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The Basics Octaves: Two notes that “sound” the same because they have frequencies in a ratio of 1 to 2, are called octaves. Since the notes “sound” the same we give them the same letter name. The reason they “sound the same” is that the higher one is the 2nd harmonic of the lower and hence you are conditioned to hearing them “sound together”. To avoid confusion we will use the following frame of reference. We will define A4 to the frequency of 440 hz. This note is commonly referred to as A 440 and has come to be the standard reference pitch over the whole world. Moving downward A3 will be 220 hz, A2 will be 110 hz, and A1 is 55 hz. We stop here because notes below 30 hz start sounding more like a motor boat than an actual sound. The note A4 has been designated the frequency 440 hz.. The piano keyboard shown below will help you to visualize the whole scheme.

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MUSICAL SCALES Just what is a musical scale? For now let us say that it is a sequence of notes that repeat themselves at regular intervals of what we now call an octave. Scales are currently viewed as sequences of notes that are bound by intervals of f and 2f where f is the frequency of an arbitrary pitch. In our modern day system we have 12 tones between f and 2f. Some traditions in the middle east have as many as 26 notes and some Asian cultures use just five notes (pentatonic scales.) Even though we have 12 notes between octaves we usually only use 8 of them in a particular scale. By the 15 th century western European musicians developed a 12 note system much like the one we use today. We shall use the keyboard shown to explain this system. Audible Keyboard

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The Well Tempered Scale To look at all possible scale systems would serve no purpose, so we will confine our studies to just two. Both of these scale systems are important if we want a clear understanding of modern tonality. To solve this problem (and inadvertently create another) early 17 th century musicians developed an ingenious scale system called the well-tempered scale. They probably thought, why not employ all 12 notes and let each be the basis for a key (i.e. a tone center). A melody could be played in any of the 12 “keys” and they would sound “the same” to the human ear. The mathematical basis for this system is to make the ratio of any two consecutive notes frequencies (say C# and D) the same. From this came the concept of the musical interval. The interval between any two consecutive notes in our 12-tone scale became known A a half step. If you go up two notes (say C# to D#) the interval is named the whole step. The scales we currently use are sequences of notes that are separated by half steps and whole steps. (The note C# is the same as Db in this system so we will make no distinction between the two.) In general to sharp a note is to raise it a half-step, which means C# is the next note above C. Clearly Db is the next note below D so we have come up with two names for the same note.) This whole system must start somewhere so the the note A4 has been designated the frequency 440 hz, causing A3 to have a frequency of 220 hz, and A5 to be 880 hz,. etc... Let us designate that common ratio with the letter r. If we multiply r by A4’s frequency twelve times, we will get the frequency of A5, which is double A4 in frequency. Let use see where this leadsl (A4)r 12 = 2(A4) and since the A4 cancels out on both sides, r 12 = 2 which tells us that r = 2 1/12 (about 1.059)

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Well-Tempered Problems Problem 1: If A3 = 220 hz, what is the frequency of (A3) # Solution: Since (A3)# is the next note above A3, we get (A3) # = (A3)r = (220 hz)(2 12 ) = 234 hz (Use your calculator here and use 2 12 rather than 1.059) Problem 2: Using A3 = 220 hz, find (G4) # Solution: (G4) # is 11 notes up from A3 so (G4) # = (A3)r 11 = (220 hz)r 11 = 220(2 11 ) 12 220(2 11/12 ) = 415 hz We can start with A4 = 440 hz and work downward one note (G4)# = (A4)/r = (440 hz)/2 12 = 415 hz Problem 3: Using the Audible Keyboard find the first three notes in the key of D for “Three Blind Mice” Solution: Hopefully you came up with F#-E-D. (*For a complete list of notes and their frequencies use the following website) Well Temperered Notes and Frequencies

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Well Tempered Intervals and Scales Chromatic Scale: If all of twelve tones are played in sequence from say C1 to C2 we call that a C chromatic scale. The notes will be C, C#, D, D#, E. F, F#, G, G#, A. A#, B and then C an octave higher. The intervals are all the same and are called intervals of a half-step. Half Step Interval: The ratio of frequencies in a half step is (2 1/12 ) which is about 1.059. Notes that are two half steps apart are called whole steps. We have to multiply by r, the common ratio of half-steps, twice in order to get a whole step. Whole Step Interval: The ratio of frequencies for a whole step interval is (2 1/12 ) 2 = 2 1/6 or about 1.059 2 The major scale is patterned after the sequence of white notes on a piano, in other words the C Major Scale. Major Scale: The intervals in a major scale are ws. ws. hs, ws, ws, ws, hs, where hs stands for a half-step interval and ws stands for a whole-step interval. Minor Scales are constructed in a similar manner but we will not delve into them in this course. For example one version of a Cminor scale has the notes C, D, Eb, F, G, A, and B. It is the lowering of the third note that gives it its “gloomy flavor”.

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The Beauty of the Well Tempered Scale Using the Well-Tempered scale let us look at the familiar tune called “Three Blind Mice”. In the key of C (C is the tone “center”) the first three notes are E-D-C. But in the key of F (F is now the tone center) the melody would be A-G-F. The reason they “sound” the same is that the ratio of the frequencies in the two melodies are the same. Try to find the notes of the melody in the key of D using the keyboard link below. Audible Keyboard The Well-Tempered musical scale, as we know it, is composed of 12 tones that repeat themselves. For example, if we start with A# (or Bb) we get A#(Bb),B, C, C#(Db), D, D#(Eb), E, F, F#(Gb), G, G#(Ab), A. The symbol # is called a sharp and denotes a note that is one half step higher, and the symbol b is called a flat and denotes a note that is one half-step lower. In order to make it easy to play in any key and have a melody be recognizable, the ratio of the frequencies of any two consecutive notes must be the same. (Musicians say the musical interval must be the same). If we let the letter of a note represent its frequency, then A#/A represents the ratio of the frequencies of the two notes.

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Before the well-tempered scale musicians used what is now called the Just Tempered Scale (sometimes called the Pythagorean scale). For simplicities sake let us look at the eight notes in the C Major scale. The scale is composed of the notes C, D, E, F, G, A and B. Before the Well Tempered scale was developed musicians used what was called the Just or Pythagorean system system of tuning. This system is based on the harmonics of a single note, which we will illustrate with the note C1. The Just-Tempered Scale C1 *f D1 9/8 f E1 5/4 f F1 4/3 f G1 3/2 f A1 5/3 f B1 13/8 f The first note we will find we will call G1 which was assigned the frequency 3/2 f. The frequency 3/2 f was found by finding the third harmonic of f and then dropping it down into the region between f and 2f. This was done by dividing by 2, giving us a note an octave lower. In a similar way E1 (5/4 f) was found by first finding the 5 th harmonic of f and then dropping it down 2 octaves by dividing by 4. In a similar way the other notes were found. The pattern is the same in all keys.. If you start with the frequency of G 660 hz then the frequency of D above G 660 is 660(3/2) = 990 hz

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C1 *f D1 9/8 f E1 5/4 f F1 4/3 f G1 3/2 f A1 5/3 f B1 13/8 f What does this whole mess give us? It gives us frequencies for a 8-note scale from C1 to B1 that are all related harmonically, hence they are “in tune” with each other. This system is beautiful in that every note has harmonics that do not “clash” with the other notes and their harmonics. The Just System has been extended to include all of the twelve notes in our current system. You can find the “complete” Just System on the next page. Well-Tempered Scale that we studied several slides back. As good as this system was it turned out to be too inflexible for composers who wanted to use more than 8 notes between the octaves in one composition. For this task they came up with the Well-Tempered Scale that we studied several slides back. *Note that f is the designated frequency of C1 Just Tempered Scale and Ratios

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More on the Just Tempered Scale The Just Tempered scale can be extended to all 12 notes of one octave. ½ steps upMusical Interval Name of note in key of C Frequency ratio to C 1 minor secondC# 25/24 2 major secondD 9/8 3 minor thirdD# 6/5 4 major thirdE 5/4 5 fourthF 4/3 6 diminished fifthF# 45/32 7 fifthG 3/2 8 minor sixthG# 8/5 9 major sixthA 5/3 10 minor seventhA# 9/5 11 major seventhB 15/8 12 octaveC 2/1

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Wow, the Well-Tempered Scale is not in tune with itself! Practically the whole world now uses the Well Tempered scale as the basis for their music, no matter what style or culture. This is certainly a unifying aspect of the system but there are some problems, notably the intonation of the system. In other words the scale is “out of tune” with itself. Let us look at the octave of notes above A 440. The “major scale” built on that note consists of A, B, C#, D, E, F#, and G#. Let us look at the frequency of F# in both systems to see how they compare. In the Equal (Well) Tempered System the frequency of F# in the octave above A 440 is 11 half-steps up, so it is 440 x (2 11/12 ) (or about 440(1.059 11 ) = (440)(1.8878) = 830.61 hz. In the Just Tempered System recall that G# = 15/8 x 440 = 825 hz. Wow, that is close to 5 hz difference! For a more complete comparision check the following link Comparison - Just vs Equal Tempered Scales This difference in the notes played in the well tempered system and the harmonics of those same notes is quite apparent. Listen to the following link and you will see the problem musicians created when they introduced the well tempered system Just or Well-Tempered - You decide! Modern guitarists have adapted a compromises between the Just and Well Tempered systems. One of these is called the True Tempered system and uses True Tempered frets. True Tempered Here is some more material on the true tempered fretting system True Tempered Fret Systems

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Sample Problems Problem 1 Find the frequency of a note a minor third above A5 880 hz and give its name in the Well Tempered system. (look back at slide 3) Solution: From slide 9 we see that a minor third is three half steps above A5 880 hz. The common ratio between any two consecutive notes is 2 1/12 which is approximately 1.059. For three half steps above 880 hz we get 880(2 1/12 ) 3 = 880(1.059) 3 = 880(1.1876) = 1045 hz approximately. Counting 3 half steps gives us C5 Problem 2: Find the frequency of the 12 th harmonic of A 880 hz and find the frequency and name of a note in the Well Tempered system that is close to that frequency. Solution: The 12 th harmonic of A 880 hz is 880(12) = 10560 hz. A note n notes above A 880 hz is 880(1.059) n. You could (and probably should) do this by trial and error and just keep plugging in whole number values of n until we get close to 10560 - or we can do some nifty mathematics. We are looking for a solution to 880(1.059) n = 10560 Dividing both sides by 880 gives (1.059) n = 12 Taking the logarithm of both sides gives us log(1.059n) = log 12 Using a property of logarithms that says log(a n ) = n log a we get n(log 1.059) = log 12 Using you calculator to evaluate the logarithm gives us n(0.02490) = 1.0792) Dividing both sides by 0.02490) gives us 43.3 or about 43 half steps. This tells us our note is 43 half steps above 880 hz. For every 12 half steps we double the frequency so if we go 3 octaves we get 880(2 3 ) = 7040 (too small) and 4 octaves gives us 880(2 4 ) = 14080 (too big) This tells us that our note is 3 octaves and 7 half steps above A 880 hz. Using a keyboard you can come up with (slide 8) we see that the note are looking for is E and we can find its frequency by the following argument; The third octave above A 880 hz is A (7040), so our answer is 7040(1.059 7 ) = 10516 hz, which is off by about 0.4% from 10560 hz. (You would hear about 16 beats per second and a distinct difference in pitch ) We are going to look at the 20 th harmonic in the homework and you will find that it is a lot further off.

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True Just Desserts Here are some groups that obviously have made some good compromises! Canticleer sings Shenandoah True Metal! The Four Freshman sing Angel Eyes The Hi-Lo's sing Black is the Color Yo-Yo Ma plays Bach

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Weekly Student Project Well Tempered & Just Intonations Possibilities: 1)Find recordings showing good comparisions of the two systems and explain the differences 2)Interview John Schneider (Pierce Music Dept. who is an expert on the Just system) (He will be a guest in class next week.) 3)Interview or invite in other experts 4)Make and compare differences in the two systems mathematically. 5)Find, explain and illustrate music examples of the Just system 6)Anything else you can think of that is relevant.

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