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So "farthing" from "fourth" or "quarter"

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This may or may not tickle your literary fancy, but the number twelve plays a vital role in the Western musical scale.

There are twelve half-steps in an octave (which gets its name from there being eight notes in the major scale). On a fretted instrument such as the guitar, there are twelve frets to an octave. The length of a vibrating string fretted at the twelfth fret is half as long as the length of the open string. This difference (and similarity) in wavelength and frequency is perceived by the listener as one octave, such as middle C and the C above that, or the opening guitar riff of The Knack's "My Sharona," or the bass guitar riff in Led Zepplin's "Immigrant Song" for you rock and roll fans.

The distance between frets on a guitar is related to the number 12 as well. For a given scale length, such as 25 inches, the first fret is placed at a distance that is the twelfth root of (0.5). (i.e. the number which when multiplied by itself twelve times equals 0.5, which happens to be 0.943874.)

Thus the first fret is placed at 25-25(0.943874) = 1.403142 inches from the nut at the top of the guitar neck, or 23.596857 inches from the bridge.

The second fret is placed at the same relative interval to the remaining length. It is placed 23.596857 * 0.943874 inches, or 22.272467 inches from the bridge.

This iterative process leads to the result that, at the twelfth fret, we have arrived at a point that is 1/2 the distance of the whole string, in this case 12.5 inches.

The musical notes that we experience when we listen to music are our brains' representation of mathematical differences and similarities in wavelength and frequency. Other cultures and musical traditions may divide the scale in other ways, and their music sounds as harmonious to them as Western music sounds to my Western ears.

An online fret calculator used by musical instrument makers is available HERE, and allows you to enter your scale length and the total number of frets on the neck (not the number of frets in an octave--it's hardwired for a 12-step scale).

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Thanks!

(perhaps mathematics, here, is a representation of what we perceive)

Last edited by inselpeter; 03/20/06 09:21 PM.
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Yes, I too remember blackjacks and fruit salads at 4 for a penny and two for ha'pence, but the farthing went obsolete in 1956.

I had forgotten to mention the sovereign and 'half sov' ~ and now I stop and think again, I remember my grandad telling me about some of the oddities, like the silver tuppence...

Hey, Bing and OT, you remember the groat, too? and the noble, angel and mark?!

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farthing might have gone from UK in 1956, but in ireland, they lingered..

as for noble's angel's and mark's, i have only read about them in historical novels...
*************************************************

another '12th' cycle are the tides, that move by rules of '12th's' --i always forget exactly how, but the volume of water that 'moves' increases by 12ths (1/12, 2/12, 3/12, 6/12th, is the progression, i think-- (and the reversed, 6/12th, 3/12th,..)

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From mav's link:
mark obsolete medieval denomination
more common in Scotland

Did Scotland used to be related to Germany?

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Quote:

Before laughing too much at the mote that used to be in someone else's eye, we should consider the some of the logs in our own, e.g.

3 teaspoons in a tablespoon
2 tablespoons in an ounce
8 ounces in a cup
2 pints in a quart
4 quarts in a gallon




I think that presenting the measuring system that way you're obscuring some sense of order that is really there. Look at it this way:

For liquid measurement:
16 tablespoons to a cup
16 cups to a gallon

A quart, being a quarter gallon, is naturally 4 cups.
A pint, being an eighth of a gallon, is 2 cups.
A firkin (honest!), being 9 gallons, at first appears to be a strange unit, but 9 gallons is equal to 144 cups.

As for tablespoons and ounces:
2 tablespoons to an ounce = 2^1 tablespoons
8 ounces to a cup = 16 tablespoons = 2^4 tablespoons
16 ounces to a pint = 32 tablespoons = 2^5 tablespoons
32 ounces to a quart = 64 tablespoons = 2^6 tablespoons
128 ounces to a gallon = 256 tablespoons = 2^8 tablespoons

Traditional dry measurement units also have an orderly progression:
1 quart = 2 pints = 2^1 pints
1 gallon = 8 pints = 2^3 pints
1 peck = 16 pints = 2^4 pints
1 bushel = 64 pints = 2^6 pints

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Well that's very odd. 1956 was the year before I was born, but I definitely remember my grandfather giving me some farthings and I'm pretty sure I spent them on blackjacks.


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You are right, Bing – although the coin was last minted in 1956 it stayed in legal tender until 1960. I should have checked my facts on an English rather than American site!

http://www.tclayton.demon.co.uk/farth.html

It’s also worth remembering that there were fractional farthings minted in the previous century: half-farthing, third-farthing, and quarter-farthing coins were minted at various times during the 1800s, but circulated only in particular British colonies and not in the UK itself for the most part. The exception was the half farthing, initially issued in 1828 for use exclusively in Ceylon but in 1842 it was made legal tender in the UK despite moans about it being far too tiny a coin to be useful. Although it only lasted until 1869 it must have been in common use since I had several examples amongst a simple coin collection as a kid.

To return to a language point, I have also been reminded that the ‘LSD system’ was not unique to Britain. At some point prior to the revolution (I don’t know when, YCLIU!) a similar pre-decimal system operated in France, also based on the Roman currency, consisting of the livre (L) sol (s) and denier (d).

http://en.wikipedia.org/wiki/Roman_currency

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Quote:

I think that presenting the measuring system that way you're obscuring some sense of order that is really there. Look at it this way:




On the other hand, you've just ignored the tricky parts, which was the point.

Here I have a simpler system.

1 pound = 1 pound.

That's it. What could be easier. Now my poor grandmother doesn't have go back to school to learn about powers of two to use all the recipes she's made for 80 years ... of course, now she can only make pound cakes but that's a small price to pay for such an elegant system of measurement

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