A Log Spiral and its Involute
Art, Music and Mathematics. Some relationships revealed.?
written by Mike Sterling for Canadian Community News
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The green lines are forces exerted on the strings of the Bernoulli-Involute Harp in pounds of tension. The magenta lines are lever arms or perpendicular distances to the pole seen as a dense spot. The stress concentration at the pole results from the tensions being multiplied and concentrated.
Charming Russian Harpist Shows the beauty and utility of the instrument.
I've been working on a new musical instrument that has beautiful properties and a pleasing shape. Click Here.
Craftsman Creates a Harp
In the process of designing a new instrument, I've had to learn something about the harp, which is so special to the human eye and ear.
One thing that bedevils harp makers is the tension that is placed upon the sound board by the strings. This tension has much to do with the massive size of the harp and how it looks. The same difficulties take place in pianos and guitars.
I did not know much about these problems, but I knew that the harp was difficult to tune and keep in tune. I now know why.
I did not start out to build a classic harp. I don't think I have the wood working skills to do so. If you take the time to review the work of the craftsman, you will gain even more respect for the instrument.
I decided on a brand new design never done before. If I fabricated it via computer, I could achieve the precision I could not achieve using rudimentary woodworking skills. Pretty cheeky, but what the heck!
The harp developed over many hundreds of years. It was only in the early 1800's that the foot pedals were introduced and the present day concert harp emerged.
There is much mystery about traditional harp making. The craftsman's video above video helps explain it, but in some ways leads to more quesitons..
I decided on an approach based upon the hidden mathematics of the chromatic scale or what is known as the 12 tone scale of the piano.
A number of insights led me to depend upon the numbers 2 and 3 and what they could do. I started with the 12th root of 2. At the heart of this is the fact that a string's pitch differs from its neighbors by the 12th root of 2. So each pitch grows or diminishes based upon the magical number. After applying it 12 times we get a doubling or halving of the pitch and thus create the 'octaves', which are not named to suit.
So, starting on the far left on the piano, the pitch rises with this number with every key.
Each string is adjacent to its neighbors in my design by 12 degrees. So, you see that 2 and 3 dictate the entire design as 12 = 22*3 and 36 = 22*32 .
The 12th root of 2 is at the heart of it all. This is not numerology, but it is functionality, which produces much beauty.
It's a 12 tone scale I'm using so adhering exactly to the scales properties makes things a lot easier. The instrument I am building will only have 3 natural 'octaves' of 12 tones.
Rather than using a sound board and box, I am using piezo crystals that are under pressure from the string's bridge, which is a new design. This pressure yields a voltage that is then captured and sent to an amplifier and speakers.
I'm not done yet and will not be for about six months. I've tested the piezo effect and have 3 strings under controlled test using only test boards and the designed bridge.
The sounds generated by it are unusual. It can be strummed or hit with a hammer. If you Click Here, you can hear what I mean even with the scratchy cheap speaker and microphone. I've only got a couple of pitches to work with, but the tests went well and as designed.
This is an example of how art, music and mathematics have close relationships. The whole idea is to capture it all in a single shape. There is much to admire in the shape and possibilities. All the variations and relationships are produced by the shape.
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Sunday, February 01, 2015