sweet numerology keyfob

(...Being a ramble through mnemonics, phonetics, information theory, pi, absolute zero, and cosmology, also touching on rutabagas and conches.)

Sometimes we need to commit a number to short-term memory, such as an street address or parking garage berth. More important numbers (for phones or credit cards, PINs or birthdays) call for long-term memorization although smart phones increasingly fill that gap. But even with a smart phone it is pleasantly useful to have the option to use one’s own brain for such things. To do this, I rely on something I call the Sweet Numerology Keyfob. Before I explain that outlandish phrase, here is some background...

Digital data is hard to memorize because digits are abstract, while our human memory works best on concrete data such as images or meaningful phrases. There are various systems to encode digits in this way, so as to make them “stick” better in the memory. The simplest systems to learn replace each digit by an word, and produce a sentence corresponding to the original sequence of digits. Here is an example where each digit N is represented by a word with N letters:

How I wish I could recollect pi!
3-- 1 4--- 1 5---- 9-------- 2-
Count the letters in each word and you’ll get a morsel of pi. There are many fine examples of this technique on Wikipedia’s Piphilology page.

But one digit per word is a waste of words. Here’s why, in terms of basic information theory: To encode a random stream of decimal digits, we will need a minimum of about 3.3 bits per digit. (Are the digits of pi really random? Probably, but nobody knows for sure.) Meanwhile, as shown by various statistical studies and human tests, English words carry 6-12 bits of information. This suggests that each word should be able to carry not one but two or three digits of “payload”. In fact, it is possible to do just that. With a small amount of practice, it is easy (and fun, in a nerdy way) to learn how.

(Here is more on the information content of English: Shannon’s pioneering study measured an upper bound of 2.3 bits per letter and speculated on an actual value near 1 bit. Experiments with human prediction have refined the upper bound to 1.2 (). The theory is fascinating. A small study of Latin and Voynichese texts shows those languages exhibit statistics consistent with 9-10 bits per word, similar to Shannon’s upper bound. Note that information content is not constant but depends on factors like style and genre; striking, artificially constructed mnemonic phrases will probably have higher entropy than narrative text.)

For representing digits by words, I use a system I learned from Kenneth Higbee’s wonderful book, Your Memory: How It Works and How to Improve It. I call it the Sweet Numerology Keyfob, but Higbee calls it the Phonetic System; it is also referred to (for no apparent reason) as the Major System. The technique is about 300 years old.

The idea is to make sounds (not letters or words) correspond to digits. In order to give just the right amount of creative freedom, vowel sounds are ignored; only consonants are significant. The following words all have the same sequence of consonant sounds, and therefore represent the same number: meteor, meter, metro, mitre, motor, mutter, amateur, emitter. The consonants are MTR, and the number happens to be 314. Because this code is perceived phonetically, spellings with a double T still contribute a single digit, because to the ear there is only one T sound. Depending on how you pronounce it, a word like hattrick or rattail probably contributes a pair of 1 digits, one for each T. My point is that your ear, not your eye, will be the judge.

Here is a table relating the digits to some of the consonants:

digitsoundexamples
0Ssea, see, sigh, so, sue, ass, ice
1Ttea, tee, tie, toe, to, at, oat
2Nneigh, knee, nigh, no, new, an, in
3Mmay, me, my, mow, moo, am, ohm
4Rray, raw, awry, row, rue, are, oar
5Llay, lee, ally, low, Louie, ale, Ollie
6SHshe, shy, show, shoe, ash, ashy
7Kkey, Kay, echo, coo, icky, auk
8Ffee, fie, foe, foo, if, off
9Ppea, pie, Poe, ape
(none)AEIOUah, eh?, I, oh!, you

As you can see from these examples, silent letters are ignored, like K in knee and GH in night. Note that the letter combination SH is a single sound, and is used to encode a digit (6). The point of giving so many examples in the right-hand column is to suggest the range of choices available for each digit. As with the “number of letters per word” code used for pi above, for any given number there will be many words to choose from when deciding on a mnemonic phrase. Longer words with more consonants can be used to encode multiple digits, giving even more choices. Using phonetics instead of letters makes it possible to use the system without attending to the written form of a word. In practice, this makes encoding and decoding easier and more reliable.

There are only a few other rules to know. First, three sounds are ignored even though they can be classified as consonants. They are W, H, and Y. (You may well ask, “why”?) Thus, the phrase happy hippo whip counts as 999. What about the other consonants? They are grouped with the ones already mentioned above, according to similarity of sound. Any consonant which vibrates the vocal cords is treated like the corresponding consonant which does not, so a hard G goes with its fraternal twin K, etc. Also, the various sounds spelled TH are grouped with T, which differs from the treatment of SH and CH. (Technically, a voiced consonant is treated as if it were unvoiced, and a fricative consonant is treated as if it were a plain stop, if it does not already have a role in the encoding.)

Here is an updated table. It is helpful to read the examples aloud. (But don’t let your family overhear.)

digitsoundexamples
0S/Zsue, zoo, ease, wheeze
1T/D/THtwo, due, the, aid, weighed, mayday
2Nno, enough, whine, anyway, Hawaiian
3Mmow, yam, home, Omaha
4Rrow, war(*), airway(*)
5Llow, wool, hollow
6SH/CH/ZH/J/(G)show, Joe, chew, huge, awash
7K/G(hard)coat, goat, wacky, wig
8F/Vfew, view, half, heavy
9P/Bpie, buy, hope, oboe
(none)AEIOUah, eh?, I, oh!, you
(none)WHYwhy, how, highway
(* 3/24 Note: See guest comments below on non-initial R, etc. Remember that hard and soft G are very different sounds!)

Although the tables above may seem daunting, their core information is simple and easy to learn, if you use your ear instead of your eye. There are many ways to remember the basic sequence of ten letters, but I like to tie it up in a phrase which both describes and encodes the technique:

SweeT Nu-MeR-oLoGy Key-FoB
0---1 2--3-4--5-6- 7---8-9
(I coined this phrase when I read Higbee’s book, in 2002.)

Now let’s use this system to encode pi, using about the same size phrase as above:

How I wish I could recollect pi!
3-- 1 4--- 1 5---- 9-------- 2-

Moderately pinch Lemuel phobic.
3-1-4-1-5- 9-26- 5-3--5 8--9-7
Both phrases are about the same length, but the second one encodes twice as many digits of pi. Not surprisingly, because it is more compressed, the second phrase sounds more arbitrary. Still, it may be easy for some of us to remember, especially if you have a mental image for someone named Lemuel. As with any such method, there are many alternative phrases encoding the same payload digits, and the user should pick whichever option works best. Here are some other possibilities, also encoding the first 14 digits of pi. Because they are longer, there is more freedom in composing them, so they might be (in some sense) slightly more logical:
Meteorite lupine, chew a lame wolfback.
3-1--4-1- 5-9-2-  6--- - 5-3- --589-7-

Motherhood, help any shallow male phobic.
3-1--4---1  5-9- -2- 6--5--- 3-5- 8--9-7.

My tired lip now shall my laugh pique.
3- 1-4-1 5-9 2-- 6--5- 3- 5--8- 9-7--
Fans of Twilight or Woody Allen might find the first or second more meaningful, respectively. I used a computer to help me build all but the last phrase. The last phrase is an example of a typical off-the-cuff product; it is as short as the others but not as vivid. As with most compression systems, the more work you put in, the better results you get. In this case, the 47 bits required to encode 14 decimal digits are distributed through 4-8 words, at an average of about 6-12 bits per word. Remember that words carry 6-12 bits each, so we are using nearly all of those carrier bits to carry payload. The channel known as Grammatical English provides the bandwidth, and we are using it efficiently.

It can be difficult to get “home run” matches like moderately for 31415, but with a little practice, phrases which combine only two or three digits per word are easy to build quickly without mechanical assistance. Memorizing a few digits using short words is the most common way I use this technique. Less commonly, I work hard to get a good vivid encoding for a number I care about, such as a family birthday or phone number.

You will notice that there is no clever tie-in to pi itself in these phrases. This is a natural effect of the high compression level of the encoding. There is not much “slack” for an author to choose words that are relevant to the subject. Although this might seem to be a defect of the phonetic system, it is easy to compensate for, by mentally adding a tie-in. If you are imagining Lemuel getting pinched, give him a pie in the face also, and you are done. This extra mental step is more than compensated by the doubled efficiency of the encoding itself.

Once I got started it was hard to stop. Eventually I got the following quasi-story which encodes the first 51 digits of pi in 20 words:

Immoderately, punchily, my wolfpack bowmen may free Ginger,
--3-1-4-1-5-  9-26--5-  3--5 8--9-7 9--3-2 3-- 84-- 6-26-4
maim via ammo hangable, sniff heavy rutabaga;
3--3 8-- -3-- --27-95-  02-8- ---8- 4-1-9-7-
the sheep imbibe my accolades.
1-- 6---9 -39-9- 3- -7--5-1-0
I can’t say why my bowmen would want to sniff root vegetables; maybe they have swine DNA. It does not have to make complete sense, as long as it is vivid and coherent enough to memorize as something that can be repeated back to oneself. The point of encoding digits this way is that most people (including me) have a much easier time memorizing 20 spoken words instead of 50 abstract digits.

That is enough pi for now. The occasion of this post (which is getting long) was a different and much simpler constant. It is used to define the Kelvin and Celsius scales with reference to the triple point of water. The magic number is 273.16, which is (by definition, and exactly) the Kelvin-scale temperature of water in its triple point. This number, along with its close neighbor 273.15, defines both the Kelvin and Celsius scales. In particular, the temperature we call “absolute zero” is -273.15 Celsius. Because the freezing point of water varies slightly depending on pressure, the Kelvin and Celsius scales are defined relative to the triple point, which is very slightly warmer than normal freezing. This hundredth of a degree only makes a difference to scientists, but it is nice to know.

Although I find that number number 273.16 interesting, it tends to slip my mind. Today I made a mnemonic to nail it in there:

Enigma dish.
-2-73- 1-6-
After all, if you were presented with a dish of triple-point water, you would have trouble telling whether it were liquid, solid, or gas: A mystery! (You’d also be inside a vacuum chamber or outside in the stratosphere, but that’s irrelevant.) And if a British Enigma machine were dropped on it, it would break. Ludicrous images are memorable ones; at least, it works for me. Similarly, the freezing point of water at normal pressures, and the exact zero point of the Celsius scale is 273.15. Possible phrases for that are (for taxpayers) income toll or (for dystopians) nuke ’em daily.

And there is a bonus here, a convenient coincidence. (Not quite a mathematical coincidence.) The digits 273 also present the temperature of the cosmos to three significant figures. More specifically, the temperature of the cosmic microwave background radiation has been measured as 2.726 degrees Kelvin. Rounding gives 2.73K, or 1% of the freezing point of water (273K). The actual ratio is not 1/100 but rather more like 10/1002. Still, it is interesting (to me at least) to note that the cold of space is two orders of magnitude colder than a cold day in most parts of the Earth. It is also interesting to think that this ratio (10/1002) is probably the same everywhere in cosmos. It is slightly more impressive, however, to people who use base 10.

Although I expect to use 273 for both purposes, the more precise value of the CMBR is of course worth remembering. It is, after all, one of the most important physical discoveries of late twentieth century. The CMBR is (to me) like the sound of the sea, unvarying across aeons. Folklore says (erroneously and/or poetically) that you can hear the sea in a shell, for example, in a conch shell. I suggest that some part of the CMBR can also be found there:

In a conch.
-2 - 7-26-
Comments:

The shell just amplifies the sound of your own blood rushing in your ears, or at least that is what I was told as a child.

Posted by guest on January 29, 2012 at 10:54 AM PST #

"Are the digits of pi really random? Probably, but nobody knows for sure". Not at all. There is a formula to calculate the Nth digit in O(1) (but in binary). Therefore PI contains only a very small constant amount of entropy.

Posted by guest on January 30, 2012 at 02:55 AM PST #

Minor error: you omitted J from the "Sound" column for 6, as the example "Joe" shows.

There is in fact a 21st consonant phoneme in English which your system does not handle, the sound NG as in "sing, singer, finger". I suggest that you treat this as a variant of N, and so encoding 2. Unfortunately, this invalidates your use of "hangable", which you treat as if it were "han-gable", thus violating your (entirely sensible) rule of using the sound rather than the spelling. On my suggestion, "sing" encodes 1-2, "singer" 1-2-4, and "finger", where the G is actually pronounced, encodes 8-2-7-4. If you adopt this, you could replace "hangable" with "knockable" or something of the sort. I've tried to find better words, but had no real luck. (A few English-speakers, notably in Birming'gam, England, and Long Guyland, New York, actually do say "hangable" as "hang-gable".)

This brings us to the most serious difficulty, that of the sound R. As long as these mnemonics are only personal and private, the variation in R does not matter. But the minute that the idea of *publishing* a mnemonic for use by other people comes up, we must confront the fact that the great majority of English-speakers in Australia, the Caribbean (except Barbados), England (except the southwest), New Zealand (except the far south), Wales, Hong Kong, and Singapore, plus many people in the United States (those in Eastern New England, much of the South, the Eastern Seaboard cities, and African-Americans everywhere), do not pronounce R except at the beginning of a syllable.

Thus for people in these regions, two of your example words, "war" and "airway", don't encode any digit at all. In particular, for these folks "motherhood" encodes 3-1-1, "tired" encodes 1-1 (the same as "tied", though not pronounced the same), and "Ginger" encodes 6-2-6, spoiling the mnemonics that use them. Very fortunately, none of the other English consonants have similar widespread variation, though a similar loss of non-initial L is beginning to take place in England and has been in place in Scotland for a long time; for some people, "wool" does not encode a digit though "low, hollow" do.

I don't know much to do about this, except to make sure that R is consonant-initial in all public mnemonics, and to change the examples to "row, Aaron, worry". You could replace "motherhood" with "mother rat" and "tired lip" with "too-red lip". Replacing "Ginger" is left as an exercise for the student.

Posted by guest on March 24, 2012 at 07:12 AM PDT #

@guest#3 Thanks for the thorough remarks. You are spot-on about /R/ and /Ng/. I personally find it convenient to Birming'gam-ize my pronunciation when using the phonetic code, but it is clearly a personal choice. Similarly, someone with a dialect that softens /R/ might adopt a "ver-ry cor-rect" accent when using the code. But it is all personal choice, as I hope I conveyed. Using someone else's mnemonics is going to be inefficient; you have to make the system your own, and that means making adjustments. I received the technique (from Higbee) in that spirit, and I'm passing it on the same way. Best wishes!

@guest#1,#2: Yes, of course. Sadly, I have lost my metaphor and irony fonts.

Posted by John Rose on March 24, 2012 at 07:52 AM PDT #

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John R. Rose

Java maven, HotSpot developer, Mac user, Scheme refugee.

Once Sun and present Oracle engineer.

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