XEVEX

Volume VI, Issue II

May, 2015

XEVEX The Ramaz Mathematics Magazine

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Volume VI, Issue II

Mathematics Behind Winning the Lottery Matthew Hirschfeld ‘17 In the New York Times article, “The Case for Buying a Powerball Ticket,” Neil Irwin, senior economics correspondent with an M.B.A. from Columbia University, discusses the odds stacked against you when playing the lottery.

Irwin begins by admitting that financially literate people are justified in their stricture on buying lottery tickets, as the odds of winning anything substantial are incredibly unfavorable. Then Irwin sternly cautions that certainly, money one cannot afford to lose should not be spent on the lottery and that “riding the lotto” is a dangerous path for compulsive gamblers or those who are impecunious.

However, he goes on to elucidate that there are certain dimensions to this game-of-chance, which the abundant warnings surrounding the lottery do not touch upon. As long as one is capable of properly viewing the lottery, which would be as a consumption good, not an investment, then buying a ticket can indeed be an entirely reasonable decision.

Firstly, if one would ever consider purchasing a lottery ticket, buying it during a Powerball jackpot like the recent one where the estimated winnings hovered around $500 million, would be the best possible time. The most substantial and widely applicable reason that buying lottery tickets is not a terrible idea is because it is quite pleasurable to imagine one’s future after arriving at vast wealth. There being some chance, however miniscule, that that fantasy could actually come to fruition, only augments to the fun. The $2 for a ticket is a relatively negligible price to pay for the

May, 2015 amusement of pondering one’s life after receiving the jackpot. If you could reap more enjoyment from this mental exercise than from whatever else you could spend $2 on, then there is nothing at all foolish about putting that money into a Powerball ticket.

Now, the reason the most opportune time to buy a lottery ticket is when there is a lofty jackpot, is because of the way the Powerball math works. Each lottery ticket would have a higher present value than at any other time.

Strictly viewing it as a math problem, you would be throwing away less money by buying a ticket when the pot is flush than when it is low. The way the Powerball works is that when there is a drawing with no jackpot winner, the money is rolled into a new jackpot. As a result, by buying a ticket for a chance at the jackpot, you are purchasing a chance to win some of the money that earlier buyers of Powerball tickets put in before you.

The Multi-State Lottery Association estimates the chances of winning the grand prize at about 1 in 175 million when the cash value of the prize is at $337.8 million. The simplest math points to the fact that a $2 ticket has an expected value of about $1.93. However, it is even more complicated than that, because that calculation does not account for the fact that there could be multiple jackpot winners, who must split the plot. Nor does it deal with the income tax one would owe on any winnings. Nevertheless, the foremost takeaway is that, while you are still throwing away money when buying a lottery ticket, you are throwing away less money, strictly economically speaking, when you purchase a ticket for an unusually sizeable jackpot 2

Volume VI, Issue II

Basketball: Do Stats Really Matter?

Ben Rabinowitz ‘16 Last month, when Charles Barkley infamously said on Inside the NBA on TNT that he was not a fan of the analytical approach to basketball, (of course in a more colorful manner), Daryl Morey—GM of the Houston Rockets and huge proponent of using analytics and statistics when constructing a team—came back firing at Barkley. He tweeted out insulting messages about how Barkley spews nonsense on TV, sparking somewhat of a verbal battle with one of the NBA’s greatest players of all time.

However, this is not just an entertaining spat uPER = (1 / MP) * [ 3P + (2/3) * AST + (2 factor * (team_AST / team_FG)) * FG + (FT *0.5 * (1 + (1 - (team_AST / team_FG)) + (2/3) * (team_AST / team_FG))) - VOP * TOV - VOP * DRB% * (FGA - FG) VOP * 0.44 * (0.44 + (0.56 * DRB%)) * (FTA - FT) + VOP * (1 DRB%) * (TRB - ORB) + VOP * DRB% * ORB + VOP * STL + VOP * DRB% * BLK x PF * ((lg_FT / lg_PF) - 0.44 * (lg_FTA / lg_PF) * VOP) ] Key: uPER= unadjusted Player Efficiency Rating MP= minutes played

between two different opinionated celebrities.

There is an important question here that is raised: do analytics and statistics matter in the NBA or is it all about talent and other qualities that can’t be measured mathematically?

There is no question that there are elements to the sport that are unquantifiable; no matter how many experts try, there are some qualities to players that can’t be measured in numbers. Two such unquantifiable attributes are clutch-ness—the ability to build team camaraderie—and team leadership, both on and off the court. These are vital parts to a player that are sometimes overlooked, simply because there is no measure of leadership; one can’t say that one player leads a team 60% more than another—it’s just not possible. However, most

May, 2015 statistics in the game are measurable. Points, rebounds, steals, assists, and blocks are all quantifiable stats that essentially assess how good or effective a player is. These statistics are very helpful when analyzing a player. However, as I mentioned before, these numbers don’t give the complete story.

Analytical assessment of players and teams is growing more common among different organizations and general managers in the NBA. One of the more popular and intriguing advanced statistics calculated is Player Efficiency Rating, or PER, created by John Hollinger. Hollinger characterized PER as the stat that “sums up all a 3P= 3 point field goals made AST= assists FG= field goals made FT= free throws made VOP = value of possession= lg_PTS / (lg_FGA lg_ORB + lg_TOV + 0.44 * lg_FTA) ORB= offensive rebounds TOV= turnovers DRB= defensive rebounds TRB= total rebounds STL= steals BK= blocks PF= personal fouls factor = (2 / 3) - (0.5 * (lg_AST / lg_FG)) / (2 * (lg_FG / lg_FT)) DRB% = (lg_TRB - lg_ORB) / lg_TRB Lg= league totals

player's positive accomplishments, subtracts the negative accomplishments, and returns a perminute rating of a player's performance”. Here is how the unadjusted PER (meaning it does not take into account team or league “pace”) is calculated: (See Above) So after seeing how complicated these advanced statistical calculations can get, the question remains, “do they matter?” Personally, I do not see the immense value in these analyses compared to the traditional eye test—when watching a player or team, simply observe whether they look good or not. However, with the analysis movement in the NBA becoming more popular, regardless of how you and I feel, these stats are here to stay.

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Volume VI, Issue II

Three Brilliant Mental Math Methods Jacob Berman ‘16 In this article I am going to explain three simple mental math methods that will wow people wherever you go. The first method is how to square numbers in your head. The second method is a more specific way to easily square multiples of five. The final method is how to multiply 11 by 10+ digit numbers and quickly get the answer.

Squaring very large numbers can sometimes be a real pain. For example, try squaring 84. To start to tackle a number like that in your mind would take forever. But if you try and relate that number to the nearest ten this problem could become very simple. In this case the closest ten is 80. We can call d the difference between 84 and 80. Now all you need to do is multiply (x-d)(x+d) and then add d . So in this case it is (80)(88) which equals 6400+640=7040 and if you add 4 you get 7056. This process is much easier than tackling 84 in your head.

The next mental math method is how to square multiples of five instantly. Lets try to square the number 55. Clearly, this is not easy to do without some mental math trick. First you get rid of the five in the unit’s digit. Then you take the remaining digit and multiply it by itself plus one. So you have (5)(5+1)=30. Now all you do is 2

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Fermi Problem Ben Kaplan ‘16 Enrico Fermi was an Italian-born physicist.

Fermi’s biggest contributions were to quantum theory, nuclear physics, particle physics, and statistical mechanics. He also participated in the Manhattan Project. A Fermi problem consists of a series of estimates which are built upon each other to arrive at an estimate for a non-trivial value. He used this method in the Manhattan Project to determine the force of a nuclear bomb.

However, the most famous Fermi problem is: How many piano tuners are there in Chicago?

There are approximately nine million people in Chicago There are on average two people per household About one out of every 20 houses have a piano

May, 2015 simply add the digits 25 to the end and you’re left with the answer—3025. This method also works with bigger numbers like 105. 105 is equivalent to (10)(10+1)=110 then if you add the suffix you get the answer—11025.

Imagine trying to multiply 11 by 9,621,576,521. It doesn’t seem to easy but soon enough it will be. First lets start off by multiplying 11 by 33. If you just add the digits in the multiplied number you get 6. Then you put that 6 in the middle and get 363. What if you multiply 11 by 55? When you add the two digits you get 10, which doesn’t quite fit in the middle of the two 5’s. All you do is you add the extra one to the hundreds digit five and you get 605.

This trick can be applied to numbers with more than two digits. Try multiplying 11 with 4,281.

You keep the initial and final digit, but you add all the digits in the middle. The answer is (4)(4+2)(2+8)(8+1)(1), which is equivalent to 47,091. Note that you’re not multiply these digits but rather keeping them in that order—and when you get a number greater than ten, it moves over to the next column to the left. The final test we’ll be doing is multiplying 11 by 9,621,576,521.

You can easily work this out with a piece of paper. The answer is (9)(9+6)(6+2)(2+1) (1+5)(5+7)(7+6)(6+5)(5+2)(2+1)(1) which is equivalent to 105,837,341,731.

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Most pianos are tuned once a year It takes two hours to tune a piano Each piano tuner works 8 hours a day, 5 days a week, 50 weeks a year If we figure out how many tunings a year in Chicago: 9,000,000 (people) ÷ 2 (people per household) ÷ 20 (Houses with pianos) × 1 (tuning a year) we get 225,000 tunings. Then if we figure out how many tunings one tuner performs in a year: ½ (pianos per hour) × 8 (hours per day) × 5 (days a week) × 50 (weeks a year) we get that each tuner tunes 1,000 pianos. Then if we divide the two values we find that there are about 225 piano tuners. In reality there are about 290 tuners, however, this shows the process and accuracy of the predictions.

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Volume VI, Issue II

Haddasah Brenner ‘17 Have you ever encountered a familiar face half way across the globe and exclaimed, “What a small world?” Is there truth in that statement?

Man has existed on earth for thousands of year, the majority of which, they survived by utilizing hunting and gathering techniques.

Nomadic and wandering, these populations never rose above 10 million. When agriculture and other characteristics of stable civilizations emerged, the numbers began to rise steadily, but for intermittent disasters of plague and war. With premature deaths decreasing, average survival rates increasing, and births remaining constant, there is no doubt that growth followed an exponential trend from then on.

The population growth can be predicted by the following formula: A=Pe , or in other words, the original population multiplied by e to the rate times time. The current rate remains approximately 1.14%, resulting in about a 75 million change per year.

According to ecological studies, a population may experience exponential growth, but as resources become limited, density dependent and independent factors will bring the population to the carrying capacity, eventually leveling off there. This population will then progress to a stage of logistic growth.

The human population is expected to do just that, though there is fear that by abusing the

environment and taking that which doesn’t rightfully belong to us, we may cheat the system, and grow past the carrying capacity.

As less developed countries, such as India, Pakistan, etc. reproduce at high rates and resources are provided to them, it is mostly they who are responsible for the exponential increase in the past years.

The population statistics published by the United Nations project that by 2050, the world population will reach a peak of about 9 billion in contrast to the current 7.6 billion. The less developed countries will discontinue high birth rates, as survival will be much easier with more sophisticated medical care and technology.

Although we are predicted to approach carrying capacity in the next century, it is not a definite number, and we cannot foretell the effects the population will have on the world as a whole. The resources utilized by humans deplete species diversity (overfishing in the ocean or deforestation for paper and other materials), as well as contamination of water, dirt, and air with waste and pollution.

The human population cannot continue to grow far past carrying capacity, but we need to avoid environmental disasters/factors that would wipe out those surpassing the limit. By education of the impending complications with exhausting supplies and the alteration of our ways of life, it is still possible to steer clear of any major population-generated difficulties.

Most Common Birthdays:

later.

Jacob Berman ‘16 What causes babies to be more frequently born on one day than another? A recent study compiled by a Harvard public policy professor has shown just how common or uncommon it is to give birth on certain days.

The most common month to be born in is September, wherein are the top ten most common birth dates. The most common birthday is September 16 . These facts clearly show that couples are busier over the Christmas break holidays, because September is nine months

Interestingly, the most uncommon times to be born are on public holidays. Although the least common birthday is February 29 , because it only occurs every four years, the other uncommon days to be born are when doctors are less available. Patients and doctors often schedule when a baby will be born. Doctors often induce patients or have to perform caesarean sections. December 24 and 25, are both very uncommon days to give birth. Additionally, Independence Day, Memorial Day, Labor Day, Valentine’s Day, Halloween, and Thanksgiving are all very uncommon birthdays.

Human Population Growth

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Volume VI, Issue II

Math of the Eruv Max Koffler ‘16 The technical requirements of the eruv— like that it must be 10 tefachim (~40 in.) high— are relatively well known. Also well known is that the eruv must not enclose a “Reshus HaRabim.” However, the controversy as to what this Reshus HaRabim actually encompasses is much less known. For a street to be considered a Reshus HaRabim, it needs to be 16 amos wide with at least 600,000 people in the domain.

There is a disagreement in how to count 600,000 people, for some say it needs to be 600,000 physically in the street at one time (Ramban) and others say it only needs to be 600,000 in the city (Rashi).

Rabbi Moshe Feinstein points out that the laws of public domain are derived from the encampment that the Jews lived in when they left Egypt. Based on this, he disagrees with both Ramban and Rashi, and suggests that the 600,000 people would need to be in an area the size of the Jewish encampment in order to constitute a public domain.

The encampment was 12 mil by 12 mil. A mil is based on the distance that an average person can walk in 18 minutes, which is about

Calculator Programming Henry Koffler ‘19 Calculators are essential to every math student’s life, whether it be simple addition, subtraction, or even graphing. What we aren’t doing is maximizing its potential. Imagine a world where you could store a formula on your calculator and it simply asks you for the input.

That’s a world we already live in. Basic calculator programming is a must nowadays; who can remember 40 formulas for a test?

Some basic formulas take second to make, like one that would solve a quadratic equation, one that would find a vertex, or one that solves for continuous compounding interest

Reduce Radicals

May, 2015 2,000 amos. R’ Avraham Chaim Na’eh states that an amah is 18.9 inches, while according to the Chazon Ish, it is 22.7 inches.

1) What was the size of the encampment?

Rabbi Avraham Chaim Na’eh: 12 mil x 2,000 = 24,000 amos 24,000 amos x 18.9 = 453,600 inches 453,600 inches / 12 = 37,800 feet 37,800 feet / 5,280 = 7.16 miles in length/width = 51.27 square miles Chazon Ish: 12 mil x 2,000 = 24,000 amos 24,000 amos x 22.7 = 544,800 inches 544,800 inches / 12 = 45,400 feet 45,400 feet / 5280 = 8.60 miles in length/width = 73.96 square miles 2) Based on 600,000 people in the encampment, what is the population density per square mile?

Rabbi Avraham Chaim Na’eh: 600,000 / 51.27 = 11,703 people per square mile Chazon Ish: 600,000 / 73.96 = 8,112 people per square mile.

PGRM:RERAD :ClrHome :0→K :Prompt D :For(I,1,5) :For(J,2,10 :If not(fPart(D/J2)):Then :D/J2→D :K+J→K :End :If K=0:Then :1→K :End :Disp “WHERE K√(D)” :Disp “K=”,K :Disp “D=”,D 6

Volume VI, Issue II

Mathematics in Medicine Matthew Hirschfeld ‘17 In this article we will be exploring the particular application of the mathematical construct of an ellipse to a certain medical process known as lithotripsy. The ellipse is a very special and practical conic section, and its most important property is its reflective property.

If you think of the boundary of an ellipse as being made from a reflective material, then a light ray emitted from one focus will reflect off the ellipse and pass through the second focus.

This is also true not only for light rays, but also for other forms of energy, including shockwaves.

Shockwaves generated at one focus will reflect off the boundary of the ellipse and pass through the second focus. This characteristic, unique to the ellipse, has inspired a useful medical application. Medical specialists have used the ellipse to create a device that effectively treats kidney stones and gallstones. A lithotripter uses shockwaves to successfully shatter a painful kidney stone (or gallstone) into tiny pieces that can be easily passed by the body. This process is known as lithotripsy.

May, 2015 energy ray reflects off a surface, the angle of incidence is equal to the angle of reflection.

Extracorporeal Shockwave Lithotripsy (ESWL) enables doctors to treat kidney and gallstones without open surgery. By employing this alternative, the risks associated with surgery are significantly reduced, there is a smaller possibility of infections, and less recovery time is required than with a surgical procedure. The mathematical properties of an ellipse provide the basis for the medical invention of the lithotripter, the instrument used in lithotripsy.

The lithotripter machine has a half ellipsoid shaped piece that rests with its opening to the patient’s side. An ellipsoid is a three dimensional representation of an ellipse. In order for the lithotripter to work using the reflective property of the ellipse, the patient’s stone must be at one focus point of the ellipsoid and the shockwave generator must be at the other. The patient is laid on the table and moved into position next to the lithotripter. Doctors use a fluoroscopic x-ray system to maintain a visual of the stone, which allows for accurate positioning of the stone as a focus. Because the stone is acting as one of the foci, it is imperative that the stone be at precisely the right distance from the focus located on the lithotripter. This is essential in order for the shockwaves to be directed onto the stone.

Hopefully, through this exploration of one of the innumerable critical and useful applications of mathematics, we can all continue to appreciate the ever-deepening profoundness of the sciences and their impact on humanity.

As illustrated in the diagram above, when an

Vertex Finder PROGRAM:VERTEX :Prompt A,B,C :(0-B)/(2A)→X :Disp X :Input “X=”,F :Disp “X=”,F,”Y=”,AFF+BF+C

Quadratic Equation Solver - PRGM:QUAD :Disp “AX2+BX+C=0” :Prompt A,B,C :B2-4AC→D :Disp (-B+√(D))/(2A)Frac :Disp (-B-√(D))/(2A)Frac :DelVar A :DelVar B :DelVar C :DelVar D 7

Volume VI, Issue II

The Golden Spiral Hadassah Brenner ‘17 I lifted the seashell with its wet surface dripping and shining in the sun; I wondered at its seeming perfection. That shell, as well as many other creations in nature, is renowned for taking on the pattern of the Fibonacci spiral.

The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34…), coined by Leonardo of Pisa (1170-1250) in Italy, is a particular series of numbers discovered and rediscovered throughout the course of history. The sequence consists of the infinite possibilities calculated by adding the two numbers before the given number (Fn=Fn-1+Fn-2). Interestingly enough, the ratio of two consecutive Fibonacci numbers approaches the golden ratio (about 1.6180339887…), a ratio of the addition of two line segments over the larger segment set equal to the larger segment over the smaller segment.

To create the “eye-appealing” golden spiral, as many have claimed it to be, one need only sequentially create boxes of width Fn and draw the swirl, as depicted in the image displayed on the right. Each square contains one fourth of a circle, with an increase by the golden ratio at each quarter turn. As shells in nature increase in size, they add chambers increasing in a growth factor of the golden ratio. This phenomenon is illustrated in the pictures above, with the golden ratio in number, next to a real shell to show the resemblance.

Not only do shells exhibit this phenomenon in nature, but also the population growth of cows, bees, and rabbits ideally expand in accordance with the Fibonacci sequence (Leonardo himself published a book on the

May, 2015 Fibonacci rabbit math problem, titled Liber Abaci, to which the solution was his own namesake). Every time these animals were once again capable of reproduction, their next generation’s population would follow the sequence.

Also notably, various plants arrange themselves in the same sequence, whether through their leaf pattern, stems, petals, or seeds.

To mention a few, lilies and irises of three petals, buttercups of five petals, and pinecone spiral designs mirror the number series. Nature conveniently utilizes the spiral to pack in their seeds and petals to the maximum, as well as grow in proper proportions.

Although our fascination of Fibonacci numbers began centuries ago, we continue to view it as a “golden” and valuable pattern, so much so that artists even incorporate it into their pieces. The distinction between the world of math and the beauty of nature has never been as obscured as it is when witnessing the golden spiral all across the universe, from its complex galaxies, to the minuscule seeds of a flower.

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Volume VI, Issue II

Architecture and Mathematics Throughout History Alex Moffson ‘16 In Ancient Greece they used the golden rectangle to design their buildings and the most famous of these building is the Parthenon. A building designed with the proportions of the golden rectangle has sides that form the ratio phi.

Many western architects find that this looks very appealing to the eye and have mimicked aspects of Greek architecture in their buildings like William Thornton who designed the Capitol Building in Washington DC.

In Islamic architecture, buildings are often decorated with patterns of tiling which use mathematical tessellations (when a flat surface is tiled on a plane using one of more geometric shapes called tiles with no overlapping or gaps).

There are many examples of these through out the Middle East but the most famous would be the Sheikh Lotfollah mosque in Isfahan, Iran and the Alhambra in Spain.

May, 2015 to the universe (such as the movement of the sun and the stars). They would use complex calculations to choose the dimensions of their buildings. It would follow astrological principles or basic aesthetics (what they found to be beautiful). Many Hindu temples like Virupaksha temple at Hampi follow this system.

During the Renaissance, the architect style placed importance on symmetry, proportions, and geometry. Many aspects of Roman architecture were used as the basis for Renaissance architecture. Orderly arrangements of columns, as well as the use of semicircular arches, hemispheres, and domes were a major part of it. A good example of this would be the Duomo in Florence Italy.

Modern architecture uses a type of synthetic geometry known as Euclidean geometry, which does not require coordinates. As opposed to older styles of architecture, modern architecture is full of disorder and far from perfectly symmetrical.

In other civilizations like India and Egypt manipulated architecture in order to correlate it

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Volume VI, Issue II

Math Education – Past and Present Avi Neiditch ‘16 Before the Cold War era, there was no set math education system. The content of the mathematics taught in schools nowadays is the same as it was then, however the mode through which it was taught was not universal.

The push for a universal system in Math during the 20th century came from the fear that Americans were falling behind in regards to the technical advances of the world. With the launch of the first satellite, Sputnik (a Soviet creation), in 1957, Americans grew concerned that they were falling behind in education in math and in the sciences, as we were losing to our arch-nemesis in the space race.

The creation of the “New Math” system consisted of more of a conceptual understanding for students, rather than just symbols and numbers.

Students learned why something happened rather than what something is. Teachers now had to explain and prove the concepts, rather than the students just being forced to accept these concepts as preestablished truths. However, this forced teachers themselves to reevaluate the fundamental reasons for certain occurrences. This eventually led to this system’s end, since many view “New Math” as largely ineffective due to lack of understanding and uniformity.

The trend of "Back to Basics" followed the "New Math" system in the 1970s and early 1980s whereby students memorized laws of math rather than understanding why something happened. They were back to just getting the answer and not understanding how they got there. The education system made a u-turn since the "New Math" system and went back to the time before it. Again, similarly to in the 1950’s

May, 2015 this system was deleterious, and the education sector, specifically the math sector, was in need of major reforms. There was a need for a solution, to deal with this lapse in the education of mathematics, which came in a short few years, with the creation of the National Council of Teachers of Mathematics.

In 1989, the National Council of Teachers of Mathematics established standards for teaching students math in a proper way. The council emphasized that students should understand what they are learning and, accordingly, teachers should explain the content as opposed to just teaching formulas.

An updated version of the standards, published in 2000, gives a much more modern way of teaching students mathematics. It addresses in better detail what the 1989 standards meant to deal with.

The standards stress the following content for each grade (k-12) in five ways: numbers and operations, algebra, geometry, measurement, and data analysis and probability. The learning process of each grade is outlined as well: problem solving, reasoning and proof, communication, connections, representation.

The pre-Sputnik math education system was unorganized. The math and content were the same, but the need for better education revealed itself as technology evolved in the 1950s and beyond. It was still the same basic math that was being taught, however, the lack of uniformity in the teaching methods, and the material for each age group led to its failure. Today, teachers stress not only receiving the correct answer, but also the process that one used to arrive at the conclusion.

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Volume VI, Issue II

May, 2015

Here is the cool part. It’s given that the A above the middle C (the 49 key from the left) has Abbey Lepor ‘16 a frequency of 440Hz. That specific A is called I hate to break it to you, but even with the newest technology, it is impossible to tune a piano the reference pitch because its frequency is a perfectly. A grand piano has a total of 88 keys: 52 whole number. With the equation below, you can find any pitch you desire on the piano keyboard: white keys and 36 black keys, which are all correlated. The keys are: A, B, C, D, E, F, and G, Pn= Pa·2(n-­‐a)/12 (Pn is the desired pitch, Pa is the reference pitch, and they each have their respective sharps, one n is the number (from the left) of the desired key half step above, and flats, one half step below. A chromatic scale is a musical scale where there are and a 49, the number of the reference pitch) From the few equations above, you might twelve notes in an octave, equally separated into think to yourself “Hey! With these amazingly semitones. For example: the progression of a accurate equations, someone can tune a piano chromatic scale starting on middle C will be: C, perfectly!” Sadly, C#, D…A#, B and that is impossible to C again. The tune a twelve-tone frequency ratio chromatic scale. Each between a key and key has its respective the next half step ratio with other keys down is the twelfth in the chromatic root of two:21/12. scale. Some divisions So…how does this within the chromatic relate to tuning a scale are more piano? Pianos are aesthetically generally tuned in pleasing, such as equal temperament, thirds and fifths, which is a way of having ratios of 5/4 tuning adjacent and 3/2, respectively. notes with the But these ratios do not match up perfectly, since a frequency ratio of 21/12. Twelve Tone Equal perfect third is four semitones =24/12=1.2599 and Temperament (TTET) is a logarithmic scale of a perfect fifth is seven semitones =27/12=1.4983. frequency where for any note the distance Adjusting those to be their ideal ratios and between it and its adjacent is the same for any maintaining the ratio between two adjacent keys is note on the piano. actually impossible. The only way to tune a piano The general properties of TTET are as perfectly if there are almost three times as many follow: We already know that the distance keys as there are on the average piano. between each key on a chromatic scale is the same. These two equations help find the smallest interval in an equal temperament scale: References: rn=p and r=p1/n https://www.whitman.edu/mathematics/SeniorPr (r is the ratio between adjacent keys and p is the ratio of the octave (2/1) into n equal parts) ojectArchive/2009/bartha.pdf The twelve tone scale is divided into 1200 http://en.wikipedia.org/wiki/Equal_temperament cents. In order to find the basic step in cents for http://en.wikipedia.org/wiki/Musical_tuning any interval in a chromatic scale is found in this http://www.yuvalnov.org/temperament/http://w equation: ww.yuvalnov.org/temperament/ c=wn (p is the width in cents (^see equation above), w/n

is the octave in cents (1200) into n parts)

The Math of Piano Tuning

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Volume VI, Issue II

May, 2015

The Constellations Hadassah Brenner ‘17 The night sky twinkles As it shines bright and brilliant Through the telescope The stars are points on the axis Of the universe Creating shapes And odd depictions Through drawing of lines Connecting the dazzling flecks Whose glow lights the world With a fiery beauty Capturing our eyes Since the beginning Of time Inspiring and evoking wonder Of the mysteries of life

Writers: Jacob Berman ‘16 Hadassah Brenner ‘17 Matthew Hirschfeld ‘17 Abbey Lepor ‘16 Henry Koffler ‘18 Max Koffler ‘16 Alex Moffson ‘16 Avi Neiditch ‘16 Ben Rabinowitz ‘16

Editors: Michael Rosenberg ‘15 Ben Kaplan ‘16

Faculty Advisor: Rabbi Stern

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