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The 39th Root of 92

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Re: The 39th Root of 92

#101

Post by hatsunearu »

int_19h wrote: post #60
kevhito wrote: post #18

Earlier quoted context omitted.

If you have never wired a house for electrical, then sure, the units seem arbitrary. But if you do, then the units seem very comfortable. 12ga for most typical 20a circuits, 14ga for light, 10ga or thicker for some heavy-duty circuits. Hey boss, we seem to be all out of 3.31mm^2 romex, all I got left is 2.08mm^2. And why did you mention cross-sectional area? Why not radius? or diameter? Or better yet, circumference?…

The "it's more natural" argument, fielded by both sides, is largely a myth - take it from someone who lived in several countries and struggled with different units. Pretty much the only reason why something feels natural to you is because that's what you are accustomed to using. There's nothing more natural about an inch than a centimeter, and nice, round, easy-to-remember numbers can be had on both scales for "natur…

I mean I hate the US customary system (i'm from a metric country and metric early-education), but jeez, AWG is fine IMO.

What's the purpose of having mm^2? You can't measure it any easier (how the heck are you gonna measure area without calipers? you're gonna need a gauge with holes in it to identify wires), nor is it easier to express the number easily.

For manufacturing purposes, expressing the radius or diameter might be good, but for using them, the AWG number is really nice and streamlined.

Re: The 39th Root of 92

#102

Post by weinzierl »

> It’s worth noting that instruments were being tuned to this scale well before the invention of logarithms. I assume it was done by ear or perhaps by geometry, not by algebra.

Well temperament wasn't known before 1681, there were predecessors that came close but none used the the twelfth root of two. Logarithms were introduced by John Napier in 1614 [1], the twelfth root of two was first calculated by Marin Mersenne in 1636 [2], well temperament was introduced by Werckmeister 1681 [3].

> Around 1600 Simon Stevin did attempt to calculate numerical values for the pitch intervals by decomposing 12th roots into combinations of square and cube roots; his results were not flawless.

He had the right idea but not the right math.

[1] https://en.wikipedia.org/wiki/John_Napier

[2] https://en.wikipedia.org/wiki/Twelfth_root_of_two

[3] https://en.wikipedia.org/wiki/Well_temperament

Re: The 39th Root of 92

#103

Post by [deleted] »

rm445 wrote: post #51
einhverfr wrote: post #28

Earlier quoted context omitted.

I am not sure. American units usually focus on divisibility and factoring of integers, while metric units focus on quick base-10 math. So they are different concerns and this is why, I think, people from elsewhere in the world have so much trouble understanding American units. If you are brought up with metric units, the units are built around the number system. A good way to understand the problem by for the HN audi…

Shout out to another base-12 fan. If only we had 12 fingers it might have happened. Arbitrary divisions of angle aren't very interesting, since the radian is so fundamental. But there is an angle system using multiples of 100: gradians. 100 is a quarter-turn and the internal angles of a triangle add up to 200. A lot of electronic calculators still offer them (the DRG button = degrees, radians, gradians).

[deleted]

Re: The 39th Root of 92

#104

Post by engi_nerd »

VLM wrote: post #84
engi_nerd wrote: post #33

Earlier quoted context omitted.

Okay, I'll play devil's advocate here. There's a huge (but not insurmountable) lock-in effect behind wire gauges. All the wire I can buy is sized in it. All the wire strippers, crimpers, pins, sockets, plugs, jacks, terminal strips, insert/removal tools and clamps I can buy are sized around the AWG standards. Bulkhead passages in ships and aircraft are sized for carrying certain numbers of wires of certain gauges in…

A somewhat more interesting problem is we live in a centrally controlled economy with many centers. There's a paperwork storm downstream of thousands of local building codes. Most are minor variations on minor details of the NEC, but it would be a huge job to harmonize and metric-ify them. For example where I live, politicians "had to do something" after someone died in a pool electrocution decades ago, so our local…

And that's just the economic "center" for commercial and residential structures. There are centers in lots of other industries. I referred to the one that I deal with directly (fighter aircraft) in my original post. I'm sure you've thought about multiplying this effort across hundreds of other industries...

Re: The 39th Root of 92

#105

Post by bit-player »

weinzierl wrote: post #102

> It’s worth noting that instruments were being tuned to this scale well before the invention of logarithms. I assume it was done by ear or perhaps by geometry, not by algebra. Well temperament wasn't known before 1681, there were predecessors that came close but none used the the twelfth root of two. Logarithms were introduced by John Napier in 1614 [1], the twelfth root of two was first calculated by Marin Mersenne…

(I'm the OP here.)

My source for the statement that "instruments were being tuned to this scale well before the invention of logarithms" is an article by Edward Dunne and Mark McConnell, "Pianos and Continued Fractions," Mathematics Magazine, Vol. 72, No. 2 (Apr., 1999), pp. 104-115. They write: "Guitars in Spain were evenly tempered at least as early as the fiftheenth century, two hundred years before Bach. And Hermanus Contractus, born 18 July 1013, invented a system of intervalic notation that anticipated equal temperatment."

My source on Simon Stevin's work is a Rudolf Rasch, "Tuning and Temperament," published as Chapter 7 in The Cambridge History of Western Music Theory." Rasch gives an interesting account of the state of root extraction before the advent of logarithms. He attributes the errors to "Stevin’s sometimes rather reckless rounding of digits after the decimal period, which are often truncated rather than rounded."

Re: The 39th Root of 92

#106

Post by billforsternz »

fennecfoxen wrote: post #76
billforsternz wrote: post #6

Earlier quoted context omitted.

I would imagine the gains would be extraordinary - the deadweight cost of interacting with the metric world must be huge. The needlessly complicated calculations involved in dealing with things like fractions of inches impose their own subtle tax on productive effort. And let's not forget the avoidance of occasional catastrophes like space probes doomed by a conversion miscalculation or an airliner running out of gas…

"Extraordinary gains?" Ordinary gains are about 4% a year (long-term returns of broad investments in the stock market, adjusted for inflation). If you buy a piece of equipment for $100 it should return $4/year (on top of maintenance costs, depreciation, and the like) otherwise you'll do better by buying a stock-market index fund. How much do you think it would cost an arbitrary manufacturing company to switch over al…

Clearly I was talking about switching the entire American society from imperial to metric. Your switch of focus to the impact on a single company and it's return on capital is certainly interesting but I am not 100% convinced it's germane. It's not necessarily a good idea to apply a Wall Street lens (quarters trump decades!) to every decision. Other countries around the world have successfully absorbed the short term pain involved and are reaping long term gains. Sticking to imperial forever means paying an imperial deadweight tax forever.

Re: The 39th Root of 92

#107

Post by continuational »

stdbrouw wrote: post #53
kevhito wrote: post #18

Earlier quoted context omitted.

If you have never wired a house for electrical, then sure, the units seem arbitrary. But if you do, then the units seem very comfortable. 12ga for most typical 20a circuits, 14ga for light, 10ga or thicker for some heavy-duty circuits. Hey boss, we seem to be all out of 3.31mm^2 romex, all I got left is 2.08mm^2. And why did you mention cross-sectional area? Why not radius? or diameter? Or better yet, circumference?…

Same deal for metric, though. I know I need 0.75mm2 for small appliances, 1.5mm2 is your run-of-the-mill cable for up to 10A (equal to 20A at 110V), 2.5mm2 and 4.5mm2 for heavy-duty circuits. I've sometimes heard the same remark about temperatures: jeez, how do you guys manage to work with something as unintuitive as degrees celcius? But really, you just get used to whatever it is you're using.

Wait, is there a temperature measurement out there that's mire intuitive than "water freezes at zero and boils at 100"?

Re: The 39th Root of 92

#108

Post by fanf2 »

jacobolus wrote: post #64
rm445 wrote: post #51

Earlier quoted context omitted.

Shout out to another base-12 fan. If only we had 12 fingers it might have happened. Arbitrary divisions of angle aren't very interesting, since the radian is so fundamental. But there is an angle system using multiples of 100: gradians. 100 is a quarter-turn and the internal angles of a triangle add up to 200. A lot of electronic calculators still offer them (the DRG button = degrees, radians, gradians).

Gradians are absolutely terrible. For everyday use, radians are also pretty impractical. I personally like just writing fractions of a full turn, but degrees/minutes/seconds aren’t too bad. There are two cases where linear divisions of arclength-measured angles make sense: (1) the angles to make regular polygons of low numbers of sides, (2) repeated binary divisions, whose cartesian coordinates can be computed easily…

Grads are part of how the metric system was constructed: the metre was originally 1/10000 the distance from the North Pole to the equator via Paris, so 1 grad of arc on a great circle is 100km. Compare the nautical mile which is 1 minute of arc.

Re: The 39th Root of 92

#109

Post by fanf2 »

hatsunearu wrote: post #101
int_19h wrote: post #60

Earlier quoted context omitted.

The "it's more natural" argument, fielded by both sides, is largely a myth - take it from someone who lived in several countries and struggled with different units. Pretty much the only reason why something feels natural to you is because that's what you are accustomed to using. There's nothing more natural about an inch than a centimeter, and nice, round, easy-to-remember numbers can be had on both scales for "natur…

I mean I hate the US customary system (i'm from a metric country and metric early-education), but jeez, AWG is fine IMO. What's the purpose of having mm^2? You can't measure it any easier (how the heck are you gonna measure area without calipers? you're gonna need a gauge with holes in it to identify wires), nor is it easier to express the number easily. For manufacturing purposes, expressing the radius or diameter m…

For manufacturing you want the cross-sectional area because that relates most directly to the quantity of material.

Re: The 39th Root of 92

#110

Post by chias »

hengheng wrote: post #54
einhverfr wrote: post #28

Earlier quoted context omitted.

I am not sure. American units usually focus on divisibility and factoring of integers, while metric units focus on quick base-10 math. So they are different concerns and this is why, I think, people from elsewhere in the world have so much trouble understanding American units. If you are brought up with metric units, the units are built around the number system. A good way to understand the problem by for the HN audi…

> American units usually focus on divisibility and factoring of integers The other day, right next to the problematic awg definition, I discovered the kcmil. Which is a kilo pi/4 Micro square inch.

Have you heard of the "cubic ton"? I am not kidding.

https://en.wikipedia.org/wiki/Cubic_ton

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