A. Pythagorean Tuning. Index. b). Cycle of 5ths hits all notes. 1. Cycle of 5ths. Physics 1200 Topic VII Tuning Theory.

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1 Note Physics 00 Topic VII Tuning Theory If some of the sounds don t play, open your audio control, make sure SW Synth volume is up! [or some reason it often gets turned down] Very rough draft Updated Oct 6, 00 Index. Pythagorean Tuning. Pythagorean Tuning. ycle of ths. Pythagorean method of tuning. nalysis of intervals (two types of semitones). Just Tuning. Triad hords. Just method of tuning. nalysis of intervals (two types of whole tones). qual Temperament. eometric Progression of requency Perception. efinition of ents (arithmetical progression). omparison of Scales. References Outline of Section. The ycle of ths (a) Perfect Intervals (P and P) (b) ycle of ths (or ths) hits all notes of chromatic scale (c) Pythagorean omma (non-closure of the circle). Pythagorean Tuning Method (a) et all notes from cycle of ths (or ths) (b) requency ratios of the notes (c) Whole Tones: whole steps between,,,, are all /. nalysis of Intervals (a) iatonic Semitone: and are the same 6/ ratio (b) hromatic Semitone other semitones are /0 (e.g. to ) Pythagorean scale hence has whole tone, but different semitones Ratio of hromatic to iatonic semitone=.06=pythagorean omma (c) Syntonic omma Thirds in Pythagorean scale sound TRRIL xample: and are ratio of /6 instead of harmonic / Ratio of Phythagorean to Harmonic third is /0=.0=syntonic comma. ycle of ths b). ycle of ths hits all notes 6 Or cycle of ths (or cycle of ths) (a) The Perfect Intervals: The Octave P is : ratio ycle of ths (red) [P is semitones] Octave can be divided by the th P (e.g. to ) is : ratio P (e.g. to ) is : ratio or to ycle of ths (green) [P is semitones]

2 b). ycle of ths (or ths) hits all notes c). Pythagorean omma P is semitones P is semitones This works because is relatively prime to or. Note, no other cycle of a single interval would hit all notes because their interval would divide [except for M of semitones] ather hristmas gets drunk after every beer nd of cycle is octaves higher, xpect Ratio: /= = ut each th raises : ratio Results in: (/) =.6 Overshot by about ¼ semitone rror is called Pythagorean omma (or diatonic comma) ( ) = = =.06. Pythagorean Tuning b. Pythagorean Scale 0 (a) onstruct all notes by P or P Method shown on the board in class Ratio: 6 6 c. Pythagorean Whole Steps Tones ( semitones, or whole steps) appear to be harmonic ratio : ll of these are to ratio to # to # to to to to to to # to to # The only one that is not is # to Pythagorean Intervals (a) Two types of semitones iatonic Semitone: # to # to 6 = =.0... to to to hromatic Semitone: to # to # to = 0 to to # =

3 b. Pythagorean omma c. Pythagorean thirds sound bad hromatic Semitone: iatonic Semitone: Pythagorean omma: = =.0... hromatic = = iatonic Major rds sound Terrible esired ratio: / ctual is slightly bigger /6 to to to to # rror is nearly a quarter tone This imperfection is due to the non-closure of the cycle of fifths. Syntonic or Ptolemaic omma: [ratio of Pythagorean to Just M] 6 = = =.0 0. Just Tuning. Triad hords 6 Probably first introduced by ioseffo Zarlino, rediscovered by Helmholtz 6 (hence known as Helmholtz Scale or Harmonic Scale) (a) Major and Minor rd ratios rom Harmonic Series system based on chords to make the rds sound harmonious (as pure harmonic ratios) Major rd (e.g. to ) is : ratio. The Triad hords. Just Tuning Method Minor rd (e.g. to ) is :6 ratio or to b. Just Intervals b. Major Triad hord c. Minor Triad hord Major hords: I. IV. V. Minor hords: II. IV. V. Made of M+m Harmonic Ratio (::6) I II III IV V VI VII Made of m+m Harmonic Ratio (0::) I II III IV V VI VII

4 . Just Tuning Method. nalysis of Just Scale 0 (a) The three major chords cover all white keys The I chord is The IV chord is The V chord is a. The Just iatonic Scale (b) Tune these chords in (::6) Tune : =, =/, =6/=/ Tune : =, so =(/6)=/, and =(/6)=/ Tune : =/ so =(/)(/)=/ and =(/)(/)=/ (c) lack Keys: could use other chords to get black keys iminish the chord to minor triad b to get b =6/. iminish the chord to minor triad b to get b =/. iminish the chord to minor triad b to get b =/. ugment the chord to major triad # to get # =/. ugment the chord to major triad # to get # =/. Ratio:.b. nalysis of Just Tuning intervals.c. Inconsistent Intervals (i) Two types of whole tones:,, are /, are smaller 0/ and come out as perfect (just) 0:: minor triads (ii) ifferent Semitones (too many!) and are 6/ b to is / if make # 6/ Then # is different / If make b to be 6/, then # is no longer same as b (enharmonic) is not however a just minor triad Some of the fifths are not perfect, n.b. is off. So, it really doesn t work globally. However, vocal groups often really use Just tuning..b. nalysis of Just Tuning intervals. qual Temperament (a) Two types of whole tones:,, are / but, are 0/ (b) One type of semitone for white keys and are 6/ (if stick to white keys) Hence if # is 6/, # is different / Make b to be 6/, then # is no longer same as b (enharmonic) This all disagrees with above method of tuning black keys. Problematic! (c) Imperfect Intervals (just tuning does not really work) and come out as perfect (just) 0:: minor triads is not however a just minor triad Some of the fifths are not perfect, n.b. is off.. eometric Progression of requency (a) requency Intervals increase with octave to is 0 Hz (but / is :) to is 0 Hz (but / is same ratio :) requency intervals double in each octave, but ratios the same (b) eometric average gives the note halfway between two notes rithmetic average of and is: ( 0 ) 0 or # + 0 / = eometric average of and is: ( 0 ) ( 0) = 0 or learly geometric average gives the halfway note we hear (c) ivide Octave geometrically into steps Semitone Ratio: Octave comes out exactly right: g = g = =.06 Whole Step Ratio ( semitones): g =. Major th ( semitones): g =. is slightly less than Pythagorean /, UT a circle of fifths or fourths will exactly close.

5 omparison: to (fifth) omparison: to ( rd ) 6 The scales really sound differently for some intervals.. Pythagorean. Just intonation. Mean-tone temperament. qual intonation The scales really sound differently for some intervals.. Pythagorean. Just intonation. Mean-tone temperament. qual intonation omparison: # to ( rd ) omparison: to (6 th ) This is really terrible in some scales:. Pythagorean. Just intonation. Mean-tone temperament. qual intonation gain ompare. Pythagorean. Just intonation. Mean-tone temperament. qual intonation omparison: # to # (6 th ). rithmetic Progression of ents 0 This interval is quite bad in some. Pythagorean. Just intonation. Mean-tone temperament. qual intonation (a) efinition of ents ach semitone is exactly 00 cents ll Octaves are exactly 00 ents ( semitones) oes not change with octave. is 00 cents, so is.

6 ..b Mathematical efinition of ents..c Intervals in ents ifference in cents between two frequencies: c c = 00Log f In terms of base 0 log: Inverse Relation: f c c = 6Log f c c f 00 f = f 0 cents Quartertone 00 cents Semitone (m) 00 M 00 m (not quite a just minor third of 6/ which would be 6 cents) 00 M (not quite a just third of / which would be 6 cents) 00 M (not really a perfect th which would be cents) 600 Tritone (minor th or augmented th ) 00 M (not really a perfect th which would be 0 cents) 00 m6 00 M6 000 m 00 M 00 P, octave. omparison of Scales Octave Note b ents semitone 0 ourth ifth Major rd 0..a Pythagorean (a) Pythagorean See table on next slide Two different semitones (0 and ) vs equal temperament s 00 ifths are 0, which are cents longer than equal temperament, which is why the cycle of ths does not close. Hence one fifth is sacrificed to have a massive error (6 cents) and will sound absolutely terrible b # ourths: same problem, they are cents short, hence will fail to close, so one interval will be too big ( cents). b Major rds: two different types, some which are only cents off from the perfect just harmonic ratio of : (6), the rest are far too big at 0 cents and will sound absolutely terrible, which is why reeks didn t use thirds b 0 0 # 0 0. omparison of Scales Octave Note Ratio ents 00 Semito ne Mth 0 0 Mth Mrd 6 6 Whole Tone 0..b Just 6 b (b) Just Temperament See table on next slide. ven for just the white keys, not all of the fifths are 0 cents (n.b. is terrible 60) Many of the Major rds are a sweat 6 cents, but this is far short of the equal temperament 00 cents, so some of the rds will come out very big (e.g. ) and will sound absolutely terrible. Since the whole point of the just tuning was to make the rds sound good, the system fails b # / / Regardless, performers will often bend the notes so that thirds are closer to just tuning because they sound better. This is easy to do with voice, but a problem if you are going to use instruments. b / / 6 6 b / 0 0 #

7 . omparison of Scales omparison of Scales How to tune Piano References 0 Tuning a perfect th is easy, you eliminate the beats. ut this would give you a : ratio (. exactly), whereas the equal temperament interval is / =. So you tune the slightly flat so that you hear a beat frequency of 0. Hertz.