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Scales and Temperament Lecture 5 #4 p. 9/11 The pitch of a musical tone is mostly determined by the frequency of the fundamental. Higher pitch is related to higher frequency. For a string the fundamental frequency is given by f0 = (1/2L) !T/", where L is the string length, T is the tension and " is the mass per unit length. Timbre is a term describing the quality of a musical note. Timbre is mostly determined by relative strengths of the harmonics. The harmonics’ frequencies are integer multiples of the fundamental: fn = n* f0. When two musical tones are heard simultaneously, they may sound pleasant (consonant) or unpleasant (dissonant). People usually perceive two tones at the same frequency (unison) to be very consonant. Usually, two tones with a frequency ratio of 2:1 (an octave) seem almost as consonant. Suppose two tones are played together with one tone’s frequency fixed, while the other tone’s frequency is swept upwards over an an octave. Typically people perceive that dissonance is minimum at unison, then increases to a maximum as the frequency ratio is a few percent, then decreases. Depending on the harmonic content of the tones, minima in the perceived dissonance may be perceived for certain ratios. Usually dissonance reaches a strong minimum as the frequency ratio hits 2:1. http://en.wikipedia.org/wiki/Consonance_and_dissonance has excellent sound files demonstrating this. Note the beats at small frequency differences. Sine waves may not show minima of dissonance, but complex waves do. Typical plots of perceived dissonance. Key observation: Minima of dissonance are strongest where frequencies are in ratios of small integers. This fact was discovered by Pythagoras around 200 BC.
"Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the ear drum in perpetual torment."  Galileo Helmholtz explained this by hypothesizing that the ear dislikes the sensation of moderately fast beats between harmonics of tones. Suppose one note’s fundamental is A = 220 Hz, and the second has frequency 330 Hz. The 3rd harmonic of 220 Hz (3*220) equals the 2nd harmonic of 330 Hz (2*330 = 660 Hz). Harmonics also coincide at 6*220 = 4*330, etc. Two tones with a 3/2 ratio tend to be consonant. If the second tone is at E = 329.6 Hz, there will be beats between the second harmonic of A (3*220) and the second harmonic of E (2*329.6 = 659.3). The beats are slow enough that these tones may only sound slightly out of tune. EEL 4930 Audio Engineering Lecture 5 #4 p. 10/11 Building Scales Because the octave (2:1) is univerally heard as consonant, all scales are made up of notes that subdivide the octave and then repeat; that is, a note with a fundamental twice that of another note is musically somehow “the same as” the note an octave below, and is given the same note name. The Pythagorean Scale After the octave, the next most agreeable ratio (and that with the next smallest integer ratio) is 3:2, which musicians (for reasons we’ll see) call the fifth. Pythagoras was a mystic and a numerologist who created a scale based on the “cycle of fifths.” Starting with a note (call it F), he successively multiplied by 3/2 to create a series of notes. The note that is a fifth above F is C, whose frequency is (3/2) that of F. All the notes are mapped (by multiplying or dividing by 2 as needed) into a single octave between C and C’ = 2*C. After C we get G = (3/2) C; then, D’ = (3/2) G = (9/4) C. This is outside the main octave and so we divide by 2 to get D = D’/2 = (9/8) C. Continuing the process, the first seven notes are F = 2/3; C = 1; G = (3/2); D = (1/2)*(3/2)2= 9/8; A =(3/2)3 = 27/16; E = (1/4)* (3/2)4 = 81/64; B = (1/2)* (81/64)*(3/2) = 243/128. Note: Ratio: Interval Ratio: C 1/1 9/8 D 9/8 9/8 E 81/64 256/ 243 F 4/3 9/8 G 3/2 9/8 A’ 27/16 9/8 B’ 243/128 C’ 2/1 256/ 243 This gives eight notes, which is why C to C’ is called an octave. These notes correspond to the eight white keys in each octave on a piano. G is the fifth white key and so the interval C to G is a fifth. The E to F and B to C interval (ratio) is called the Pythagorean semitone or halfstep. The other interval, 9/8, is called a Pythagorean whole tone. If you continue the (3/2) process above B, you get a note that is (3/2)*(243/128) = 729/512, which is between F and G; that note is called F#. Similar math leads to notes in the CD, DE, FG, GA and AB gaps; these correspond to the black keys. This makes for 12 keys per octave, with unequal ratios for several sizes of semitones. The Pythagorean scale has some nice numerical properties, and it works fine musically if only one note is played at a time. Also, if you restart the process from another note, you can get back many of the same notes. This would let you play in more than one key signature (group of related notes) without needing to add lots of extra keys on a keyboard. However, the largeinteger ratios sound out of tune when two tones are played at the same time (in harmony). [One problem comes from the fact that if you keep going around the cycle of fifths, you get within 1.5% of C, but you never hit C again. The difference between these notes is called the Pythagorean Comma. You can show that it is equal to 312/ 219 = 531441/524288 = ~1.0136.] EEL 4930 Audio Engineering
The Just Scale Lecture 5 #4, p. 11/11 To solve these problems, another scale came into common use in the 16th century or so, called Just Tuning. It replaces some of the notes with lower wholenumber ratios. Note: Ratio: Interval Ratio: C 1/1 9/8 D 9/8 10/9 E 5/4 16/15 F 4/3 9/8 G 3/2 10/9 A’ 5/3 9/8 B’ 15/8 C’ 2/1 16/15 This scale sounds great with most notes played along with C. However, if you recreate this scale on another home note (to produce a new key), few of the notes would be the same as these. (There used to be keyboards with 16 or more keys per octave.) This means that you can’t stray too far from the home key, or else things sound very dissonant. The EvenTempered Scale No choice of notes (temperament) works ideally in all keys. To allow modulation freely from one key to another, a scale was developed that compromised by dividing the octave into 12 equalratio steps. Thus, the ratio between two adjacent notes is the twelfth root of two. Frequency ratio = 21/12 ! 1.05946 If you multiply the frequency of C by this ratio, you get C#, Repeating this gives D, and after 12 repititions we arrive at the 2:1 ratio that is the octave. None of the resulting intervals are pure. In this equal temperament, the perfect fifth from C to G, ideally 3/2, is 27/12 ! 1.4983 instead. The major third from C to E is an especially poor approximation. Ideally it would be 5/4 = 1.2500, but in equal temperament it is 21/3 ! 1.2599, and the error can be pretty noticeable. The Cent As a way to keep track of these ratios, the cent is a useful concept. It is based on a log scale, with the octave divided into 1200 equal parts on a log scale, 100 cents per tempered half step, 200 cents per tempered whole step, etc. The defining relationship is Cents = 1200 log(Ratio)/log(2) Thus a ratio of 3/2, the perfect fifth, is 702 cents. In comparison, the tempered fifth is seven tempered halfsteps, or 700 cents. Differences of less than about 10 cents are hard to detect. The error for the tempered major third (C to E, or 4 halfsteps) is worse: The ideal ratio (in just tuning) is 5/4, or 1200*log(1.25)/log(2) = 386 cents, whereas the tempered ratio is 1200*log 24/12 log(2) = 400 cents. The error of about 14 cents can be quite audible. The Pythagogean major third is 81/64, or 1200*log(81/64)/log(2) = 408 cents (error = 22 cents), which is even worse than the tempered interval. Discussion of scales, with audio files: http://www.music.sc.edu/fs/bain/atmi02/hs/indexaudio.html Applet for demonstrating lots of tuning issues: http://mrhall.org/music/javatuner/javatuner.htm ...
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 Spring '08
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 Frequency, Ratio, Octave, Semitone, Musical tuning, Just intonation

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