The Interplay of Music and Physics 2/2

and in fact different cultures will divide the scale in different ways with different kinds of consonant intervals but these fundamental simple whole-number ratios appear to be universal and again that is connected to the way that year the brain want to hear those simple whole numbers you call that a musical interval so which is thought mathematically minded in the audience is how do we make a scale now we've seen the intervals the octave the perfect fifth perfect fourth and so we can build a scale out of those in fact pythagoreans built all of their scales entirely on these intervals have no others so they built all their scales by fifths and octaves so in order to come to another pitch they would multiply or divide by three halves that ratio of 3 to 2 or they multiplied or divided by 2 the ratio of 2 to 1 or the octave and so all of their adjacent notes were related either by a ratio 98 in the frequency or a ratio 256 or 243 I told it that grants love their numbers so if we have all of the different frequencies so if we start reading see scale so we start with our lowest frequency we call that one and then the d the next pitch up is in a ratio of nine to eight with the with the sea and we get there by multiplying by three halves twice and dividing by two and we can go all the way up the scale here we get the perfect fourth on the perfect fifth and all the way finally up to the octave so simply by using ratios of 3 2 or 2/1 the Pythagorean built up with our scale and it was based on the perfect fifth of the octave in the fourth Oh this gives you a perfectly respectable and usable scale but it has the problem and the problem is that suppose you start at C at very low C 32 Hertz and you go up my fifths you get you know C 2 GH d 2 and so forth and these are the frequencies and so you end up after you've gone up this high you end up with what we would call be sharp and that's a 4150 2 Hertz just you just bend over if you start on the same note but you all know why octaves multiply by 2 you go so from C to C to C see you come up to what you would call C and it's not the same frequency as that be sure now those of you play the piano you know that you know B sharp and C are the same note the note that is above B natural which would be B sharp is C but if you start at C and arrive at that note by two different roots you don't get the same pitch and these two are very noticeably different it doesn't take a trained ear to hear the difference and so the oh and that's a simple matter of mathematics that if three-halves to the twelfth power is not equal to two to the seventh power so you're never going to get this to close and this just this discrepancy is referred to as the and it's an interval that comes because when you go up like this and you go back is you don't end up with the same go so if you want to be a scale that you are going to end up with the same note we have to do some adjusting and that's the up the practice that's known as temporary or adjusting the scale and so how are we gonna be well the ear demands the pure lock the octave is really the most fundamental range of that frequency so we need to keep that so we have to adjust everything else to get rid of this Pythagorean comma but keep the octaves pure and there's different ways of doing them so here are here's the Pythagorean and we see so we start here and C and we go up to to see it twice that frequency we keep our fifths pure and we allow the other notes we adjust the other notes to get rid of that so the pythagoreans kept the fifth fewer and let the up let the other pitches be a little adjustment in the 16th 17th centuries that was the practice of me to up Everest where they wanted to keep one have the octaves pure but they also wanted to help the thirds to be pleasing and constant are pretty major third so they would allow the fifth from the fourths to be adjustable so they would change the pitches of the fifth importance on the scale that that's the modern practice is let's hold equal terminals where you keep the octave the octave and you adjust all the other notes so that they have the same interval between every two successive notes every two successive studies tell us and so the interval between successive notes the frequency interval between successive notes is the 12th root of 2 or 1.05 not but the frequency of c-sharp is one point two nine five times the frequency of see the frequency D does that say that a multiple or frequency of c-sharp and so forth so you taking that Factory and comma and you just spread equally over all of the pitches so all of the pitches in equal temperament are equally a lacuna are equally and it's simply a compromise that you have to make in order to count the different keys sound the same you know no piano you can play a piece at C and then you can play it safe decent t-shirt then you can play it and if it's not on the same pitch will go out but but you won't receive any difference in the intervals between the DOS the relationships are just one to each other if you have for example hard to you know two-dimensional temper that you play and see you plan C sharp and it will sound very different and those that work in nice ratios to each other and see won't be in Nice racers to each other the C sharp so you don't have the freedom to modulate but that's why equal-tempered has become the standard an opportunity that's because it allows you to play in any key you like and it will always sound in the same it's not the same as it did in the other key but different even in western music at different times people use different tunings and so if you listen to composers who use these tunings took advantage over if they wanted something to sound nice and pleasing heavy music they would use it in a key in which these intervals sounded pleasing if they wanted them to trust emotional distress and love happiness of Rage 1 up they be then modulated quickie in which these intervals down to the sembra then worked in the same ratios and so you could actually use that in a an expressive sense um the practice of modifying instruments to make them more useful better sound or whatever like you know for that better developing the piano following after the harpsichord is an ongoing process that's still routine use to today um here's an example of a flute player of underwater food but the holes instead of being round or square and if you think about the details of the the pressure distributions within the air column you can see yes that would make some a difference and people who play the flute can hear the difference it's so it's not a dramatic difference but it is thick and this hasn't caught on to weapons made by a flute maker in Asheville North Carolina and he's he's sold a number of them but has certainly taken the world by storm um Dizzy Gillespie modified this instrument you didn't accidentally at least and but he said afterwards that he liked the sound better in the accidentally modified instrument in the original instrument so thereafter when he kind of new trumpet me he didn't like that because he liked it and again its salt but the pressure distribution is going to change if you put a pin in that - and so you get a slightly different sound one that he found to be more pleasing and new instruments are still being made new instruments the pen she needs work look what we think it was now but the standard clue usual clue was really a complete reworking of the flutes that existed in the mid 19th century by terrible vote and so he completely reworked it based on acute liquidity principles and made it in fact much easier to play much easier combination of keys without having to move your fingers around on the flute the saxophone yes a given instrument that was invented in the 19th century didn't exist before by at all Sachs Tennessee saxophone and it has a very logical system he is actually relatively easy instrument to play from that point of view so so new instruments continue to be made and some of them become part of our ordinary concert and other kinds of music jar like these two instruments other instruments have not yet quite gotten there yet and so I want to show you a few both really notes that have been made by students in my class one of the assignments in the class that I teach is each student must make a stringed instrument and a wind instrument out of whatever they can find lying around their dorm and then the last day of class we have a concert in which the students play their compositions and it's interest to show you few examples this is the pre olegmon space it's made out of a real box that's a spoon support it's in hearts of court accepts that set of strings that has spoons and that one actually as you can see this is the slid over the copper sled and if you go to my website he and the bridge and then there's professor flame Joe it's made out of the bucket and the plant hanger and some other events and if you go to my web site you can hear duet between this little Abigail if we would come into this category but it was not made by the students here's some more the same three do out of a Halloween decoration a soda ball a Dixie cup and a lava cake and this thing had to notice one of them will see sharpened the composition that she produced actually but the head of the composition division in the music department came to our concert which she does every year and he said given what she was working she did a very good job it sounded like that's what she could do the bottle blow purse would get a lot of instruments that are based on blowing across the tops of bottles he assured me that he was not responsible for emptying these bottles there's a freshman it would have been illegal this one is a sort of a funeral and it played in an a he played it in a an arrangement waltz which was very good pan types of course are a very good instruments in this and in fact when this young lady named her Pam thought she was a flute player doesn't work really well she could play quite beautifully but she learned a real world lesson being making this because she sat down who worked out all of the weight that she needed or the appropriate pitches that she wanted she figured out what scales who wanted to tune into and all that and so she went into the hardware store and said please mr. barber sir ma'am would you please cut me copper tubing in these lights I need that interest yes this guy well you didn't believe me when I told her that about Lee cement glue and so a corporate minister but she was a drummer player she made a pleasure Paris and plays we've already trained Iran it sounded wonderful so these new instruments are probably not coming to any concert hall near any festival but who knows what new instruments will be developed in the future that will be somewhat more effective and that may turn up in different kinds of music that you will be listening to in the future so with that I will just leave you with thank a few people my colleague professor grant Witek and of course the students of the class and dog Aylor as a professor clarinet clarinet spectrum I showed you Mike Schultz is the principal oboist of the North Carolina Symphony he kindly provided the OCO spectrum those of you who are interested in learning more about this subject I have three things to recommend to you for a book that's aimed at the general public is called measured tones by Australian physicist and musician if you want the hardcore stuff the real details with all the Bessel functions and all the wonderful mathematics I recommend this book for you and then to have a conversation with one of the great minds of the 19th century I recommend to you Helmholtz is on the sensation of tone it's available Zadora paperback and it was a work that all candidates for musical degrees were expected to study in the 19th century and my colleagues in the music department actually wish that that were still true but they find it a little hard to enforce that but if you're interested I highly recommend using and so with that I'll come to the close of a rather flying tour to musical acoustics is why we've received certain pitches as being pleasant and other pitches not and as hardly felt has to do with the two pitches told together sounded together if they have a lot of frequencies in common the people are working on the auditory pathway and how's that work that's not other students the wonderful subject you research for Silas yes please play like you for knowing what the treble be able to go up display like that mozart was across it in summary differ because Mozart of course was writing his peace knowing the acoustic characteristics of the deal and so I'm not American except reporting with this I certainly don't know well but if it were to if someone were to play to heard of the viola in that piece the balance between the two instruments now would be very different it would not be been about not might be pleasing but it would not be the same because Mozart knew with the PLO syllable and he knew how to use it most effectively to accomplish in his musical ideas so now I'm not aware of anyone having made recording of that piece double streams he's passing over well I mean a double string person what number missiles would build strength and the two are not tuned to the same so that that to get those combinations of sounds digits depending on the frequency for tuning the honking is mostly because the reticle is smaller than it should be in order to amplify those would be older that's out there but it's undersized worth hitches and so you tend to get more of a higher frequencies which gives it and that's because the actual distribution of the pressure and low pressure is some more complicated than I was giving you the first simple version and then once you start doing across fingerings the pressure distribution gets a little more complicated such that you can bring out frequencies you never has no frequencies that are simply is into a short leash which you and the size of the holes manage to so how big they are the diameter the hole relative to the diameter of the tube will make a difference in frequencies so what you're doing is if you go back slide here okay what you're doing is okay okay the question is what the phenomenon of harmonics on the oilmen where you get these these low string can make a very high pitch by just gently touching the string right position what you're doing is if you touch the string in the middle right in the center this lowest frequency vibration no longer can happen because your finger is keeping after happening so now the lowest frequency vibration is the red one which is an octave higher so the picture here is the octave higher and it has that sort of thin and in substantial sound because any of these frequencies with a maximum vibration in the middle will be suppressed so you'll get only the 2nd the 4th the 6th so it's a much thinner material and of course you can do that you can do that not just the middle but if you put your finger here you're going to suppress both the first of the second but not the third and so now you're going to get the optimum to fifth and of course there's two places you can do that and put your finger to different places and it will say come on yeah it's hard I was young as much string work with charities guitar it's not so far the question regarding that I played the flute in order to get the full tone you produced over time with the same fingering so you practice the overcome and then you should complain yes what isn't that physics okay it's a matter if they don't think yeah secretly they may be that you're preferentially excited that second why exactly don't you will probably expect the vibration from blow brothers so that you enhance that higher so you can actually you can make that but you can play the second octave without configured as you're safe by enhancing that second partial and suppressing the preferable all the details of that very complicated well they think they didn't really understand the place but they had their fresh drinks and they would change the length with Aparicio in fact we have our students do this we take my words and they sit there and they rediscover all these simulant factors discoveries as they didn't understand the way physical sparkly now but they could look at the ratios of string lengths they were taking them all the same and then just change and those are just ratios a small thing trying to do it oh yeah writing it out would be very complex but visualize I mean literally there's what used to be it on the strength so you can see ok that's twice as long as that so having the physical object I mean you just keep keep coming down to two thirds or gaps and those are pretty easy to do but the only ratios they used so that made it do