Scales

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1349155.  Tue May 26, 2020 6:17 am

The other day second-born and I were discussing mathematical music (the patterns and relationships in music which often have a mathematical beauty as well) when we got into a bit of a silly argument (unusual for me, you'll agree). That started a thought process that led to an area that I think qualifies as Quite Interesting. The "scales" thing is a bit artificial - it was just to shoehorn it ito the S-series stuff!

The starting point is this:

1. How many notes are there in an octave? [klaxon is 8 - the actual answer is seven]

Just in case you think I'm talking about tones rather than notes let's ask that question too:

2. How many whole tones are there in an octave? [klaxon is 7 - the actual answer is six]

If you don't believe me try counting them. Then count them for two octaves and divide by two, three octaves and divide by 3 etc. When I looked into this I discovered that it is an instance of a thing mathematicians call "The Fence Post Problem" which leads to the "Off By One Error" (OBOE) [detailed by Matt Parker in his book "Humble Pi", which is the source for much of what follows]. The example they give is to imagine a 100-foot run of fencing made from 10-foot panels. To do this you obviously order ten panels and ten posts, but you find you're one short and need eleven posts - the difference in specifying the quantity by the number of boundaries vs the number of intervals. Count the wrong one and the answer will be off by one - the OBOE.

So in our octave there are eight boundaries, but only 7 notes because in the scale of C the last C is actually a fence post not a fence panel. I found this error crops up all over the place - there was a Transport for London poster next to escalators that were under repair explaining that all escalators were fully reconditioned twice before being replaced, and that doubled their life. It actually *trebles* their life (use-replace vs use-recon-use-recon-use) because in this case the use was the fence post and the reconditioning was the fence panel. There's the way that a 29-year-old person is in their 30th year of life because we count panels (years) for age but posts (years on the planet) for the nth year (although that one could be argued as a rounding convention).

Matt Parker points to OBOEs as far back as a contemporary of Julius Caesar who, in an essay on architecture, warns that for a temple that was twice as long as it is wide the number of columns needed for the long dimension is not just double the number needed for the short dimension. The actual relationship (which he didn't have the notation to write down) is the more complicated:

n' = 2n-1

He also cites the early pontifs putting leap years every three years because they counted the start of the year (fence posts) - so initially if 2001 was a leap year they'd put the next one in 2004. This was corrected later, it seems.

Obviously we don't do anything that stupid today, because we all understand numbers from an early age, or do we?

Imagine I hire cleaners for my offices. Each floor takes an hour to clean, and I occupy floors 8-12 of Nowhere Tower so I contract the cleaners from 8-12. But they never get to the final floor! we count floors by panels and time by fence posts. so the required time suffers an OBOE.

I get a product on 7 day approval. I buy it at 16:55 on a monday. I decide I don't like it so I take it back at 09:10 the following monday. In many cases the return will be refused because they would say I have had it for more than 7 days, when clearly I hadn't - they were counting day boundaries (fence posts) rather than days (panels). There have been legal arguments about this, to the point where laws have been passed to remove the ambiguity:

 Massachusetts Rules of civil procedure 9:(a) wrote: In computing any period of time prescribed or allowed by these rules, by order of court, or by any applicable statute or rule, the day of the act, event, or default after which the designated period of time begins to run shall not be included

Ask someone to count the time between two events by counting out loud and they will almost invariably start with one rather than zero - fence posts again. Even more basic - how many numbers can you count to using the fingers of one hand (just using "finger or no-finger" - forget finger joints or finger positions)? "Five" gets you a klaxon - the answer is "six" (or perhaps "seven" in Norfolk - Ed) because "no fingers" (zero) is actually a number too.

So the bottom line is that when we think modern humans are intelligent and numerate we are probably wrong, because we can't even agree basic stuff like how to count...

PDR

 1349157.  Tue May 26, 2020 6:22 am The Matt Parker you mention -- is this the Matt Parker who I have met as part of the Spoken Nerd collective?

 1349159.  Tue May 26, 2020 6:31 am That's him. I bought a copy of one of his books at a Evening of the Spoken Nerd gig, and when I started researching this subject the book title came up - I was about to order it when I realised it was already sitting on the bookshelf not 10 feet (fence panel number system) from where I was lounging... PDR

 1349160.  Tue May 26, 2020 6:33 am You get the OBOE when working with handsaws too - Americans use points per inch (PPI) and the rest of the world tends to use teeth per inch (TPI).

 1349660.  Tue Jun 02, 2020 3:26 pm Ah, I misunderstood, and thought you might be talking about scales (cosmic and microscopic) of the rather marvellous Powers of Ten film.

1372615.  Sat Jan 23, 2021 10:12 am

Interesting post about the OBOE (which I've noticed many times myself), but your introductory example doesn't quite work.

 PDR wrote: The starting point is this: 1. How many notes are there in an octave? [klaxon is 8 - the actual answer is seven]

Off the top of my head I'd have answered neither 8 nor 7, but 12 - because of the 12 notes of the chromatic scale.

But actually I'd say that the question is ill-formed, because an octave is a musical interval (a difference in pitch between two sounds), whereas musical notes are the sounds themselves (or the symbols used to represent them).

If the question had been "how many notes in a (major or minor) scale?", then the answer would undoubtedly have been 8, because a scale includes both the bottom and the top notes. So in this instance it's not an OBOE - it's the correct answer.

I think the question you were actually looking for was "how many steps are in an octave?", to which the answer is seven (five whole tones and two semitones). A "step" is the interval between two consecutive notes of the scale.

 Quote: 2. How many whole tones are there in an octave? [klaxon is 7 - the actual answer is six]

That one makes perfect sense, because a whole tone is an interval, and intervals can be divided into smaller intervals.

 Quote: So in our octave there are eight boundaries, but only 7 notes because in the scale of C the last C is actually a fence post not a fence panel.

Not quite. The notes are the "fence posts" (boundaries); the steps are the "fence panels". Eight notes, seven steps.

Incidentally the same applies to other musical intervals. C to G is called a (perfect) fifth, not a fourth, because if you play the notes of the scale from C to G there are five of them, even though there are only four steps.

 Quote: Matt Parker points to OBOEs as far back as a contemporary of Julius Caesar who, in an essay on architecture, warns that for a temple that was twice as long as it is wide the number of columns needed for the long dimension is not just double the number needed for the short dimension.

That's probably because the OBOE was actually built into the Roman method of counting. For example, in most months the Ides fell on the 13th; but the 11th was referred to as "ante diem tertium Idus" (the 3rd day before the Ides). I'm not surprised they got confused!

 Quote: Even more basic - how many numbers can you count to using the fingers of one hand (just using "finger or no-finger" - forget finger joints or finger positions)?

I make it 32 (2^5), but I don't think that's what you meant...

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