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#1
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Five eggs and the twelfth harmonic.
My supermarket is selling eggs in boxes of 5 and 10. This European
decimalisation thing has obviously overstepped the mark. The boxes don't work properly. There is cubic close packing, and hexagonal close packing. 5 and 10 are really bad numbers. We don't even have 10 fingers. It's a dreadful historical mistake. Octal would have been fine. Dozens are the best sharing numbers by far, obviously. In a recent discussion with Iain I probably made the mistake of thinking numbers were slightly different when the most likely cause was rounding error. Can anyone confirm that, for an even-tempered scale of n intervals per octave, the mth harmonic of any note will be itself be a note in the same scale if n/m is an integer, or if m is a power of 2? Sorry for the clumsiness of expression. Hence 12 intervals in the most common scale, and the aural assimilation of 2,3,4,6,8 and 12H. I guess ppl argue, somewhere still, about whether the development of our taste for modern harmony has shaped our senses, or whether the history of music was bound to end up this way because of how our ears work. I'm inclined to think there is a dynamic relationship between the two. Action and reaction are equal and opposite. Music theory and psychoacoustics are opposites in some ways, and equal in others. But music is the action. Music comes first. I still have a picture on my wall of the first release of the Yamaha R1 motorcycle, which to my eyes was a revolution in style. After years of copying and then surpassing European and US engineering, the R1 quite suddenly, but nonetheless subtly, subsumed western style into something totally Japanese. No longer at our cutting edge, they became their own. The R1 has Japanese eyes. They also appear, in this process of redefinition, to have lost immediate view of their own history. My son brought a girl from Japan and, desperate for some point of intercultural commonality but depending for my knowledge of her country upon what I have gleaned from playing "Samurai, Total War", I quizzed her on the subject of sea shanties. I played her "Sailor song", sung by Sandy Denny with Fairport Convention, being the only approximation to the genre in my collection, whilst Tom attempted to translate the story. She explained that most Japanese people are more concerned about earthquakes. Sea shanties are not part of popular consciousness. Further the origin of fish troubles them no more than current affairs, to which they seem oblivious. I suspect continuity of tradition is maintained only through the feudal clan system, now a preserve of the great and good. They must be the people who buy the astronomically priced super-high-end Japanese audio stuff. Hopefully China won't make the same mistake. Traditionally they use fewer notes than us, but still within the same structure. There is less history of the complications arising from many instruments playing different notes simultaneously, so although their scale is like ours it is more pure...closer to the ideal even temper, I am told. A popular, caring culture of simplicity and purity should suit valve audio perfectly. I wonder how long we might wait for an amplifier with a Chinese soul? Anyway, I read that intermodulation distortion is bad because it is not harmonic. So what...why isn't harmonic distortion bad because it is not intermodular? What would a scale built around sums and differences, rather than multiples, sound like? Just as some harmonic distortion has a musical effect within the common western scale, is there some kind of music within which IM would be transparent? In the absence of progress in the history of music, it won't be long before these Europeans decimalise it. Time will be next. Dancing won't be much fun. It's a while since I have done any dancing. It's not sausages and chips that's making us fat. We've stopped dancing. Ian |
#2
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
"Ian Iveson" wrote in message . .. My supermarket is selling eggs in boxes of 5 and 10. This European decimalisation thing has obviously overstepped the mark. The boxes don't work properly. There is cubic close packing, and hexagonal close packing. 5 and 10 are really bad numbers. The answer is quite simple buy five and one is a double yolker, buy ten and two have double Yolk's. next thing you'll be reporting a 10 hour CLOCK bassett |
#3
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
Ian Iveson wrote:
My supermarket is selling eggs in boxes of 5 and 10. This European decimalisation thing has obviously overstepped the mark. The boxes don't work properly. There is cubic close packing, and hexagonal close packing. 5 and 10 are really bad numbers. No doubt sold at the same price as 6 & 12 were before? |
#4
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
"Ian Iveson" wrote in
message Anyway, I read that intermodulation distortion is bad because it is not harmonic. So what...why isn't harmonic distortion bad because it is not intermodular? I think you are missing a critical piece of knowlege - audio equipment does not have harmonic or intermodulation distortion. Those are just abstractions and mesruement techniuqes that we use to characterize a certain class of imperfections in audio gear. The general class of imperfection in audio gear that leads to the abstractions we call harmonic and intermodulation distortion, is called curvature of the input-output curve. An ideal piece of high fidelity gear will have a straight-line input-output curve. IOW, if you put in 1.000... volt and get out 2.000... volts, then if you put in 2.000... volts you will get 4.000... volts. If the input-output curve is nonlinear, if you put in 1.000... volt and get out 2.000... volts, then if you put in 2.000... volts you might get 5 volts, or 4.1 volts or 3.9998 volts, but not 4.000... volts. As a general rule, any piece of audio gear that produces spurious harmonic signals while amplifying music, will concurrently generate spurious intermodulation signals. So, trying to separate the two can be a big waste of time, and lead to all sorts of thoughts that are not productive. |
#5
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
On Apr 11, 11:54 pm, "Ian Iveson"
wrote: I guess ppl argue, somewhere still, about whether the development of our taste for modern harmony has shaped our senses, or whether the history of music was bound to end up this way because of how our ears work. I'm inclined to think there is a dynamic relationship between the two. Action and reaction are equal and opposite. Music theory and psychoacoustics are opposites in some ways, and equal in others. But music is the action. Music comes first. Just as some harmonic distortion has a musical effect within the common western scale, is there some kind of music within which IM would be transparent? One issue that you haven't raised is the problem of Equal Temperament. This modern tuning system is already a distortion of the natural harmonic set of overtones. I don't have enough electronics knowledge to answer your question, but I am certain that tuning systems would have to be taken into account as part of the answer. |
#6
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
Ian Iveson wrote: Can anyone confirm that, for an even-tempered scale of n intervals per octave, the mth harmonic of any note will be itself be a note in the same scale if n/m is an integer, or if m is a power of 2? Sorry for the clumsiness of expression. Uh ? Graham |
#7
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
Ian Iveson wrote: Can anyone confirm that, for an even-tempered scale of n intervals per octave, the mth harmonic of any note will be itself be a note in the same scale if n/m is an integer, or if m is a power of 2? Sorry for the clumsiness of expression. It may well be. I fear that transposition to the sopranino Sarusaphone and the counter-clockwise ukulele may prove to be the exceptions to the rule. Iain |
#8
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
The discussion of harmonics with respect to instruments is quite
fascinating, and a bit a diversion for those interested. Tuned string instruments use a 'just' or diatonic tuning--that uses perfect fifths or fourths between the strings. This results in harmonics that create natural resonance from string to string. Every good string player knows if the instrument is out of tune by playing just one string--the harmonics give it away. Percussion instrumnets, like the piano, use a tempered [close to, but not the exactly the same as Pythagorean, which is based on the 12th root of 2] tuning which flats the fifths (or sharps the fourths) ever so slightly. This is not percieved as out of tune (for most people) becuase the three strings per note in the upper register muddy the out-of-tuneness of the sound. In the bottom register where there is just one string per note--our sense of pitch is not as accurate. Harmonics are totally different on piano--and have much less impact. However, I know piano tuners who use special high-tension strings for the bass just to create extra sympathetic harmonic resonance in the upper registers. They also tune the piano slightly different as well, but I'm not sure what the algorithm is. A harspsichord is a string instrument because the strings are plucked-not struck with a hammer. It can be tuned either 'tempered' or 'just', but it sounds out of tune if a tempered tuning is used. When I was the concert master of the college symphony and on tour, it was my job to tune the harpsichord for each concert. Since it was used only for one piece in the first half and one in the second half (most of the time) I would retune the harpsichord for each specific performance so that it was diatonically tuned to the correct key for that performance. If one piece was in Emaj and the second in Cmin, that meant a lot of work--as almost every note had to change. Going from Amin to Dmaj on the other hand was just a quick touch up since D is the dominant in the key A. An interesting side note is that on a 'just' tuned instrument--there can be problems when playing unaccompianed. For example: in the E minor Partita, first movement, for the unaccompianed violin (piece every one is familiar with in these days of ring tones) there is a modulation from E minor to B major to F# , back to E minor. The key of F# minor uses no open strings, so the violinist by ear will naturally tune F# to be a perfect fith to B major. Unfortunately, if you do that, when you transition back to E minor--the open E string is now out of tune with respect to the F# key you were just playing. Thus the violinist has to slightly flat the transition to F# from B in order to keep the following modulation back to E in tune. On a piano of course, this problem would not exist. On other interesting side note is that genetically there are people who sense of accuracy of pitch is so acute that they can not stand to listen to music. Everything sounds out of tune to them. These are usually the children of two classical concert musicians. At the other end of the scale is a rock concert. The volume in the bass is usually turned up until the distortion is so high the center of the note cannot even be found. Can anyone confirm that, for an even-tempered scale of n intervals per octave, the mth harmonic of any note will be itself be a note in the same scale if n/m is an integer, or if m is a power of 2? Sorry for the clumsiness of expression. Hence 12 intervals in the most common scale, and the aural assimilation of 2,3,4,6,8 and 12H. I guess ppl argue, somewhere still, about whether the development of our taste for modern harmony has shaped our senses, or whether the history of music was bound to end up this way because of how our ears work. I'm inclined to think there is a dynamic relationship between the two. Action and reaction are equal and opposite. Music theory and psychoacoustics are opposites in some ways, and equal in others. But music is the action. Music comes first. I still have a picture on my wall of the first release of the Yamaha R1 motorcycle, which to my eyes was a revolution in style. After years of copying and then surpassing European and US engineering, the R1 quite suddenly, but nonetheless subtly, subsumed western style into something totally Japanese. No longer at our cutting edge, they became their own. The R1 has Japanese eyes. They also appear, in this process of redefinition, to have lost immediate view of their own history. My son brought a girl from Japan and, desperate for some point of intercultural commonality but depending for my knowledge of her country upon what I have gleaned from playing "Samurai, Total War", I quizzed her on the subject of sea shanties. I played her "Sailor song", sung by Sandy Denny with Fairport Convention, being the only approximation to the genre in my collection, whilst Tom attempted to translate the story. She explained that most Japanese people are more concerned about earthquakes. Sea shanties are not part of popular consciousness. Further the origin of fish troubles them no more than current affairs, to which they seem oblivious. I suspect continuity of tradition is maintained only through the feudal clan system, now a preserve of the great and good. They must be the people who buy the astronomically priced super-high-end Japanese audio stuff. Hopefully China won't make the same mistake. Traditionally they use fewer notes than us, but still within the same structure. There is less history of the complications arising from many instruments playing different notes simultaneously, so although their scale is like ours it is more pure...closer to the ideal even temper, I am told. A popular, caring culture of simplicity and purity should suit valve audio perfectly. I wonder how long we might wait for an amplifier with a Chinese soul? Anyway, I read that intermodulation distortion is bad because it is not harmonic. So what...why isn't harmonic distortion bad because it is not intermodular? What would a scale built around sums and differences, rather than multiples, sound like? Just as some harmonic distortion has a musical effect within the common western scale, is there some kind of music within which IM would be transparent? In the absence of progress in the history of music, it won't be long before these Europeans decimalise it. Time will be next. Dancing won't be much fun. It's a while since I have done any dancing. It's not sausages and chips that's making us fat. We've stopped dancing. Ian |
#9
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
Arny said:
Anyway, I read that intermodulation distortion is bad because it is not harmonic. So what...why isn't harmonic distortion bad because it is not intermodular? I think you are missing a critical piece of knowledge - audio equipment does not have harmonic or intermodulation distortion. Those are just abstractions and measurement techniques that we use to characterize a certain class of imperfections in audio gear. The general class of imperfection in audio gear that leads to the abstractions we call harmonic and intermodulation distortion, is called curvature of the input-output curve. An ideal piece of high fidelity gear will have a straight-line input-output curve. IOW, if you put in 1.000... volt and get out 2.000... volts, then if you put in 2.000... volts you will get 4.000... volts. If the input-output curve is nonlinear, if you put in 1.000... volt and get out 2.000... volts, then if you put in 2.000... volts you might get 5 volts, or 4.1 volts or 3.9998 volts, but not 4.000... volts. As a general rule, any piece of audio gear that produces spurious harmonic signals while amplifying music, will concurrently generate spurious intermodulation signals. So, trying to separate the two can be a big waste of time, and lead to all sorts of thoughts that are not productive. I am hoping someone else might query your first paragraph, Arny, which I wouldn't have expected from you. I'm shocked As an act of devotion, I passed it through my American spellchecker. I'll give it another day or so, if you don't mind, save to say that *all* thoughts are potentially productive, and to agree with your implied definition of a non-linear system, from which it follows that, in the output of such a system, the IMD and HD it produces are interrelated (dammit). Thanks, Ian |
#10
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
Eeyore grunted
Can anyone confirm that, for an even-tempered scale of n intervals per octave, the mth harmonic of any note will be itself be a note in the same scale if n/m is an integer, or if m is a power of 2? Sorry for the clumsiness of expression. Uh ? I'm a prophet, not a dictionary. Ian |
#11
Posted to rec.audio.tubes
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Five eggs and the twelfth harmonic.
coffeedj wrote
The discussion of harmonics with respect to instruments is quite fascinating, and a bit a diversion for those interested. Tuned string instruments use a 'just' or diatonic tuning--that uses perfect fifths or fourths between the strings. This results in harmonics that create natural resonance from string to string. Every good string player knows if the instrument is out of tune by playing just one string--the harmonics give it away. Percussion instrumnets, like the piano, use a tempered [close to, but not the exactly the same as Pythagorean, which is based on the 12th root of 2] tuning which flats the fifths (or sharps the fourths) ever so slightly. This is not percieved as out of tune (for most people) becuase the three strings per note in the upper register muddy the out-of-tuneness of the sound. In the bottom register where there is just one string per note--our sense of pitch is not as accurate. Harmonics are totally different on piano--and have much less impact. However, I know piano tuners who use special high-tension strings for the bass just to create extra sympathetic harmonic resonance in the upper registers. They also tune the piano slightly different as well, but I'm not sure what the algorithm is. A harspsichord is a string instrument because the strings are plucked-not struck with a hammer. It can be tuned either 'tempered' or 'just', but it sounds out of tune if a tempered tuning is used. When I was the concert master of the college symphony and on tour, it was my job to tune the harpsichord for each concert. Since it was used only for one piece in the first half and one in the second half (most of the time) I would retune the harpsichord for each specific performance so that it was diatonically tuned to the correct key for that performance. If one piece was in Emaj and the second in Cmin, that meant a lot of work--as almost every note had to change. Going from Amin to Dmaj on the other hand was just a quick touch up since D is the dominant in the key A. An interesting side note is that on a 'just' tuned instrument--there can be problems when playing unaccompianed. For example: in the E minor Partita, first movement, for the unaccompianed violin (piece every one is familiar with in these days of ring tones) there is a modulation from E minor to B major to F# , back to E minor. The key of F# minor uses no open strings, so the violinist by ear will naturally tune F# to be a perfect fith to B major. Unfortunately, if you do that, when you transition back to E minor--the open E string is now out of tune with respect to the F# key you were just playing. Thus the violinist has to slightly flat the transition to F# from B in order to keep the following modulation back to E in tune. On a piano of course, this problem would not exist. On other interesting side note is that genetically there are people who sense of accuracy of pitch is so acute that they can not stand to listen to music. Everything sounds out of tune to them. These are usually the children of two classical concert musicians. At the other end of the scale is a rock concert. The volume in the bass is usually turned up until the distortion is so high the center of the note cannot even be found. Thanks, in my library of real-world intelligence. It seemed to me I should find out something about what I'm trying to amplify. cheers, Ian |
#12
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Five eggs and the twelfth harmonic.
Arny said
Anyway, I read that intermodulation distortion is bad because it is not harmonic. So what...why isn't harmonic distortion bad because it is not intermodular? I think you are missing a critical piece of knowlege - audio equipment does not have harmonic or intermodulation distortion. Those are just abstractions and mesruement techniuqes that we use to characterize a certain class of imperfections in audio gear. I'm still surprised you said this, and still hoping someone else might respond. Last time I tried to make the distinction there was clamour from the defenders of the faith. The general class of imperfection in audio gear that leads to the abstractions we call harmonic and intermodulation distortion, is called curvature of the input-output curve. Urgh. That's an awkward and ugly expression! Never come across it before, thankfully. An ideal piece of high fidelity gear will have a straight-line input-output curve. IOW, if you put in 1.000... volt and get out 2.000... volts, then if you put in 2.000... volts you will get 4.000... volts. If the input-output curve is nonlinear, if you put in 1.000... volt and get out 2.000... volts, then if you put in 2.000... volts you might get 5 volts, or 4.1 volts or 3.9998 volts, but not 4.000... volts. Good. Close to a textbook definition of "non-linear". As a general rule, any piece of audio gear that produces spurious harmonic signals while amplifying music, will concurrently generate spurious intermodulation signals. So, trying to separate the two can be a big waste of time, and lead to all sorts of thoughts that are not productive. A reduction to absurdity was implicit, I hoped. So in my scale, based on intermodulation rather than harmonic intervals, the IM distortion may be assimilated, but the attendant and inescapable harmonics all sound unmusical. Damn, and I've just about finished working out the fret spacing. Back to the drawing board. Incidentally, is a "spurious harmonic signal" the same thing as "just an abstraction"? cheers, Ian |
#13
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Five eggs and the twelfth harmonic.
"Ian Iveson" wrote in
message . uk Arny said As a general rule, any piece of audio gear that produces spurious harmonic signals while amplifying music, will concurrently generate spurious intermodulation signals. So, trying to separate the two can be a big waste of time, and lead to all sorts of thoughts that are not productive. A reduction to absurdity was implicit, I hoped. So in my scale, based on intermodulation rather than harmonic intervals, the IM distortion may be assimilated, but the attendant and inescapable harmonics all sound unmusical. More like the reverse. The harmonics may be assimilated, but the attendant and inescapable IM is very likely to sound unmusical. The harmonics may be assimilated because they occur at the same frequencies as the harmonics that are part of the musical input signal. A musical instrument may also generate some intermodulation products, but they are generally smaller than the harmonics. The low-order harmonics in some musical signals may exceed the amplitude of the fundamentals. Incidentally, is a "spurious harmonic signal" the same thing as "just an abstraction"? The spurious harmonic signal is a real signal, just like the spurious intermodulation signal. Nothing abstract about either of them. |
#14
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Five eggs and the twelfth harmonic.
Arny Krueger wrote
...So in my scale, based on intermodulation rather than harmonic intervals, the IM distortion may be assimilated, but the attendant and inescapable harmonics all sound unmusical. More like the reverse. The harmonics may be assimilated, but the attendant and inescapable IM is very likely to sound unmusical. Sorry Arny, but I don't think you are quite following my argument. That's OK, though, I was only musing and as you said, more or less, not all thoughts are useful or interesting to everybody. With respect to a scale based on intermodulation, rather than harmony, IM would be assimilated, and harmonics would stick out like sore thumbs. I am interested in opposites per se, but I must admit it is a minority pursuit. So I find it remarkable that, to define a harmonic scale, the difference between the top and bottom notes is a given, and the only basic decision that needs to made is how many intervals should divide it. Twelve turns out to be the optimum number, as far as history has so far decided, but other numbers are feasible. For a scale based on intermodulation, in contrast, the intervals choose themselves, and the only decision that needs to be made is the difference between the top and bottom notes. Now I am just beginning to wonder to what extent the intermodulation products arising from harmonic intervals are themselves harmonious. I know I'm barking. It may not even be the right tree. Oh well, ho hum, etc. As you were... The harmonics may be assimilated because they occur at the same frequencies as the harmonics that are part of the musical input signal. A musical instrument may also generate some intermodulation products, but they are generally smaller than the harmonics. The low-order harmonics in some musical signals may exceed the amplitude of the fundamentals. This seems much the same as what I wrote in the first place, that you seemed to disagree with. Incidentally, is a "spurious harmonic signal" the same thing as "just an abstraction"? The spurious harmonic signal is a real signal, just like the spurious intermodulation signal. Nothing abstract about either of them. But I don't see how you square this with the rest of what you wrote, e.g. "I think you are missing a critical piece of knowlege - audio equipment does not have harmonic or intermodulation distortion. Those are just abstractions and mesruement techniuqes that we use to characterize a certain class of imperfections in audio gear. "The general class of imperfection in audio gear that leads to the abstractions we call harmonic and intermodulation distortion..." I tried some time ago to discuss the difference between identity and equivalence, particularly with respect to Fourier. That went quite badly I recall. cheers, Ian |
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