^n = 2 for > any n > 1. In other words, no rational scale with equal intervals? > > Is that true? It is true that there is no rational nth root of 2 - no rational number x such that x^n = 2 - but the irrational numbers can be approximated to any desired degree of accuracy (i.e., for audible frequencies we can distinguish frequencies that are different by about 0.2 Hz) with a rational number. As I have mentioned elsewhere, the scale we use is a compromise between having enough notes available to produce an interesting variety of chords and NOT having pianos with more keys than a Chinese typewriter. Tom Davidson Richmond, VA > <snip> > > > What do you mean by "integer ratio rule"? > > > > If you mean frequencies in ratios of whole integers, obviously the > > smallest possible interval is the octave. No, that can't be it. Okay, I may have been guilty of sloppy language, Franz stated my position very well though. Since the time of the ancient Greeks, it has been known that intervals whose frequencies were related to each other in the form of ratios of small integers such as 2/1, 3/2, 4/3, 5/3, etc., were perceived as harmonious by humans. As Franz also noted, forming a scale out of those ratios is impossible without compromise, therefore a large set of compromise systems has been developed. The current compromise using 12 roots of 2 works very well and is called equal temperament. There are other compromises used for special purposes. One way of defining these simply is to ask the question, IF A = 440, THEN what is C? And now we have an interesting point of confusion caused by the people on this list. Middle C is the C below A 440. If you use your 2^(n/12) rule for finding the frequency, you will get 263 and change. If you use the rules for just temperament you'll get slightly different but close. So you can answer the question, IF A = 220 Hz, THEN what is C, by 263 Hz; or you can look on a physicist's desk and pick up his tuning fork. It is clearly stamped 256 Hz. So is the one in the doctor's office. Everywhere in science people have ignored physicial reality and simply defined middle C to be a convenient power of 2, that would not be called C in any known temperament system. No one has accepted responsibility for this travesty and I am STILL waiting for an apology! > Everywhere in science people have ignored physicial reality > and simply defined middle C to be a convenient power of 2, that would > not be called C in any known temperament system. > > No one has accepted responsibility for this travesty and I am STILL > waiting for an apology! :-) Thing is, there is no basis in physical reality for locking A to 440. The overall scaling of the system is arbitrary and has wandered all over the real axis over the years. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein <snip> > And now we have an interesting point of confusion caused by the people > on this list. [quoted text clipped - 5 lines] > Hz; or you can look on a physicist's desk and pick up his tuning > fork. It is clearly stamped 256 Hz. You are assuming (unjustifiably) that physicists and concert musicians use the same scale. These two groups define their scales differently becaue they have different needs to which the scale will be applied. The physicists' scale is *defined* in terms of middle C being 2^8 Hz, which results in much convenience of measurements. The musicians' scale is *defined* in terms of concert A being 440 Hz, which results in aesthetic rewards for the musicians. > So is the one in the doctor's > office. Everywhere in science people have ignored physicial reality > and simply defined middle C to be a convenient power of 2, that would > not be called C in any known temperament system. There is no disregard for "physical reality" involved, merely a different *convention*. The frequencies of 265 Hz and 263.33 Hz are different, but not to a degree that would involve any disregard for physical reality. It is not as if we are deiling with a photoemission work energy threshold or something like that. The scales are as different !and as equivalent! as Arial and Courier fonts. Just pick the one you like and stick with it. If you are working with several other people, use they one THEY agree on. > No one has accepted responsibility for this travesty and I am STILL > waiting for an apology! The "travesty" is entirely self-inflicted. Any time you are willing to sit down and re-examine your own assumptions and realize that no one else here has even been made aware of what your assumptions *are (and are therefore not responsible for their adverse consequences to your aesthetics), then we will be here to witness and accept your apology to yourself. Tom Davidson Richmond, VA > > It is true that when the ratios of frequency are in simple whole > > numbers the ear assigns some qualties to them. That description of [quoted text clipped - 12 lines] > > Bob Kolker Bob, the simple fact of the matter is that when notes separated by an exact 2:1 octave are played together, they are locked together in phase and you can verify this by watching them on an oscilloscope. In other words, the resultant waveform is that of: Sin (2 * Pi * f * T) + Sin (2 * Pi * 2f * T) No neuroscientific explanation is needed here, just a little physics. Where neuroscience could prove helpful is to expain why chord structures have the psychological impact on us that the do, with major chords generally sounding "bright" and minor generally chords being perceived as morose or depressing. Harry C. > > > It is true that when the ratios of frequency are in simple whole > > > numbers the ear assigns some qualties to them. That description of [quoted text clipped - 29 lines] > > Harry C. Whats even wierder is the fact that a Major chord contains a major third interval and a minor third interval. Reverse them and it becomes a Minor chord. Play the minor chord in a second inversion and voila...another Major chord. So what makes a minor sound minor? You have both intervals in both Chord qualitys. Rick Fitzpatrick I think we can conclude from this that the "tonic" has special quality. (Meanwhile, I think it's possible to hear other interesting qualities if we make the tonic more ambiguous.) Still... what makes it all sound so 'interesting' or emotive to us in the first place? --Ken. >Whats even wierder is the fact that a Major chord contains a major >third interval and a minor third interval. Reverse them and it becomes >a Minor chord. Play the minor chord in a second inversion and >voila...another Major chord. So what makes a minor sound minor? You >have both intervals in both Chord qualitys. >Rick Fitzpatrick ***** Heisenberg Certainty Corollary: Free Will isn't just a personal responsibility... it's THE LAW! ***** > > > It is true that when the ratios of frequency are in simple whole > > > numbers the ear assigns some qualties to them. That description of [quoted text clipped - 27 lines] > chords generally sounding "bright" and minor generally chords being > perceived as morose or depressing. I don't find minor chords either morose or depressing. I find them "haunting", for want of a better word. Franz > > "Robert J. Kolker" <robert_kolker@hotmail.com> wrote in message > news:<5o0qc.12793$gr.1177559@attbi_s52>... [quoted text clipped - 48 lines] > > Franz That also works for me, as in Bach's Toccata and Fugue in D Minor. Still, Wagner's "Death of Siegfried" is about a morose as depressing as it gets. Harry C. > > "Robert J. Kolker" <robert_kolker@hotmail.com> wrote in message > news:<5o0qc.12793$gr.1177559@attbi_s52>... [quoted text clipped - 46 lines] > I don't find minor chords either morose or depressing. I find them > "haunting", for want of a better word. And what makes you think your interpretation of "haunting" is not the same as the previous posters interpretation of "morose" or "depressing"? > Franz > > > It is true that when the ratios of frequency are in simple whole > > > numbers the ear assigns some qualties to them. That description of [quoted text clipped - 29 lines] > > Harry C. The brain perceives minors with negative emotions because of it's destructive wave interference pattern. Majors on the otherhand are constructive, and perceived positively. > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 3 lines] > an infinity of colors (or however many the human eye can distinguish within > the EM spectrum). You got more and better answers to the original question than I would've thought possible. However nobody thought to mention what would have been quite topical on a physics newsgroup: One way in which the light-analogy fails is the fact that we do not actually perceive a full octave of light. The lowest red and the most highest violet that are still visible by a human are just not quite a factor of two from each other in frequency. The obvious question becomes: what of those animals that can see a little into the near-UV or near-IR? How do they perceive colors that are exactly one octave from each other? Do they see a "commonality" akin to the acoustic commonality between C and C'? It sure appears as if the sequence from green through blue, indigo and violet could quite naturally be extended "back to red" (which has certainly been done with subjective color-representations like Goethe's color circle). If so, what is the apparent natural selective pressure to limiting the human sensory system through just less than one octave? Would the added confusion about the "two types of red" or "two types or violet" outweigh the (probably marginal) benefits of directly experiencing a wider visual spectrum? In this context I'd like to note that there is a "violet-like" color that is called "purple" (and is generated in pigment by admixing a certain red component to violet). One question that pops into my mind in this context is whether it might be possible to build a "wave-band compressor" that transmits red unchanged and all shorter wavelengths redshifted by an amount proportional to their wavelength so that violet input becomes blue output and near-uv input gets shifted into the visible violet. How would the world look like through such a filter? Or would it be mundane and uninteresting? Finally: while the assignment of the "colors of the rainbow" sure appears to be rather arbitrary, I can't fail to note that there appear to be seven steps to the octave and thus I reserve the right not to be surprized if the same "even integer ratios" that simplify acoustic processing turn out to be responsible for governing visual processing as well. Very little is really invented completely anew in signal-processing, mostly new circuitry is merely upscaled and adopted things that were successful in the past somewhere else. Nothing to see here. Just some rambling. Go back to your lives, citizen. cordially Y.T. -- Remove YourClothes before emailing me. > > I'm a biologist, with no background in music and limited physics, so excuse > > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 20 lines] > naturally be extended "back to red" (which has certainly been done > with subjective color-representations like Goethe's color circle). No "octave-like"analogy between visual and aural perception can be made. Sounds are perceived by virtue of a system performing a frequency analysis in the ear. Colours are perceived by virtue oif the relative responses of only three detectors in the eye. [snip] Franz > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. The seven notes are convention. And by convention, most musical instruments are built to seven notes. It is a system of sound reproduction deemed delightfully capable of all the composers desires. Modern musical composition for the flute includes all sorts of extra noises, by the way. Literally claks and flutterring given a symbol. Douglas Eagleson Gaithersburg, MD USA <snip repost> > The seven notes are convention. Agreed. A, B, C, D, E, F, and G. These names are arbitrary. > And by convention, most musical > instruments are built to seven notes. There are actually 12 semi-tones in one octave. The fact is that 12 semitones allows producing all chords in any key to a degree of accuracy acceptable to most untrained ears. While most woodwinds are engineered to this specification, strings and some brass instruments allow for "in-between" notes. Percussion instruments (except for the keyboards) are not built to "notes" per se, although skilled drummers can modulate the tones of their skins, and polytonal bells are also known and used. The mathematics of harmony (frequency ratios of 3:4:5 for major chords, 12:15:20 for minor chords, etc.) are such that only a very large finite number of fractional tones per octave can accommodate all the chords in all the keys *exactly*, but 12 semitones per octave will come pretty close, and provides a good compromise between chord production and managability of the scale. Other musical systems (notably in India, China, and Japan) have a different arrangement with a different number of fractional tones in the octave. > It is a system of sound reproduction deemed delightfully capable of > all the composers desires. > > Modern musical composition for the flute includes all sorts of extra > noises, by the way. Literally claks and flutterring given a symbol. Trombonists slide notes, violinists and guitarists "squeeze" notes as much as a quarter tone, drummers tweak the screws. The word "glissando" leaps to mind. > Douglas Eagleson > Gaithersburg, MD USA\ Tom Davidson Richmond, VA > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. The 7 letters A-G are actually note-names rather than notes. As you suggest, they are an arbitrary convention developed over centuries of western musical tradition. On the piano, the several different 'A's are of course different notes. They have the same name for several reasons, mainly for musical convenience - to a musician, a chord of two notes an octave apart (octave = eight notes, so they have the same name) is the least discordant combination of notes, and notes with the same name have some similar qualities in terms of their usage in harmony and melody. There is normally no confusion as to which note is being referred to - "the A below middle C", "top A" and similar descriptions are used to distinguish between them when necessary. I did not follow the discussion. Nonetheless I am sure, nobody dealt with tones as not necessarily immediately related to frequency but joint autocorrelation instead. I would like everybode to try and use available software as to calculate and display the autocorrelation function of an octave, a major and a minor third, fifth, second, etc. Please compare it with what you heared and tell me the outcome. Eckard Blumschein > I'm a biologist, with no background in music and limited physics, so excuse > me if this is a stupid question. Why are there 7 discrete notes? The pitch [quoted text clipped - 17 lines] > 7 notes and the octaves relate to the continuous spectrum of air wave > frequency? Thanks in advance for any info. The 7 letters A-G are actually note-names rather than notes. As you suggest, they are an arbitrary convention developed over centuries of western musical tradition. On the piano, the several different 'A's are of course different notes. They have the same name for several reasons, mainly for musical convenience - to a musician, a chord of two notes an octave apart (octave = 8 notes, so they have the same name) is the least discordant combination of notes, and notes with the same name have some similar qualities in terms of their usage in harmony and melody. There is normally no confusion as to which note is being referred to - "the A below middle C", "top A" and similar descriptions are used to distinguish between them when necessary. Top posted!! -- Only 5 notes for Chinese music > > I'm a biologist, with no background in music and limited physics, so excuse > > me if this is a stupid question. Why are there 7 discrete notes? The pitch <snip> I don't want to enter into an argument about the relative merits of top-posting or bottom-posting - both AOL and Google bottom-post by default and it is pointless attempting to beat the formatting machines at their own game. There are infinitely many 'discrete notes' (440 Hz, 440.0001000... Hz etc)- notes with the same name (eg A=440Hz, A=880Hz) are not the same note. > Top posted!! -- Only 5 notes for Chinese music > [quoted text clipped - 6 lines] > > <snip> |
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