User:Mopedmeredith/sandbox
Musical form[edit]
Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order.[1] The common types of form known as binary and ternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.
The idea of creating a piece of music with no use of repetition has proved to be a difficult task. Even within a random collection of tones and rhythms there almost always exists some sort of repetition or tangible form to a structured piece of music. John P. Costas inadvertently solved this issue while working on developing a perfect sonar ping for the United States Navy. Working with esteemed discrete mathematician Solomon W. Golomb and using the prime number mathematics developed by Évariste Galois, Costas was able to create a pattern of tones or "pings" that contain no form of repetition within an 88x88 Costas array. Scott Rickard took this idea even further and translated this pattern into the 88 keys of a piano and has engineered "the ugliest possible piece of music, with the beautiful mathematics of the Golomb ruler." [2]
Beauty in experience[edit]
Mathematical beauty is difficult to portray to a person unfamiliar with the laws and components of a certain level of mathematics because the beauty of math often lies in the complexity and manipulation of numbers and symbols. Take for instance Euler's formula described above, in order to understand the beauty of its structure, one must first understand the properties of e, i, and pi. These mathematical elements are often not presented until higher mathematical courses and thus it takes greater mathematical knowledge and experience before one can begin to understand and appreciate the beauty within the formula.
When describing mathematics to the general society, the majority will relate this information to the utility of mathematics in science, engineering, and business models. While this is where the bulk of mathematics is used within many cultures, there is a whole other side of mathematics that exists within a philosophical and theoretical realm of study that often is overlooked by the immediate practical use of how society can benefit from certain mathematical principles. Within the research and development of new mathematics, mathematical beauty is often found in the fluidity of bridging existing mathematical ideas or concepts into new formulas and equations. Because theoretical mathematicians often deal with a small branch or specific concept of mathematics, one might need specific experience in that small niche of higher mathematics in order to truly understand and appreciate the beauty in these new mathematical ideas.
The most intense experience of mathematical beauty for most mathematicians comes from actively engaging in mathematics. It is very difficult to enjoy or appreciate mathematics in a purely passive way—in mathematics there is no real analogy of the role of the spectator, audience, or viewer.[3] Bertrand Russell referred to this principle as the austere beauty of mathematics.
John Wallis[edit]
Hobbes so evidently opposed the existing academic arrangements, assailed the system of the original universities in "Leviathan", and then went on to publish "De Corpore" (which contained not only tendentious view on mathematics, but also an unacceptable proof of the squaring of the circle). This all led mathematicians to target him for polemics and sparked John Wallis to become one of his most persistent opponents. From 1655, the publishing date of his DeCorpe, Hobbes and Wallis went round after round trying to disprove the other work. After years of debate, the spat over proving the squaring of the circle gained such notoriety that this feud has become one of the most infamous in mathematical history.
Multiplication[edit]
To demonstrate how to use Napier’s Bones for multiplication, three examples of increasing difficulty are explained below.
Example 1[edit]
Problem: Multiply 425 by 6 (425 x 6 = ?)
Start by placing the bones corresponding to the leading number of the problem into the board. If a 0 is used in this number, a space is left between the bones corresponding to where the 0 digit would be. In this example, the bones 4, 2, and 5 are placed in the correct order as shown below.
Looking at the first column, choose the number wishing to multiply by. In this example, that number is 6. The row this number is located in is the only row needed to perform the remaining calculations and thus the rest of the board is cleared below to allow more clarity in the remaining steps.
Starting at the right side of the row, evaluate the diagonal columns by adding the numbers that share the same diagonal column. Single numbers simply remain that number.
Once the diagonal columns have been evaluated, one must simply read from left to right the numbers calculated for each diagonal column. For this example, reading the results of the summations from left to right produces the final answer of 2550.
Therefore: The solution to multiplying 425 by 6 is 2550. (425 x 6 = 2550)
Example 2[edit]
When multiplying by larger single digits, it is common that upon adding a diagonal column, the sum of the numbers result in a number that is 10 or greater. The following example demonstrates how to properly carry over the tens place when this occurs.
Problem: Multiply 6785 by 8 (6785x8=?)
Begin just as in Example 1 above and place in the board the corresponding bones to the leading number of the problem. For this example, the bones 6, 7, 8, and 5 are placed in the proper order as shown below.
In the first column, find the number wishing to multiply by. In this example, that number is 8. With only needing to use the row 8 is located in for the remaining calculations, the rest of the board below has been cleared for clarity in explaining the remaining steps.
Just as before, start at the right side of the row and evaluate each diagonal column. If the sum of a diagonal column equals 10 or greater, the tens place of this sum must be carried over and added along with the numbers in the diagonal column to the immediate left as demonstrated below.
After each diagonal column has been evaluated, the calculated numbers can be read from left to right to produce a final answer. Reading the results of the summations from left to right, in this example, produces a final answer of 54280.
Therefore: The solution to multiplying 6785 by 8 is 54280. (6785 x 8 = 54280)
Example 3[edit]
Problem: Multiply 825 by 913 (825 x 913 = ?)
Begin once again by placing the corresponding bones to the leading number into the board. For this example the bones 8, 2, and 5 are placed in the proper order as shown below.
When the number wishing to multiply by contains multiple digits, multiple rows must be reviewed. For the sake of this example, the rows for 9, 1, and 3 have been removed from the board, as seen below, for easier evaluation.
Evaluate each row individually, adding each diagonal column as explained in the previous examples. Reading these sums from left to right will produce the numbers needed for the long hand addition calculations to follow. For this example, Row 9, Row 1, and Row 3 were evaluated separately to produce the results shown below.
For the final step of the solution, begin by writing the numbers being multiplied one over the other, drawing a line under the second number.
825 x 913
Starting with the right most digit of the second number, place the results from the rows in sequential order as seen from right to left under each other while utilizing a 0 for place holders.
825 x 913 2475 8250 742500
The rows and place holders can then be summed to produce a final answer.
825 x 913 2475 8250 +742500 753225
In this example, the final answer produced is 753225.
Therefore: The solution to multiplying 825 by 913 is 753225. (825 x 913 = 753225)
References[edit]
- ^ Imogen Holst, The ABC of Music, Oxford 1963, p.100
- ^ "Scott Rickard: The beautiful math behind the ugliest music". TED: Ideas Worth Spreading. TED Conferences, LLC. Retrieved 19 December 2013. (primary source)
- ^ Phillips, George (2005). "Preface". Mathematics Is Not a Spectator Sport. Springer Science+Business Media. ISBN 0-387-25528-1. Retrieved 2008-08-22.
"...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all doers, not spectators.