Peterik Chapter 9: Making Use of Melody in Songwriting
Table of Contents
Introduction
Today we reach Chapter 9 of Peterik's book -- the melody chapter. Therefore this will be a great time to relearn what EDL scales are and how they are used in Mocha. I'll fix the melody part of our TI music generator in order to take advantage of what we learn about melody from Peterik. Finally, even though I don't know yet what classes I'll teach -- and hence what songs I'll want to perform -- for fall, I already have an idea of which EDL's I'll be using to compose those songs.
Rapoport Question of the Day
Today on her Daily Epsilon of Math, Rebecca Rapoport writes:
How many 0s are at the end of the decimal representation of 100! ?
Here 100! refers to 100 factorial. Factorials, by definition, have lots of factors (as they are found by multiplying lots of numbers together), including many factors of ten. Thus factorials end in many 0s.
Since 10 = 2 * 5, we are looking for factors of two and five. We notice that there are 20 multiples of five up to 100, and four of those are also multiples of 25, supplying an extra four factors of five, bringing us to 24, It's easy to find at least 24 factors of two as well -- the first 24 even numbers, so that gives us 24 factors of ten.
Therefore the desired number of 0s is 24 -- and of course, today's date is the 24th.
Let's generalize this procedure -- let z(n) be the number of terminal zeroes of n!, so z(100) = 24. We know that each factor of ten requires a two and a five -- but there will always be many more twos than fives, so z(n) is really controlled by the number of fives. We see that each multiple of five gives us a five, each multiple of 25 an extra five, each multiple of 5^3 an additional five, and so on:
z(n) = floor(n/5) + floor(n/5^2) + floor(n/5^3) + ...
Notice that if we remove each of the floor functions, then this becomes a geometric series with initial term n/5 and common ratio 1/5, so the sum would be n/4. But because of the floor functions, only finitely many terms are nonzero, and so z(n) is always less than n/4 (for n > 0). We can come very close to n/4 though -- after all, z(100) = 24, which is just barely less than 100/4 = 25. In general, z(n) is very close to n/4 when n is a power of five, or such a power times two, three, or four (100 = 5^2 * 4).
In the limit, we see that the limit of z(n)/n as n approaches infinity is indeed 1/4.
The Factorial Problem in Dozenal Base
The factorial problem can be generalized to other number bases. In any base, n! will tend to end in many 0s, since n! accumulates factors of all numbers, including the base itself.
Define z_b(n) to be the number of terminal zeroes of n! in base b. We've already calculated z_10(n) for decimal above -- and since we only needed to look at the fives, z_10(n) = z_5(n) for base five as well.
Let's also define Z(b) to be the limit of z_b(n) as n approaches infinity, so Z(10) = Z(5) = 1/4. If the base b is prime, then it's easy to find Z(b) using geometric series:
z_p(n) = floor(p/b) + floor(p/b^2) + floor(p/b^3) + ...
Z(p) = 1/(p - 1)
Thus for binary, Z(2) = 1. (In binary, if n is a power of two, then n! ends in n -1 zeroes.) For ternary, we see that Z(3) = 1/2, then Z(5) = 1/4, Z(7) = 1/6, Z(11) = 1/10, and so on.
It's also easy to find Z(b) if b is a prime power. For example, in base four, we know that any number that ends in k zeroes also ends in 2k zeroes (possibly 2k + 1) in binary, and any number that ends in k zeroes in octal ends in at least 3k zeroes in binary. So this gives us:
Z(p^i) = 1/(i(p - 1))
Z(4) = 1/2, Z(8) = 1/3, Z(9) = 1/6
As for numbers which are neither primes nor prime powers, we see that b = 10 is controlled by the number of fives, and likewise b = 6 is controlled by the number of threes, since there will always be enough twos to cover the threes or fives. In other words, if we factor b = p^i * ... * q^j:
Z(p^i * ... * q^j) = min(1/(i(p - 1)), ..., 1/(j(q - 1))
In other words, we factor the base into primes and prime powers and find Z for each of the component primes and prime powers. Whichever Z is the lowest equals Z for the full base, since the lowest Z is in control of how many factors of the base there are.
Now let's consider the dozenal base -- a base so popular that it has its own forum, Dozens Online. As it turns out, something odd happens with dozenal factorials that occurs in no other human-scale base. To find Z(12), we first find Z(3) and Z(4):
Z(3) = 1/2
Z(4) = 1/2
In other words, neither the threes nor the fours are in control, because in the long run, the number of each factor tends to be equal! In other words, sometimes Z_3(n) > Z_4(n) (for example, if n is a power of three) and sometimes Z_3(n) < Z_4(n) (for example, if n is a power of four). If we want to find the exact value Z_12(n), then we must calculate both Z_3(n) and Z_4(n). (Contrast this with decimal, where we can just ignore Z_2(n) and calculate only Z_5(n).)
Number of zeros in which n! ends, for ternary, quaternary, and dozenal:
n base 3 base 4 base 12
1 0 0 0
2 0 0 0
3 1 0 0
4 1 1 1
5 1 1 1
6 2 2 2
7 2 2 2
8 2 3 2
9 4 3 3
10 4 4 4
11 4 4 4
12 5 5 5
Notice how ternary takes the lead at n = 3, with quaternary tying it up at n = 4. The 0s in each base are equal until quaternary pulls ahead at n = 8 -- and then ternary leapfrogs at n = 9 since there are two factors of three there. (Another leapfrogging will happen soon -- ternary gains its sixth zero at n = 15, and then quaternary gains two zeros for a total of seven at n = 16.) In any case, the number of dozenal zeroes matches the smaller of the number of zeroes in bases three and four. In the long run, the ternary and quaternary zeroes will be equal.
Again, dozenal is the only human-scale base where this happens. Since Z(5) = Z(9) = Z(16) = 1/4, it follows that the next bases where this phenomenon occurs are bases 5 * 9 = 45 and 5 * 16 = 80. Both base 45 and base 80 are well beyond the human-scale range.
Peterik Chapter 9: Making Use of Melody in Songwriting
Chapter 9 of Jim Peterik's Songwriting for Dummies is called "Making Use of Melody in Songwriting," and here's how it begins:
"You've plunked down your hard-earned money to buy this book, and chances are you have some songs stirring in your soul already and a melody or two percolating in your brain."
To me, the melody of a song is the part that a soloist sings, or that a lead guitarist plays. In this book, we learn that some melodies are more successful than others. Let's find out why:
"The melodies that last, the ones people never tire of, are usually simple, but they have something unique about them that keep folks coming back for more. How do you create a great melody?"
Peterik explains the technical parts, the basics of every melody:
"Notice the intervals between notes, how many are jammed into one measure, the range (the distance from the highest note to the lowest), and the notes relationship to the chord (in the key of A, the note A is the root or tonic note of the chord)."
So how do we find our melodies? Let's see:
"Some people write their best melodies while working at something else -- washing the car, shopping, driving, working in the yard, working out, hiking, or any pleasurable or charitable pursuit."
The author suggests that once a melody pops up in our head, we can record it with a tape player:
"Then play it back, experimenting with different melodies over the chords. Be sure to record the combination of music and melody on yet another recorder or on another track."
Peterik discusses how he once came up with his melody -- his co-writers Bill Syniar and Barbara Unger came up with the chords and rhythm, and he tried different melodies over that bass line:
"We wrote 'Under the Spell' for The Doobie Brothers in this fashion, as well as 'Blinded by Emotion' and 'You Still Shock Me' for the melodic rock band Mecca."
We should strive to make our songs more memorable:
"Certain songwriters can make you feel a certain way, and the true masters always seem to have a very specific mood."
There are certain ways to make our melodies fit the mood. For example, the author explains that major keys sound more upbeat and optimistic, while minor keys are darker and more mournful. But he does mention a caveat:
"In certain situations, it's advisable to mate an optimistic set of chord changes with a dark and evocative melody that is technically at odds with the mood of the music."
We can also find melodies that fit the genre we're writing in. For example, Peterik explains how he wrote a certain song for rock 'n' roll:
"However, one of my biggest hits, 'Vehicle,' uses a narrow five-note range, similar to the early rock 'n' roll and blues songs, and achieves its impact through the changing chords (Eb minor to Bb minor) to give the melody its variety. When you are writing a song, make sure it's singable."
The author keeps range in mind as he moves on to the pop rock genre:
"Melodies in the rock genre range from the blues-based, two or three note walls (some call it screaming on key) by Robert Plant of Led Zeppelin ("Whole Lotta Love," written by Led Zeppelin, and 'The Lemon Song,' written by Led Zeppelin and Chester Burnett) to the octave leaps of Steve Perry, lead singer of melodic rock pioneers Journey ('Open Arms,' written by Steve Perry and Jonathan Cain, and 'Faithfully,' written by Jonathan Cain)."
Peterik warns us about urban music or rap, which is much less melodic:
"It's very attitude-based, where too much melody and too many chord changes can serve only to break the hypnotic groove of this type of music."
At this point, the author branches out to the various sections of each song. For the verse, he looks at Sheryl Crow's "If It Makes You Happy":
"The verse, however, is never a tune-out factor (when your hand involuntarily goes reaching to find another station); because even though it's short on melody, it's long on intriguing images ('scrape the mold off the bread and serve you French toast again')."
We move on to the pre-chorus section:
"In Elton John's classic song from 1992, 'The One,' the pre-chorus songs suspiciously like a chorus the first time you hear it ('In the instant that you love someone, in the second that the hammer hits'), as the suspended chords of the verse give way to pure majors and the melody starts to soar."
Let's proceed with the main part of the song, the chorus:
"In 'Day of Freedom' (written by Cindy Morgan, Brent Bourgeois, and Chris Eaton; sung by the young Christian artist, Rachael Lampa), the song features an Eastern-influenced melody in the verse (complete with sitar and tabla arrangement touches), a brief two-bar channel (pre-choruses need not be long to be effective,) and then hits its sprawling chorus featuring long notes stretched over the 4 minor chord to the 5 seventh, and the words 'We're all children of the faith, and though we walk in different ways, let's live our lives to celebrate the day of freedom.'"
Finally, we have the bridge section of the song:
"The bridge is your last chance to say anything lyrically, chordally, and melodically that has been left unsaid. If you have nothing left to say in your song, you don't really need a bridge."
This takes us to the next "Practice Makes Perfect" section.
Returning to the Mocha EDL Scales
Once again, we're done with the musical 2016-17 project for now. As I said in my last post, that was really an underhanded way to prepare songs for my new charter school Math 8 position, only for me to end up not taking that job. I still wish to write some songs for the upcoming school year, but I don't know what class that will be (and I might not know until just before the first day of school). There's no purpose in continuing to write songs that I might never perform.
Instead, we're going to return back to melodies in EDL scales. In particular, in today's post I'll decide in which EDL scales I'll be composing this year's songs. There was some mention of EDL's in my most recent post as well as my June 28th post, but today I'm writing about EDL in earnest.
First of all, let's review what EDL stands for -- Equal Division of Length. Here "length" refers to, say, a length of string. Look at a guitar and consider its frets -- the frets nearer the nut are far apart, while those closer to the bridge are closer together. Therefore the guitar strings are not fretted to EDL.
If we were to divide the string equally, then we'd obtain an EDL scale. For example, last month I wrote about one of our simplest EDL scales, 12EDL:
The 12EDL Octave:
Degree Ratio Cents Note
12 1/1 0 white A
11 12/11 151 lavender B
10 6/5 316 green C
9 4/3 498 white D
8 3/2 702 white E
7 12/7 933 red F#
6 2/1 1200 white A
Here is what each column in this chart stands for:
- The Degree tells us what part of the string is vibrating. For 12EDL we're dividing the string into ten parts, so Degree 12 means the entire open string is vibrating. Degree 11 means we are fretting the string so that 11/12 of the string is vibrating, 10 means 10/12, and so on.
- The Ratio gives the ratio between the full open string length and the vibrating part, so this is the reciprocal of the fretted string length. Simple ratios correspond to well-known musical intervals, so 2/1 is the octave, 3/2 a perfect fifth, 4/3 a perfect fourth, 5/4 a major third, and so on.
- The Cents are defined so that 1200 cents an octave. Our standard scale is 12EDO, which divides the octave into 12 semitones of 100 cents each. In EDO's the numbers of cents between each step are equal, while in EDL's the lengths of string between each step are equal. The formulas for conversion are ratio = 2^(cents/1200) and cents = log(ratio)/log(2)*1200.
- The Notes here are assigned colors -- Tall Kite's color notation. They refer to factors that appear in the ratio. If there are only factors of 2 and 3 then the color is "white" (abbreviated "wa"), a 5 in the denominator is "green" ("gu"), a 7 on the bottom is "red" ("ru"), and 11 is "lavender" ("lu").
There are also colors for factors in the numerator -- 5 is "yellow" ("yo") and 7 is "blue" ("zo" -- the "z" stands for "azure," to avoid mistaking the "b" in "blue" for a flat symbol). But Mocha's EDL scales involve factors on the bottom, so "yellow" and "blue" are unplayable in Mocha. Because the factors appear under the fraction bar, they are called "undertones" or "utonal," and have a "u" in their color syllables ("gu," "ru," "lu"), as opposed to "overtones" or "otonal" color syllables ("yo," "zo").
Here's a short program in Mocha that plays EDL scales -- type in 12 to indicate 12EDL:
https://www.haplessgenius.com/mocha/
Line 10 contains the line N=8. This tells us which undertone (that is, which key) to play our scale in -- if we had N = 1, then the highest notes (Degrees 12-6) would be played. Instead, using N = 8 allows the scale to play three octaves lower, starting at Degree 12 * 8 = 96. Notice that in our last post, we found out that the standard note A440 appears to be Degree 95, so our white A ("wa A") is very close to this. I often use N = 16 instead, so that the scale begins near A220 (just below middle C).
Oh, and the last magic number in the Sound command 4 corresponds to quarter notes.
Which EDL Scales Will I Play?
Here is my final decision -- during the year, I will write in a different EDL scale depending on what time of the year it is:
If by chance I ever teach a summer school position, then its songs will be in 20EDL. By following this pattern, it means that I get to compose in progressively more complex EDL's throughout the year -- and I get to learn more about particular mood of each scale.
Once again, when I was writing songs earlier this summer, I was looking at a pacing plan for eighth grade Illustrative Mathematics pacing plan and assigning an EDL based on what month we might reach that lesson. (I know, you're tired of hearing about that teaching position that I didn't take, but so much of how I wrote the songs was tied up in that job!) According to that pacing plan, the 8.NS lessons on irrational numbers are covered last, hence their songs were 18EDL, while the 8.G lessons on geometry are covered first, hence their songs were 10EDL. Of course, now that I'm not taking that job, I must wait until I find out my actual assignment for the upcoming school year.
At any rate, we know that 10EDL and 12EDL will cover the first semester, so I should at least pay close attention to these two simplest EDL's. As 12EDL appears earlier in this post, let's look at 10EDL:
The 10EDL Octave:
Degree Ratio Cents Note
10 1/1 0 green C
9 10/9 182 white D
8 5/4 386 white E
7 10/7 617 red F#
6 5/3 884 white A
5 2/1 1200 green C
According to Peterik's book, major scales indicate a happy mood while minor scales indicate a sadder or darker mood. When looking at EDL scales, it helps to consider the third, as it's the third that makes a scale sound major or minor.
Here are the thirds available in each EDL:
I might change my mind by the time we get to September. But as for now, I'd rather keep 10EDL in October and 12EDL in December for a simple reason -- October is the 10th month while December is the 12th month. It also maintains my main goal of starting with the simplest EDL, 10EDL and moving up to the higher EDL's.
So as of now, our first EDL will be 10EDL. Once again, the notes of this scale are C-D-E-F#-A-C, which is very similar to the major pentatonic scale C-D-E-G-A-C. Depending on the songs, it's possible that I might end up singing red F# as G because I'm expecting a pentatonic scale, even though the song as played in Mocha is red F#. Once again, the main rule of music is that it should sound good -- and so whether I sing/play F# or G depends on what sounds good in that particular song.
EDL's and Kite Colors
When Kite first came up with his color notation, he intended it to refer to both intervals (for example, a yellow/yo 3rd) and notes (for example, a yellow/yo E). The interval colors are clearly relative (to the root of the interval), but as it happens, so are the note colors. For Kite, the tonic of any scale must be white or wa. So if a song is in the key of C, it must be wa C. Then all the other notes are relative to the tonic -- yo E, for example, is a yo 3rd above the tonic wa C.
But in our 10EDL scale above, the tonic is green/gu C, not wa C. This is because I prefer to color my notes absolutely, not relatively. To me, Degree 160 is gu C and Degree 162 is wa C -- and since 162 is not a multiple of 10, it can't be the tonic of a playable 10EDL scale in Mocha. Indeed, a 10EDL with a wa tonic would require a yo 3rd, but no yo note is playable in Mocha.
In many ways, the wa notes of Kite are similar to the white keys on a piano. Insisting that the tonic of a scale must be wa is like insisting that it must be a white key -- so keys like Bb major and F# minor would no longer be available. Indeed, it's like insisting that the tonic of F# minor is really F (since the tonic must be a white key), even if it's played as Concert F#.
So to me, wa C is only Degree 162 (or 81), and wa A is only Degree 96 (or 192, or 48 etc. -- octave equivalence is implied here). Any other Degree must be colored other than wa, even if it's the tonic.
Of course, most musicians don't use Mocha, so to insist that wa A is only Degree 96 (Sound 165) makes no sense to someone who isn't on Mocha. If we're going to declare that wa A is one particular note, then that note should be defined universally -- using Hertz, not Mocha Degrees.
In my last post, I wrote that Degree * Hertz = 41862, so that Degree 95 is A440. Using this formula, we see that Degree 96 is around 436 Hz, so this would be our wa A.
But notice that while Degrees are a utonal system, Hertz are otonal -- and the otonal colors (yellow and blue) are based on factors. So while Concert E is around 330 Hz, we'd prefer our yo E to be 320 Hz, since yellow/yo indicates five multiplied by a 3-smooth number. Likewise, blue/zo ought to mean seven multiplied by a 3-smooth number.
In order for this to work, the wa notes need to be the 3-smooth numbers themselves -- so in particular, 256 Hz needs to correspond to some wa note. Since 256 Hz is closer to Concert C than any other concert note (Concert C = 262 Hz, Concert B = 247 Hz), it follows that the powers of two are C's. It turns out that the corresponding wa A (three perfect 3/2 fifths above C) must be 432 Hz. Thus, even though our chosen wa A is 436 Hz, if I were constructing this system from scratch (rather than basing it on an old Mocha program), I'd choose wa A = 432 Hz.
Of course, the frequencies 256 Hz and 432 Hz have appeared on YouTube and other sites. The fringe politician Lyndon Larouche once suggested that C256 and A432 are more in accord with "the natural frequency" of our planet, and so music should be composed using C256 or A432 instead of A440.
If 432 Hz were to become wa A, then what note would 440 Hz become? We notice that 440 Hz is 11/8 over 320 Hz, which in turn is 5/4 over 256 Hz, which is middle wa C. Then 320 Hz is a yo 3rd above wa C, so it's yo E -- but what about 11/8? The syllable for 11 over is "lo," so 440 Hz is loyo. But, as Kite points out, it's ambiguous whether 11/8 should be a perfect 4th (making it an A -- a perfect fourth above E) or an augmented fourth (making it A#). But as there's such a strong tradition of referring to 440 Hz as A, we should let the name A prevail. Therefore if 432 Hz is wa A, 440 Hz is loyo A.
To make this work, Degree * Hertz would need to be 41472, which is 144 * 288. This means that the note wa D, at Degree 144, is 288 Hz. Technically speaking, one octave lower than this would be Degree 288 at 144 Hz, but the lowest note in Mocha is Degree 260.
Only factors of 41472 -- the wa notes C, G, D, A, E -- are the only notes that can be a whole number of both Degrees and Hertz. (Yes, these would form a C pentatonic scale, except that they are all wa.) In particular, wa F is Degree 243 but not a whole number of Hertz, and vice versa for wa B.
Melodies and Ranges
Peterik wrote about the melodic range throughout this chapter. Certainly I must consider range when preparing these EDL songs.
First of all, let me consider my own vocal range. My strongest range is the octave below middle C -- that is, from c to c' or C3 to C4. This puts me in the baritone-bass range (like most male vocalists). On the low side, I can definitely sing B2, and possibly A2, but G2 is hard for me to sing strong. On the high side, I can definitely sing D4, but E4 is a little tricky. I can't really get F4 (without making it falsetto).
Notice that our 10EDL scale goes from C to C. The first line of the code above has N = 8, which produces the octave C5 to C6. Changing this to N = 16 makes the octave C4 to C5. (The lowest note playable on Mocha is around E3 -- I must always sing at least one octave lower than Mocha.) And our 12EDL scale goes from A to A, so I should sing this as A2 to A3 -- one octave below N = 16.
But it gets tricky once we reach 14EDL, because it naturally goes from F# to F#. For me, F#2 to F#3 is a bit low, but F#3 to F#4 is on the high side. At this, one might point out that we don't need to enter either N = 8 or N = 16. For example, N = 12 would place the range from B2 to B3 -- on target. The problem is that when I'm learning melodies, I naturally want to sing Degree 14 as F# as for the longest time, I had almost perfect pitch. Changing 14 to a note other than F# (and the other notes, such as Degree 12 away from A, Degree 9 away from D, and so on) will throw me off.
Meanwhile, 16EDL ranges from E to E, but it leads to a different problem -- the fact that it has only a neutral third (16/13) rather than a major or minor third. The fact that I had almost perfect pitch -- for 12EDO that is -- throws me off when trying something new like a neutral third. In addition, notice that I don't have any true EDL instruments, so all of these scales are being approximated on the guitar, where I'd be forced to play either a major or minor third. (Of course 14EDO and 18EDO also have Degree 13, but neither one of these place Degree 13 on the third, where it's easily noticed.)
It's possible to solve both of these problems -- practice singing lower notes such as F#2, and then declare my range to be from F#2 to D4. Then this puts all of 14EDL (which has two plausible thirds, one minor and one major) in range, while kicks 16EDL (with its problematic neutral third) out. If I try this, then I can sing my 14EDL songs from F#2 to F#3 in January and February, but leaves me nothing to sing for 16EDL in March and April.
But there's something else for us to consider. So far, we've been setting the range to one octave. Yet as we learn in Peterik, most songs don't have an exact octave range. In fact, he points out that in my songs, the verse might have a "five-note range" (that is, a perfect fifth), while the chorus might contain a wider, higher range. Based on many familiar songs, I notice that common ranges include an octave plus a second (a ninth), or an octave plus a third (a tenth). "The Star Spangled Banner," our national anthem, is notoriously hard to sing to its wide range, an octave plus a fifth (a twelfth or tritave).
How we might handle this depends on the particular EDL scale. For example, 10EDL resembles a pentatonic scale, and we expect a pentatonic song at least to contain all five notes C-D-E-G-A. So its minimum range is a sixth. We might extend 10EDL beyond the octave C3-C4 to maybe one note on either side, such as A2 or D4. It's the higher EDL's like 18EDL where we might choose to restrict the range to a fifth, since there are so many more notes available.
By the way, if I were to extend 10EDL down to A2 -- Degree 12 -- I'd still label the song as a 10EDL song rather than 12EDL. I want the number in the EDL name to be the tonic, so calling it 10EDL emphasizes that the tonic is Degree 10, not Degree 12.
Keep this in mind as I might label some songs as 15EDL, where Degree 15 (F3) is the tonic. Such a song might have a range from, say Degree 20 (C3) to Degree 9 (D4), and it contains a true major chord on the tonic (Degree 15 F, Degree 12 A, Degree 10 C). Even though the lowest note is Degree 20, I'll label it as 15EDL to emphasize the tonic. (Hmmm -- perhaps one way to solve the 16EDL problem is to play 15EDL instead.)
A More Melodic TI Program
So let me write a new TI program that takes all this into account. Instead of entering a single value, we'll ask for a range, and then we'll choose random notes within this range:
:Disp "LOWEST"
So instead of entering an EDL, we can enter 20 and 9 for the 15EDL scale described above. (It doesn't matter whether we enter 20 then 9 or 9 then 20, since randInt works properly either way.) In such a song, we might go 20 to 12 (a sixth) for the verse, 20 to 10 (an octave) for the pre-chorus, and then the full range 20 to 9 (a ninth) for the chorus.
Conclusion
That was an exciting World Championship Track Meet in Oregon this year. I especially enjoyed seeing world records set in both women's hurdle events (100 and 400) -- congrats to Sydney McLaughlin (USA) and Tobi Amusan (Nigeria). And since these were actually the delayed 2021 Worlds, the next championships are only a year away!
Speaking of hurdles, there was some controversy on the men's side. In the heats, Devon Allen (USA) was DQ'd for a false start, and so were some of the women.
Let me try explain what happened here. There are three things that a hurdler does at the start of a race:
B. React to the gun.
C. Start to run.
Both events B and C must happen after A, or else it's considered a false start. But unfortunately, event B is invisible. So instead, event B is considered to occur a fixed time before event C -- the "reaction time," which is considered to be 0.1 second. So if event C happens less than 0.1 second after event A, then it's assumed that BAC occurred instead of ABC, and so the sprinter or hurdler is DQ'd.
For Devon Allen, C occurred 0.099 seconds after A. Therefore Allen was DQ'd, since the working assumption is that B occurred 0.001 second before A. (Event B, if your recall, is invisible -- no replay can ever show B occurring at all, much less before A.)
Some people wonder why there even is a reaction time rule anyway -- they believe that as long as event C occurs after A, it shouldn't matter when B occurred. The reason is that if we allowed this, then track would turn into a event of luck rather than skill. Someone might win a medal or even break the world record because they were lucky enough to place A between B and C -- they anticipated the gun yet managed not to move before the gun, while their opponent is dutifully waiting for the gun. In order to keep all medals and records legit, we must sure that only ABC is legal, not BAC.
The other point of contention is where the value 0.1 second comes from -- perhaps it really is possible for a human being to react in less than 0.1 second. It suddenly becomes a sorites paradox (or heap paradox) problem -- a race is legal if there's a "heap" of time between A and C, and yet 0.1 seconds is considered a "heap," but 0.099 seconds isn't.
The value 0.1 second comes from experiment, but the experiment was likely conducted before clocks accurate to a millisecond were invented. The value 0.1 second (100 milliseconds) is clearly a round value -- it's 100 ms and not 99 ms for the same reason that most people celebrate 100th anniversaries, not 99th anniversaries. But is it an accurate value for the human reaction time?
If we lived in a dozenal (base 12) world, the reaction time would likely be 1/12 second, not 1/10. Then Allen's (and the other women's) reaction times would be legal, with no DQ's (as 1/12 s = 83 ms).
(Indeed, this is what happened with the magic musical constant earlier in this thread. I stated that my preferred value is 41472 = 20000 dozenal, though its measured value is 41862 = 20286 dozenal.)
So a new experiment should find a more accurate value of the reaction time, and this time should be the official reaction time in ms (even if it's some odd prime number, rather than a round 100 ms). It's possible, for example, for the new experiment to reveal that women have a faster reaction time than men, so this should be taken into account as well.
It's likely that there's a probabilistic range -- for example, at 110 ms there's a 5% chance that the runner false started, but at 95 ms there's a 95% chance of a false start. So the question is, are we more fearful or false negatives or false positives? (Here "positive" means legal, so a false negative is an innocent ABC runner who was DQ'd, and a false positive is a guilty BAC runner who gets a medal or record.)
If false negatives are the problem, then we should choose the 95% value. If false positives are the problem, then we should choose the 5% value, but perhaps have some mitigating factor for the athletes, such as restoring the old "first false start is charged to the field, second false start is DQ" rule.
I taught Statistics in high school last year, so I'm apply the Stats that I taught to this real-world problem.
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