Baker's octave
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A recent xkcd:
Mouseover title: "169 is a baker's gross."
Everyone knows that a "baker's dozen" is 13, one more than a standard dozen, though Wikipedia told me a new story about its origin:
In England, when selling certain goods, bakers were obliged to sell goods by the dozen at a specific weight or quality (or a specific average weight). During this time, bakers who sold a dozen units that failed to meet this requirement could be penalized with a fine. Therefore, to avoid risking this penalty, some bakers included an extra unit to be sure the minimum weight was met, bringing the total to 13 units or what is now commonly known as a baker's dozen.
Randall's comic extends the generalization to other things whose normal count is 12: inches in a foot, hours from midnight to noon, faces of a dodecahedron, semitones in an octave, members of a jury, stars on the EU flag, the number of protons in magnesium's nucleus.
But as explainxkcd explains, Randall gets the musical example wrong:
In the 12-tone music systems, octaves contain 12 half-steps, also known as semitones (a half-step is the distance between adjacent notes, such as F and F#). A 'baker’s octave' would have 13 semitones (corresponding to a minor ninth) and cause problems in musical composition, as octaves (of the baker’s variety) would be dissonant, instead of being consonant. However, Randall's musical notation actually shows a major ninth, with fourteen semitones. If he wanted thirteen semitones, Randall could have used D♭ instead of D, or drawn a bass clef instead of a treble clef. Another way is to shift couple of notes up to make it E and F or one note down to make it B and C, both pairs are actually 13 semitones away from each other.
The "baker's octave" concept, without the error, resonates with a wonderful insight from Bill Sethares that's been on my to-blog list ever since I learned about it in a recent presentation ("Three eras of Pythagoreanism") by Dmitri Tymoczko.
A sketch of the musical math follows — see Tymoczko's slides and Sethares' book for the full story. But you might want to start by skipping ahead to the musical examples, to decide whether the details are worth it.
Here's a quick explanation taken from three of Dmitri's slides:
- Consonance is not produced by numbers themselves.
- Instead, it arises through a process of dissonance minimization that depends on the structure of a sound, which in turn depends on the physics of the instrument involved.
- It is a minimum in a continuous space.
- A vibrating body produces partials according to its detailed physical construction.
- Consonance is produced when the partials of two sounds coincide (or are sufficiently distant so as not to interfere).
- Familiar instruments vibrate harmonically:
–f, 2f, 3f, 4f, … - The consonance of whole-number frequency ratios is due to the spectrum of the sound.
- Therefore, nonharmonic sounds would sound consonant at different, non-whole-number ratios.
In the material for his 1998 book Tuning, Timbre, Spectrum, Scale, Sethares observes that instruments with inharmonic partials, such as bells, are not only possible but common; and he works out what a 12-step scale based on an inharmonic "circle of fifths" would be like, in particular if the ratio of successive partials were 2.1 (= 13 semitones in a conventional 12-step scale — a baker's octave!) rather than 2.
As a clue about what that means, note that 2/1 partials take us step-by-step around the circle of fifths. If we start at 100 Hz, for example, the second overtone is at 3*100=300 Hz, which is one 2/1 octave up from 150 Hz, whose ratio of 3/2 relative to the starting point yields (the first) perfect fifth. See "Pavarotti and the crack to chaos" (1/9/2007) for one take on the rest of the story. Sethares works through the analogous scale construction for a different (2.1/1) overtone ratio.
Here's Sethares' video, illustrating how a bell-like instrument with 2.1/1 stretched partials sounds dissonant playing two notes separated by a traditional 2/1 octave, but consonant when the notes are in a 2.1/1 ratio:
Given the choice between an instrument with 2/1 (harmonic) and one with 2.1/1 (inharmonic) partials, and the parallel choice between a scale based on harmonic (2/1) partials and one based on inharnomic (2.1/1) partials, there are then four possible variants of a given piece of music. (These examples are taken from the CD accompanying Sethares' book.)
- Familiar scale (based on 2/1 partials) +
Familiar instrument with harmonic (2/1) partials:
- Familiar scale (based on 2/1 partials) +
Instrument with stretched (2.1/1) partials:
- Stretched scale (based on 2.1/1 partials) +
Instrument with stretched (2.1/1) partials:
- Stretched scale (based on 2.1/1 partials) +
Familiar instrument with harmonic (2/1) partials:
Philip Taylor said,
May 25, 2025 @ 9:01 am
""In England, when selling certain goods, bakers were obliged to sell goods by the dozen […] some bakers included an extra unit to be sure the minimum weight was met" — not unlike the tobacco/cigarette industry, where a small excess of tobacco in a cigarette was less likely to provoke a complaint that a small deficit, and so the control electronics were set to ensure that the correct average content was achieved while the tolerances were asymetric. See, for example, https://www.facebook.com/groups/1796303300588719/posts/3971824286369932/.
Robert Coren said,
May 25, 2025 @ 9:39 am
I have on occasion jocularly referred to 11 as "a lawyer's dozen".
JMGN said,
May 25, 2025 @ 1:50 pm
@Robert
Care to expand on it a bit? I can't grap that one…
D.O. said,
May 25, 2025 @ 3:18 pm
Riddle: how's baker's dozen, jury, EU, and Mg are different from baker's noon, new year, and octave? And where do foot and dodecahedron fit?
Answer: Sometimes 12 different objects are combined to form a dozen in other cases one thing is divided on 12 parts. Dodecahedron and foot do not fit with either. If we used a lunisolar calendar, then we would have real baker's years.
Rudi said,
May 25, 2025 @ 3:19 pm
There's also the "Colundi Sequences" (https://daily.bandcamp.com/lists/colundi-aleksi-perala-interview), but since Aphex Twin is involved I'm not sure in which way to take them serious.
Julian said,
May 25, 2025 @ 4:51 pm
Reminds me of a joke.
Conductor of a small community choir is taking them through a new piece. Modern music with lots of odd ethnic rhythms. Not just four four.
Choir is having a bit of trouble coping with that.
Conductor: " look! It's easy counting in seven eight! Like this: "one, two, three, four, five, six, seven. One, two, three, four, five, six, seven. One, two, three, four, five, six, seven…."
AntC said,
May 25, 2025 @ 5:55 pm
@Julian, sorry I'm not getting it.
"four four" means four beats of crochets aka 'quarter notes', hence written on the stave as 4 over 4.
"seven eight" means seven beats of quavers aka 'eighth notes', hence written on the stave as 7 over 8. In practice seven usually phrased as 4 beats + 3 or some such. (Compare 'Take Five' phrased as 3 + 2. Also by Brubeck 'Blue Rondo à la Turk' 9/8 "three measures of 2+2+2+3 followed by one measure of 3+3+3 and the cycle then repeats." [wp]) 7/8 appears for example in Shostakovich's 2nd Piano Concerto, last movement, second theme. (Arguably counted as 3 + 1 (unstressed) + 3.)
AntC said,
May 25, 2025 @ 6:01 pm
Consonance is not produced by numbers themselves.
Indeed. Hence Bach's work is titled for the 'Well-Tempered Clavier', not the Equal-Tempered Clavier which all modern pianos and electronic keyboards are tuned to.
You can find some very nerdy videos on YouTube of people re-tuning their claviers for each of the 48.
MC said,
May 25, 2025 @ 7:18 pm
AntC – The joke is that it takes longer to say seven. That's why when you count 7/8 you need to break it down into 3+4 or some such combo of smaller numbers that are all one syllable each. Joke works better spoken.
Julian said,
May 25, 2025 @ 11:02 pm
@MC
In other words, if you read it out loud including the pause that you would naturally put on the full stop, you're actually counting in eight.
Chas Belov said,
May 25, 2025 @ 11:18 pm
@Julian: I don't get how it's a joke.
Chas Belov said,
May 25, 2025 @ 11:22 pm
Oops, missed the other posts. But I don't say seven in twice the time it would take one two. If I count to eight, I still start eight the same time after saying seven as I said two after saying one.
Chas Belov said,
May 25, 2025 @ 11:24 pm
That is, if I count to ten in rhythm there are longer pauses after 1-6, 8, 9, and 10 than after 7 so that everything is said starting on a constant beat.
Chas Belov said,
May 25, 2025 @ 11:26 pm
Urk, that didn't come out right. Let's try again.
If I count to eight, I still start eight the same time after starting seven as I started two after starting one.
Chas Belov said,
May 25, 2025 @ 11:28 pm
That said, yes, it's perfectly typical to break it up 3+4 or 4+3. But it's not required and the fact that seven has two syllables has nothing to do with it.
I am not a musician and this is not musical advice.
AntC said,
May 25, 2025 @ 11:31 pm
@Julian, you appear not to be a musician. If I'm counting for the purposes of conveying a rhythm, I'd contract to se'en, so as to be one syllable. No musician would put a pause on any full stop, "naturally" or otherwise. Any more than you'd put a pause on a full stop when reciting poetry.
Chas Belov said,
May 25, 2025 @ 11:37 pm
Or in counting the notes via notation:
7
4 ♩♩♩♩♩♩♫♩♩♩♩♩♩♫♩♩♩♩♩♩♫
Chas Belov said,
May 25, 2025 @ 11:39 pm
Sigh. And for some reason the bar lines aren't showing up. Please picture a bar line separating each ♫ from the following ♩.
Chas Belov said,
May 26, 2025 @ 12:33 am
When a dog barks at me and their human apologizes, I usually reply "Dogs gonna dog."
Chas Belov said,
May 26, 2025 @ 12:33 am
Oops, posted that on the wrong tab, sorry.
Nat J said,
May 26, 2025 @ 3:58 am
Three Eras of Pythagoreanism is a fantastic presentation! It seems relevant to a question that’s long perplexed me: what kinds of facts are musical facts? Mathematical? Physical? Psychological? Or just sui generis?
At one point Tymoczko questions the reason for integer exponents. But I would think the inverse square in gravity must be rather different from the square in uniform acceleration. I’d think gravity’s exponent should derive from dimensionality. In which case it would necessarily be integer-valued. But doesn’t Tymoczko suggest that the integer exponents are accidental approximations imposed by people? Provocative and fascinating at any rate.
Robert Coren said,
May 26, 2025 @ 9:14 am
Re "lawyer's dozen": Not knowing the origin story recounted above, I assumed that the idea of a "baker's dozen" being 13 was that bakers had a reputation for generosity, which lawyers stereotypically do not. ("Miser's dozen" might have been closer to the mark, but, unlike "baker" and "lawyer", "miser" is not a profession.)
KevinM said,
May 26, 2025 @ 2:33 pm
Re: the 7-beat count-off joke. Beats aren't syllables. So, for example, near the beginning of the Beatles' Love is All You Need, the 7th beat actually contains two notes, and a dotted rhythm at that.
ajay said,
May 27, 2025 @ 6:28 am
There's also the "purser's pound" of 14 ounces.
The 18th century Navy issued rations by the purser's pound. This was because the purser was personally financially responsible for all the stores on the ship, and had put up a massive financial bond in advance as a guarantee before the voyage started.
If he'd signed for 800lb of bread, he had to be able to prove, with documentary evidence, that he had issued out 800lb of bread to the crew, who had regarded it as fit to eat and then eaten it. If he had only issued 750lb he would be expected to pay for the other 50lb, or prove either that he still had it, or that it had been properly condemned as unfit for use by a court of inquiry. Disputes over accounts could take years. Pursers lived in constant fear of bankruptcy. (NAM Rodger notes that a common joke among sailors was that albatrosses were the ghosts of naval pursers, condemned to follow their ships for eternity in the hope of finally getting their accounts signed off.)
But, of course, rations get wasted. They get eaten by rats, or spilled, or whatever. The Admiralty reckoned that a competent purser should be able to keep wastage down to 12%, and introduced a very simple way of ensuring this – the 14-ounce pound. If you got 800 16-ounce pounds of bread, and could prove that you had issued 800 14-ounce pounds of bread to the crew, you were fine.
ajay said,
May 27, 2025 @ 6:30 am
And, I suppose, the still-extant practice of selling horses in guineas (one guinea = £1.05, or one pound and one shilling in old money). You buy a horse for 400 guineas. The seller gets £400. The other 5% is the auctioneer's commission.
Tom said,
May 30, 2025 @ 8:12 pm
Baker's Noon is funny, but like Foot and Dodecahedron doesn't fit the pattern established by Baker's Dozen.
Also, Baker's New Year's Eve introduces an interesting calendar in which New Year's Eve would be skipped periodically, whenever the New Year cycle coincided with the Leap Year cycle.