Everyone has heard sounds. Sound is ineluctable. It pervades every country of human being. The first sound most of us hear each twenty-four hours is that of the dismay clock impolitely agitating us from sleep. From that point on, our twenty-four hours is filled with countless sounds. The java shaper, the auto door, the ambulance Siren, the pen on paper: each makes a distinguishable sound which we must construe. We decide immediately if a sound is “ normal ” or if it signals danger. There are sounds which we have trained ourselves to disregard and others to which we pay close attending. One type of sound to which we pay attending is music. To state “ everyone likes music ” is by and large true. No uncertainty there are those who care small for it, but on the whole, it is normally so. To do the statement, nevertheless, that “ everyone likes the same sort of music ” is an obvious falsity. One demand merely look in their nearest Wal-Mart to see the many types of music available. From Rap to Rock, from New Age to Classical, each of these genres claims to be music. What is the common denominator which makes Mozart ‘s symphonic musics and Tupac ‘s poetry musical? The reply is that both of these instrumentalists made their music from graduated tables. Many people have had the chance to “ slam around ” on a keyboard. It is dubious that these people questioned why the keyboard is arranged as it is. Why are n’t at that place black keys between all of the white 1s? Why are the black keys in groups of 2 and 3s? The reply once more is, because of graduated tables. So, what precisely are graduated tables, and how do they exercise such influence on music? I will reply these inquiries in the class of this work.
Before we can discourse existent graduated tables, we must take a expression at some of their edifice blocks. Since graduated tables are nil more than a aggregation of single sounds, allow us first look more closely at “ sound. ” A “ sound ” is “ the esthesis experienced when the encephalon interprets quivers within the construction of the ear caused by rapid quivers of air force per unit area ” ( Webster ‘s 2:948 ) . To explicate it in other footings: a sound can be thought of as a moving ridge traveling through the air. A sound moving ridge is measured in several different ways ; nevertheless, here we will utilize merely one measuring: the moving ridge ‘s frequence. Another term for frequence is the figure of the moving ridge ‘s “ rhythms per second ” ( Christiansen III-6 ) . A rhythm is the completion of one moving ridge from crest to trough and back to crest. When covering with music, frequence is renamed “ Hz ” ( Hz ) . From Dr. Christiansen, we learn that worlds are able to hear sounds from 20Hz to every bit high as 20,000Hz ( IV-1 ) . We get music from the combination of assorted single sounds within this frequence scope.
Another of import portion of graduated tables is the interval. An interval is the distance between two notes. For illustration from 256Hz to 342Hz, we have an interval of 86Hz. There are many types of intervals which we will discourse in this essay. The most of import one is the Octave interval. It equals a doubling of the original pitch or a 2:1 ratio. ( E.g. 440Hz to 880Hz ) Other intervals which are used in musical graduated tables include the Semitone, the Whole Tone, the Major 3rd, the Perfect 4th, the Perfect 5th, the Major 6th, and the Major 7th. The ratios of these intervals vary between graduated tables. The lone invariables we will see is the Octave ( 2:1 ) and the Perfect 5th with a ratio of 3:2. In order to mensurate the difference between close frequences such as 220Hz and 222Hz, we will utilize a step known as “ cents. ” For each Octave, there are 1,200 cents. In order to happen the difference in cents between our two frequences of 220Hz and 222Hz, we use the undermentioned expression: the figure of cents = 3,986.3 x log ( f2/f1 ) . Using this expression, we find that there is a difference of 15.7 cents between the aforesaid frequences ( Johnston 350-352 ) . Harmonizing to Cartwright et al. , the human ear can observe differences in frequences every bit low as 3 or 4 cents ( 53 ) . Knowing this, we can infer that the difference of 15.7 cents between 220Hz and 222Hz is clearly hearable.
A concluding similarity of the graduated tables we will analyze is the division of whole and semitone intervals. It will be seen that all the graduated tables we will analyze hold five whole tones and two half steps. Besides, to show and compare the strengths and failings of each graduated table, we will reassign them to the modern keyboard. Our keyboard will stretch from a depression of 27.5 Hz up to 4,186Hz. The frequences run left to compensate in order of increasing magnitude. As is seen, this scope of frequences is clearly within the human hearing capacity. It covers a scope of seven octaves plus two whole tone intervals. With this background information, allow us get down to look at some existent musical graduated tables.
The first graduated table we shall discourse is the oldest recorded graduated table in history. It was discovered by the well known Greek philosopher and mathematician Pythagoras. Of class, there was music prior to Pythagoras, but instrumentalists merely learned to play by ear. They knew what sounded good and what did non. Pythagoras, harmonizing to history, was inspired to experiment with musical intervals one twenty-four hours after go throughing near a blacksmith ‘s store. Upon hearing the changing Pings of the Smith ‘s cock, he began to inquire why there was a difference in the sounds. Pythagoras went on to experiment with assorted lengths of strings, different size bells, and spectacless filled with H2O. The basic intervals and related ratios which he discovered are the Octave ( 2:1 ) , the Perfect 5th ( 3:2 ) and the Perfect 4th ( 4:3 ) . The graduated table formed from his experiments is known as the Pythagorean Scale ( Johnston 4-5 ) .
The Pythagoreans, who were adherents of Pythagoras and his instructions, felt the 3:2 ratio was really of import. In fact, it is possible to build every bit separated intervals inside the Octave interval utilizing merely the 3:2 ratio. To understand this, allow us look at Johnston ‘s account of the topic on pages seven to nine of his book. In his words, “ Take an bing ratio and multiply or divide by 3:2. If the figure is greater than 2 so halve it ; if it is less so 1 so duplicate it. ” To show this, allow ‘s get down with a note of 100Hz. We will name this note “ C ” . ( Please note that 100Hz does non be the existent “ C ” tone. We will utilize 100 Hz merely for easiness of computation. ) In order to put parametric quantities for this graduated table, we must foremost happen the frequence one Octave higher so our C note. Multiply 100Hz by 2:1. This gives us 200Hz which we will name C ‘ ( C prime ) . So so, all frequences of the graduated table which we will build must be within 100Hz – 200Hz. Perform the undermentioned operation: 100Hz / 3:2 = 66.66. This frequence is below 100Hz, so we must multiply by 2. 66.66Hz ten 2 = 133.3Hz. This is the frequence we will give to the note of ‘F ‘ . We could besides multiply by 3:2 squared: 100Hz ten ( 3:2 ) ( 3:2 ) = 225. Since this is greater so 200Hz, we must split by 2. 225 /2 = 112.5Hz. This will be ‘D ‘ . If we were to go on multiplying or spliting by 3:2 so multiplying or spliting by 2 as necessary, we would happen a set of frequences with distinguishable ratios spliting them. Take a expression at Example 1 below. This is the graduated table which Pythagoras and his followings assembled. Notice that the form of multiplying or spliting by 3:2 and so by 2 outputs notes divided by precisely two intervals: 9:8 and 256:243. These are known as whole tones ( tungsten ) and half steps ( s ) severally. Notice besides that if we multiply the whole and half steps together we get the intervals from C to C ‘ shown below the frequences. For illustration: ( 9:8 ) ( 9:8 ) ( 256:243 ) = 4:3
Example 1:
Multiply by: 9:8 9:8 256:243 9:8 9:8 9:8 256:243
C -w- D -w- E -s- F -w- G -w- A -w- B -s- C ‘
Hertz: 100 112.5 126.6 133.3 150 168.8 189.8 200
Pythagorean Time intervals: 4:3 3:2 2:1
Other Time intervals: 9:8 81:64 27:16 243:128
The Pythagoreans developed single graduated table manners utilizing this form of whole tones and half steps. A modern equivalent to the manner is the “ Key. ” For illustration, we say that a vocal is written in “ A Minor ” or “ F Major ” etcetera. Let ‘s take a expression at two of the Pythagorean manners: the Dorian and Lydian Modes ( Lawrence 283 ) . The tonic ( opening ) note frequences come from Example 1.
Example 2:
Lydian Mode:
Multiply By: 9:8 9:8 9:8 256:243 9:8 9:8 256:243
F -w- G -w- A -w- B -s- C -w- D -w- E -s- F ‘
Hertz: 133.3 150 168.7 189.8 200 224.9 253.1 266.6
Pythagorean Time intervals: 729:512 3:2 2:1
Other Time intervals: 9:8 81:64 27:16 243:128
Example 3:
Dorian Mode:
Multiply By: 9:8 256:243 9:8 9:8 9:8 256:243 9:8
D -w- E -s- F -w- G -w- A -w- B -s- C -w- D
Hertz: 112.5 126.6 133.3 150 168.8 189.8 200 225
Pythagorean Time intervals: 4:3 3:2 2:1
Other Time intervals: 9:8 37:27 27:16 243:128
Notice that the same ratios for whole tones and half steps are used in these manners. Notice besides that the Perfect 5th interval ratio of 3:2 and the Octave ratio of 2:1 are preserved. However, in the Lydian Mode, the Perfect 4th interval alterations from a simple 4:3 in Examples 1 & A ; 3 to a really complex 729:512 ratio. Of all the manners which can be derived from the Pythagorean Scale, the Pythagoreans felt that the Dorian Mode was superior, based on the fact that the Dorian divides the whole and half steps in a absolutely symmetrical manner ( w-s-w-w-w-s-w ) ( Johnston 9 ) . Note that if we were to reassign these manners to a modern keyboard, we would hold merely the white keys. This fact demonstrates one of the jobs with the Pythagorean Scale.
The job of holding merely white keys arises when a musician wants to permute a vocal to a higher or lower quinine water on a individual keyboard. For illustration, a vocalist performs best in the quinine water of D ( Dorian Mode ) ; nevertheless, the vocal he wishes to sing is written in tonic F ( Lydian Mode ) . The musician wants to fit the music played on the keyboard to what he is singing. Therefore, he must permute or alter the vocal from the higher frequence Lydian into the lower frequence Dorian Mode. To do account of this easier, the Moveable Doh system will be used for this illustration.
In short, the Moveable Doh system begins on a tonic key ( for illustration D in the Dorian Mode ) . D is called Doh ; E is Ray ; and so on through Me, Fah, Soh, Lah, Te, and back to Doh. Therefore, alternatively of stating a vocal is in the key of D but the first note is F, we can state the vocal is in the key of D with the first note of the tune being Me. This system helps to divide the quinine water or get downing cardinal from the notes of the vocal.
Returning to permuting, imagine that the vocal being transposed from the Lydian to Dorian manner has a tune which begins Doh-Fa, the Perfect 4th interval. By deducting the frequence of Doh ( F ) from that of Fa ( B ) , we find the two notes are detached 56.5Hz or 612 cents. Now, we play the vocal once more in Dorian Mode. The vocal sounds wholly different after merely the first two notes. To happen the ground for this, we once more subtract the frequences of Doh and Fa. This clip Doh is D, and Fa is G. After deducting, we find the notes are 37.5Hz ( 498 cents ) apart. So the instrumentalist has several picks. He can a. ) play the music in the Dorian manner even though the vocal will sound really different, B. ) keep the vocal in the Lydian Mode but travel it an full Octave lower which will do it hard to sing or c. ) retune the full keyboard so the Lydian Mode is played get downing at the same frequence as the Dorian Mode. Obviously none of these are really good picks. This, nevertheless, is what instrumentalists had to cover with for 100s of old ages after the creative activity of the Pythagorean Scale.
A 2nd job with the Pythagorean Scale is found in what is called the Pythagorean Circle of Fifths ( Lawrence 282 ) . The Circle of Fifths is best represented by a line which moves outward in of all time spread outing spirals. Imagine it, possibly, as a coiled rattler. On this line, we see the notes of the musical graduated table ( A, B, C, Daˆ¦ ) repeated multiple times. In theory, if one were to pick a note ( Example: Degree centigrade at 100Hz ) and multiply it by the Perfect 5th ratio of ( 3:2 ) 12, it should make the C note precisely seven octaves higher so the original. ( Seven Octaves equal the figure of Octaves intervals on a normal keyboard. ) In simpler footings: f1 ( 3:2 ) 12 = f1 ( 2:1 ) 7. The job is that if we do the math the two frequences do non fit. Notice: f1 ( 3:2 ) 12 = 1,2974.6Hz and f1 ( 2:1 ) 7 = 12,800Hz. Following the cent expression, we find that this is a disagreement of 23.5 cents. This mismatch is known as the Pythagorean Comma ( Lloyd 53 ) . One may state that it is possible to rectify the job by adding the black keys to the keyboard as modern 1s have. However, there is a job. To what frequence should this middle cardinal be tuned? The most practical solution would apparently be to do the new key a half step higher so C ‘ . Notice though what happens if we do that: 12,800Hz x 256:243 = 13,484.8Hz. This is much greater so the border we needed. There is another job with doing the black key one half step higher so C. Let ‘s utilize the graduated table from Example 1. If we insert a black key between C and D and do it a half step above C, theoretically, it should besides be one half step below D. This is non the instance. Notice: 100Hz x 256:243 = 105.3Hz. To take down D by the same ratio, we divide as follows: 112.5Hz / 256:243 = 106.8Hz. 106.8Hz and 105.3Hz are separated by precisely 23.5 cents- the Pythagorean Comma! Now what is to be done? Two black keys could be added. The first would be C # ( C crisp ) one half step higher so C. The other key would be Db ( D flat ) one half step below D. However, besides the fact that this would do a keyboard really cumbrous and hard to play, there are other disagreements to see. As Lawrence says, “ The mismatch aˆ¦ is but the tip of the iceberg. ” Let ‘s expression at one more point of disagreement along the Circle of Fifths. Theory states that ( 3:2 ) 8 ( 81:64 ) should be ( 2:1 ) 5 ( Lloyd 53 ) . ( 81:64 is an interval known today the Major 3rd. ) Notice:
100Hz ( 3:2 ) 8 ( 81:64 ) = 3,243.7Hz and 100Hz ( 2:1 ) 5 = 3,200Hz. This is a difference of 43.7Hz which once more equals the Pythagorean Comma of 23.5 cents. The point here is that there is no manner to give an single key to each of the points without a disagreement. So so, what was the solution?
A spot of betterment was made in graduated tables in the fourteenth century with the credence of the Just Scale. This graduated table is based in portion on the Ptolemaic System named for its Godhead Ptolemy ( Johnston 12-13, 18-19 ) . Ptolemy argued for the extension of simple intervals which the Pythagoreans had discovered. In add-on to the 2:1, 3:2, and 4:3 ratios, he suggested a simplified Major 3rd interval with a ratio of 5:4. Ptolemy ‘s graduated table is found in the same mode as the Pythagorean Scale. The lone difference is that both the 3:2 and 5:4 ratios are used in its building. Let ‘s get down once more at 100Hz. We follow the same form of multiplying or spliting 100Hz by 3:2 or 5:4 ; so we divide or multiply by 2 as necessary to convey the new frequence inside the Octave.
Example 4:
Merely Scale:
Multiply By: 9:8 10:9 16:15 9:8 10:9 9:8 16:15
C -w- D -w- E -s – F -w- G -w- A -w- B -s- C ‘
Hertz: 100 112.5 125 133.3 150 166.7 187.5 200
Ptolemaic Time intervals: 5:4 4:3 3:2 2:1
Other Time intervals: 9:8 5:3 15:8
Notice the stopping point relationship this bears to the Pythagorean Scale. The differences in frequences are really little. However, in footings of cents, they are every bit much as 21.7 cents different. Clearly, if played at the same time, the disagreements would be heard. Notice besides that there are two ratios for the whole tone intervals. The 9:8 interval remains from Pythagoras, but a new whole tone of 10:9 has been used. Besides, the complex Pythagorean half step of 256:243 has been replaced with a much simpler 16:15 ratio. The little alterations in ratios and frequences made music more harmonic or harmonious ( Johnston 20 ) , but the job of heterotaxy remained unchanged. It was non until the seventeenth century that a solution to the job of heterotaxy was created.
Harmonizing to Johnston, composers realized a demand for a graduated table which had an equal sum of spacing between each key. At this clip, keyboards had seven keys to the Octave. The spacing was “ w-w-s-w-w-w-s. ” One possibility would be to split the Octave every bit across the seven keys. In other words, happen a figure that equals 2 when multiplied by itself seven times, or merely happen the 7th root of 2, which equals 1.104. We now multiply this by 100 to happen the per centum of addition from one key to the following. ( 1.104 x 100 = 110.4 % ) In this instance if C equals 100Hz, D would be 100Hz ten 110.4 % ( 110.4Hz ) , E would be 110.4Hz ten 110.4 % ( 121.9Hz ) , and so on. However, holding seven every bit separated whole tone intervals leaves really small room for imaginativeness and invention. There needed to be a better, more flexible method. Rather than hold lone seven keys, it was decided to add a new key between each whole tone. The new key would efficaciously infix a half step interval on top of the whole tone doing it possible to play
C- C # ( semitone interval ) or C- D ( whole tone interval ) . These new keys are the black keys on the keyboards of today. With their add-on, the Octave moved from seven keys to twelve. Now we have the inquiry of how to split the Octave equally between them. Again, we use the square root, merely now we find the 12th root of 2 ( Christiansen X-8 ) . The 12th root of 2 is equal to 1.0594. Multiply by 100 to happen the per centum of addition. ( 1.0594 x 100 = 105.9 % ) This graduated table is known today as the Equally Tempered Scale. To put up this graduated table, allow ‘s get down with 100Hz and multiply 12 times by 105.95 % .
Example 5:
Equally Tempered Scale:
C C # D D # E F F # G G # A A # B C
Hertz:100 105.9 112.3 118.9 126.0 133.5 141.4 149.8 158.7 168.2 178.2 188.8 200
On modern keyboards, ( Pictured below in Example 6 ) the white keys are tuned to the tonic C. So, if a vocal is in tonic C, it is possible to play the full vocal without touching the black keys but merely if the vocal has no sharps or flats. Another of import note is that the infinites in the white keys which contain no black key are already separated by a half step interval.
The Equally Tempered Scale is an first-class graduated table for permuting vocals and/or altering to a different quinine water in the center of a vocal. Let ‘s take a expression at permuting. Suppose we have a vocal in the quinine water of C. The first four notes of the tune are E – G – C – D. Since the white keys are already tuned to tonic C, these four notes are played merely on the white keys. We wish to permute down to tonic A. We will utilize the keys toward the centre of Example 6.
Example 6:
Since tonic C has two whole tones between C and E, we must hold the same interval in quinine water A. Notice on the keyboard that there is a whole tone between A and B, but merely a half step between B and C. Therefore, we must travel a half step further and get down our vocal on the black key between C and D. This key ( C # ) is the first note of our vocal. The 2nd note in the original is G. E – G is an increasing interval of one half step and one whole tone. If we start on C # and increase the same distance, we find the 2nd note will be E. Now the original goes down 3 and A? tones from G – C. In our converse vocal, this interval takes us from E down to A. Finally, the original goes from C – Calciferol: an interval of one whole tone. Traveling up one whole tone from A, we get B. To repeat what we have: the original notes played in tonic C were E – G – C – D. Our converse notes in quinine water Angstrom are C # – Tocopherol – A – Bacillus. If these sets of notes were played side by side, they would sound the same except for quinine water A sounding lower in frequence so tonic C. Here so is the benefit of utilizing the Equally Tempered Scale. The possibilities of permuting vocals or altering tonic in the center of a musical piece are about limitless.
We have seen in this essay the importance of graduated tables. It is surely possible make music without cognition of graduated tables. However, without graduated tables, all music would hold to be taught and played merely by hearing and memorisation. Scales make it possible for us to play the same vocal on assorted instruments. Knowledge of graduated tables is similar to the ability to read. Merely as we learn the ideas of work forces from the books which they write, so can knowledge of graduated tables give us the ability to play music from 100s of old ages ago. The patterned advance of graduated tables discussed in this essay shows the superior qualities of the Equally Tempered Scale ; nevertheless, it is non intended that the Pythagorean Scale or the Just Scale should look unimportant. Quite the opposite is true. It is work forces such as Pythagoras and Ptolemy who laid the basis for all the music we have today. Though their graduated tables may non be used by many people today, it is their graduated tables which opened the door to the musical diverseness we now have. Through their work, we can analyze assorted genres to compare what makes each one distinct from the other. For, in world, whether the music is Rap, Rock, Christian, or Classical, they all use the same sounds. It is merely the mixture of those sounds which makes them different. Scales give us the tool to put each genre on paper for survey, for narration, and for enjoyment.