Re: Two interesting projects
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Hello Jeff,? In the pendulum frequency formula, the factor ¡®k¡¯ includes the force of gravity. For a stretched string, the ¡®k factor becomes a function of the string tension. Of course, it¡¯s not as simple as that because stringed instruments ¨C harp, guitar, piano, violin ¨C anchor the stretched string at both ends. At the fundamental frequency, the string is one half-wavelength long, with nodes at each end. Natural strings, those with uniform distribution of mass per unit length, will therefore only produce odd harmonics ¨C 3rd, 5th, 7th ¡ Violinists can choose to emphasise or attenuate specific harmonics by choosing exactly where to bow or pluck. Hence, the ¡®sweetness¡¯ of the violin¡¯s sound depends almost entirely on the violinist¡¯s skill. You can do the same with your guitar.??? In order to suppress that most disharmonious 7th harmonic on the piano, the hammer strikes at the node of the 7th harmonic. Re-tuning a vintage piano, that is, one designed for A = 420 Hz, to bring it up to modern standard pitch of A = 440 Hz, the additional tension on the harp will shorten the harp and automatically shift the hammer strike point away from the node of the 7th harmonic. The tension on the harp is measured in tonnes ¨C several! The sound board to which the harp is attached may well crack under such insult. Good piano tuners often refuse to raise the pitch of older pianos.? The formula for the fundamental frequency, f1, of a string anchored at both ends is:? f1 = ? ¡Ì(T / mL) where: T = string tension m = string mass L = string length? Now you can see why I referred to the pendulum formula ¨C I was using a mathematician¡¯s shorthand.? For over-wound strings, the mass per unit length is not uniform, and hence the harmonics are not exactly 3rd, 5th and so on. The amount and style of over-winding gives each piano manufacturer¡¯s design its specific ¡®voice¡¯. Over time, the repeated striking of the over-wound strings results in relaxation of the over-winding pattern, and the note sounds increasingly dull. A cunning piano tuner will then release the tension on the dull string, rotate the string at the end furthest from the hammer, re-tension and re-tune to get some of the previous harmonic richness back; this method is cheaper than getting a new string wound.? For string basses, some players like to use over-wound strings because of the smaller tension on the tuning peg. A further benefit is a ¡®richer¡¯ sound from such over-wound strings ¨C the string excites more air. Purists will still prefer non-over-wound strings. You know that over-wound strings are available for tenor and bass guitars.? More grist for your mental mill.? Now that you know about the equally-tempered scale, you know that G should actually be 199.59977 Hz. So, if your tuning fork is exactly 196.0000 Hz, and if you could tune accurately to that tuning fork, fellow musicians whose instruments had been fine tuned to the equally-tempered scale would sound about 0.4 Hz flatter than you. This would be experienced as a beat approximately every 2.5 seconds. Unless you use an electronic device to stretch tone level over several seconds (eg, a sustain pedal), neither you nor your fellow musos would know. However, if you play a G a couple of octaves higher, the 0.982 Hz beat will be more noticeable over a shorter time period, ie, less than a second. Some string players attempt to get around this problem by rocking the ball of their finger over the string on the finger board, or fret board, to induce a frequency tremolo.? When I tune my piano, I use an electronic tuning fork that I have compared, using my oscilloscope, with the tones from WWV. Even when I get within 1 cent, I still listen for beats for 13 seconds. I don¡¯t look forward to tuning my piano because it takes me several hours. My piano invariably goes flat. I pull up the middle octave of 12 notes, the harp compresses, all the adjacent notes go flat. So, I then do the octaves above and below the middle octave, and then retune that middle octave. And so on, leap-frogging back and forth.? Cheers, Brian.? From:
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[mailto:[email protected]] On Behalf Of Jeff
Green ? On Fri, Nov 25, 2022 at 11:50 PM, Brian wrote:
Thank you, you answered
questions I never knew I had. ? The math for a double body pendulum is .... interesting.? ? One thing the equation leaves out is tension. As I think I said, I play guitar (with perhaps more energy then skill.) So I am very familiar with tuning. ? I have a 196Hz tuning fork, open ¡°G¡±, that I used until my EE friend showed me a $25(US) Deltalabs CT-30 clamp on tuner. ? Guitarists with ¡®the touch¡¯ can produce harmonics on the 3rd and 6th fret. A good way to check proper fret placement is to compare the harmonic touching lightly above 12th fret with the note produced on the 12th fret. ? The bridge on most acoustic guitars is slanted so the higher strings have a short overall length to allow for the difference in wound versus straight strings. ? I thought all the strings in a piano had multiple strings. Our daughter once told me ¡°Daddy I love you but you are not to open the lid on my piano!¡± ? For some reason she thought I might play with the innards. I¡¯ve been foolish more times in my life then I like to admit, but even I¡¯m not silly enough to mess with the inside of a delicate piece of equipment. ? I¡¯ve played with a Brazilian fretless guitar once and once was enough. One would need perfect pitch to find the note and thinking of making chords on a fretless guitar make me want to sob.? ? If you want to cause minds to melt, go into ¡°transposing instruments.¡± It is easy for guitar, the music is written one octave higher then it¡¯s played. ? Or delve into the never ending war over ¡°Is 440Hz right?¡± ? The Baroque crowd goes for an ¡°A¡± of 415Hz and considers the rest of us to be heretics. ? My EE friend sent me these links.
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We humans
can find the oddest things to fight over. I can¡¯t wait to ask our daughter about the Rhodes Piano, I wonder if she¡¯s ever used one. She says the Clavinova digital piano is almost a real piano. The feel of the keyboard comes very close to the feel of a real piano. ? What I find odd is the Treaty of Versailles has a codicil that sets concert pitch: ¡°Article 282 Section 22 of the Treaty of Versailles: "Convention of November 16 and 19, 1885, regarding the establishment of a concert pitch."? ? A friend tracked down a PNG of the entire treaty and I have a PNG copy of ¡°Article 282 Section 22.¡±?I can place the PNG in the files section but it isn¡¯t informative because if refers to a previous international congress. My friend is trying to track down a copy. ? This offers one explanation for A=440Hz:
"The B.B.C. tuning-note is derived from an oscillator controlled by a piezo-electric crystal that vibrates with a frequency of one million Hz. This is reduced to a frequency of 1,000 Hz by electronic dividers; it is then multiplied eleven times and divided by twenty-five, so producing the required frequency of 440 Hz. As 439 Hz is a prime number a frequency of 439 Hz could not be broadcast by such means as this."
BE, MBA, PhD, CPEng,?APEC Engineer, IntPE(Aus),?FIEAust MD, Clarke & Associates P/L Email sent using Optus Webmail |
Brian