Part 16.

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In this way great pianos have a superior consonant grouping than studio pianos. ... Piano and harpsichord tuning is not stamped by sans beat connections, but rather ...
Slide 1

Section 16 Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano

Slide 2

The Just Scale All interims are number proportions in recurrence Major Scale Minor Scale

Slide 3

A Note of Caution

Slide 4

Major Scale Minor Scale Notes on the Just Scale The D compares to the upper D in the pair found in Chapter 15. Additionally, the tones here (with the exception of D and B) were the same found in the sans beat Chromatic scale in Chapter 15. Here we utilize the lower D from part 15 and the upper Ab . In music hypothesis two other minor seventh are perceived, the grave seventh (16/9) and the symphonious minor seventh (7/4).

Slide 5

Notes on Just Scales These equitable scales work (great symphonious tunings) the length of the piece has close to two sharps or pads. The accompanying notes apply to organ tuning Organs produce maintained tones and consonant connections are effortlessly listened

Slide 6

The Equal-Tempered Scale Each octave included twelve equivalent recurrence interims The octave is the main genuinely symphonious relationship (recurrence is multiplied) Each interim is = 1.05946 The fifth interim is near the simply fifth = 1.49831 though the simply fifth is 1.5 The Difference is Only fifths and octaves are utilized for tuning

Slide 7

The Perfect Fifth Three times the recurrence of the tonic down an octave 3*f o/2 The third consonant of the tonic equivalents the second symphonious of the fifth 3*f o = 2*f fifth The fifth must be tuned down around 2 pennies

Slide 8

Tuning Fifths (Organ) C 4 = 261.63 Hz G 4 has a recurrence of 1.49831*C 4 or 392.00 Hz Use the second run of fifths to get the beat recurrence 3(261.63) – 2(392.00) = 0.89 Hz

Slide 9

Notes of Fifth Tuning The following table above demonstrates the complete tuning in fifths from C 4 through C 5 . A trap is to utilize a metronome set to around the right beat recurrence to get acclimated to listening for the beats. Whatever is left of the console is tuned by sans beat octaves from the notes we have tuned in this way.

Slide 10

The Changing Beat Pattern

Slide 11

Just and Equal-Tempered

Slide 12

Notes on Organ Tuning Certain interims sound smoother (or rougher) than others. In playing music we at times harp on any two notes in length enough to see exact tuning. Harmonies made of three or more notes (equivalent tempered) make a more "tuned" impact than the two note interims would suggest. Maybe one purpose behind the unpredictable harmonies of music since Beethoven.

Slide 13

The Circle of Fifths

Slide 14

Key Signature Derived from the Circle of Fifths

Slide 15

Pythagorean Comma Start from C and tune flawless 5ths the distance around to B#. C and B# are not in order. An immaculate fifth is 702 pennies. 702+702+702+702+702+702+702+702+702+702+702+702= 8424 pennies An octave is 1200 pennies. 1200+1200+1200+1200+1200+1200+1200= 8400 pennies 8424 - 8400 = 24 pennies = Pythagorean Comma

Slide 16

Pythagorean Comma More Precisely

Slide 17

A Well-Tempered Tuning Werckmeister III Created by Andreas Werckmeister in 1691 – valuable for elaborate organ, harpsichord, and so forth. The framework contains eight unadulterated fifths, the remaining fifths being smoothed by ¼ the Pythagorean Comma.

Slide 18

Werckmeister III Start with a reference note (C 4 ) Tune a beat free significant third above (E 4 ). Develop a progression of contracted fifths so we wind up back at E 6 , which will tune with E 4 . The interim is found by isolating the Pythagorean Comma into four equivalent amounts of (23.46/4 = 5.865). So rather than the ideal fifths being 702 pennies, they are 696.1 pennies.

Slide 19

The Shrunken Fifth Using the pennies mini-computer, the fifth interim will be 1.49492696. The only interim of the ideal fifth is 1.5, so every fifth is around 5.9 pennies short. The initial six stages in the tuning are… The ideal fifth above C 4 would have a recurrence of 392.45 Hz, so we tune for a beat recurrence of (392.45 – 391.12) 1.33 Hz. Different sections in the last segment above are computed in a comparative manner.

Slide 20

Werckmeister III (the ideal fifths) Recall that the recently tuned fifths delivered an E 6 tuned in to E 4 . The following step is to retune E 6 to be an impeccable fifth over the A 5 officially decided. That would put it at a recurrence of 1311.05 Hz. This additionally retunes the E 4 to 327.76 Hz. The new E 6 can now be utilized to tune B 6 a flawless fifth above it at 1966.58 Hz. Beginning from C 4 again tune flawless fifths descending to Gb. We raise the pitch an octave intermittently to stay in the focal point of the console.

Slide 21

Downward Perfect Fifths Column bounced demonstrate octave changes

Slide 22

Gathering Results into One Octave

Slide 23

Werckmeister Circle of Fifths Numbers in the interims allude to contrasts from the ideal interim. The ¼ alludes to ¼ of the Pythagorean Comma.

Slide 24

Notes Bach\'s tuning was comparative (he partitioned the Pythagorean Comma into five sections). Both of these all around tempered tunings admits to each of the 24 keys (major and minor). This is the premise of Bach\'s Das Wohltemperirte Clavier.

Slide 25

Comparison Table The following slide is like Table 16.1 I demonstrate the fair interims and the Werckmeister frequencies that are produced with an assortment of tonics. Penny contrasts in these two tunings are additionally pulled out that the minor interims are all level in the Werckmeister III

Slide 26

Just - Werckmeister III

Slide 27

Musical Implications The table plainly demonstrates that transposing yields diverse flavor or temperament Modulating to another key likewise creates distinctive states of mind contingent upon the key that was simply cleared out. Measure up to demeanor loses these progressions.

Slide 28

Density = d r L Physics of Vibrating Strings Flexible Strings Stretched between unbending backings, the recurrence of symphonious n is… Clearly, f n = nf 1

Slide 29

Some Dependencies as L  , f  (longer strings  lower tones) as r  , f  (bigger strings  lower tones) as T  , f  (more strain  higher tones)

Slide 30

Physics of Vibrating Strings Hinged Bars Because strings are under pressure, they are solid and tackle a percentage of the properties on dainty bars. The frequencies of the sounds are… All the images have their same significance and Y = Young\'s Modulus, is a measure of the versatility of the string. Unmistakably, for a bar f n = n 2 f 1

Slide 31

Some Dependencies twofold the length and the recurrence is up two octaves the inverse conduct of the string, as r  , f 

Slide 32

Real Strings We have to consolidate the string and bar conditions Felix Savart found…

Slide 33

The firmness (bar) commitment is fairly little contrasted with the strain commitment in genuine strings. We can inexact the above work as… And is little (around 0.00016)

Slide 34

Departures from Harmonic Series The ideal Harmonic Series is nf 1 We can ensure takeoffs from the ideal arrangement are little in the event that we make J little Using long strings (build L) Make the strings rigid (expansion T, the pressure) Make the strings thin (diminishing r)

Slide 35

Sample Series

Slide 36

Physics of Vibrating Strings The Termination Strings act more like cinched associations with the end focuses as opposed to pivoted associations. The clip has the impact of shortening the string length to L c . L c is identified with J. The impact of the end is little .

Slide 37

Physics of Vibrating Strings The Bridge and Sounding Board We utilize a model where the string is solidly tied down toward one side and can move uninhibitedly on a vertical pole at the flip side between springs F S is the string characteristic recurrence F M is the normal recurrence of the piece and spring to which the string is associated.

Slide 38

Results The string + mass goes about as a basic string would that is prolonged by a length C. The somewhat more length of the string gives a marginally bring down recurrence contrasted with what we would have gotten if the string were solidly tied down.

Slide 39

Change the Frequency

Slide 40

Results For the instance of F S > F M , part 10 proposes that the mass on the spring slacks the string by up to one-half cycle. As the string pulls up the mass is moving down and the other way around. The string goes about as if it were abbreviated by a length C. The abbreviated length raises the pitch over a solidly tied down string.

Slide 41

Example of a Guitar String Consider the D string of a guitar and its initial a few. The proportions demonstrate a propensity to become bigger in view of the impacts of string firmness. Anomalies in the grouping happen when one of the guitar body thunderous frequencies happens to be close to one of the partials.

Slide 42

Bigger is Better Larger sounding sheets have covering resonances, which have a tendency to weaken the anomalies. Along these lines stupendous pianos have a superior consonant succession than studio pianos.

Slide 43

Pitch of a Single String Sound Because the piano string has a marginally inharmonic arrangement, the apparent pitch of a key may fluctuate from an instrument with entirely symphonious groupings.

Slide 44

The Piano Tuner\'s Octave When a tuner tunes an octave to "sound right" we find there are still beats, yet there is a lessening in the "tonal waste." For the C 4 – C 5 octave this is accomplished when the central of C 5 is 3 pennies higher than 2*C 4 .

Slide 45

A Real Piano Tuning Below I list the initial couple of partials of C 4 and the subsequent C 5 and its partials. Note: The qualities utilized here for the C 4 partials are the same as were utilized already to contrast piano with organ tuning and presented the inharmonic element J. Additionally see that the C 5 is not 3 pennies sharp of the second consonant of C 4 .

Slide 46

"Impeccable" Fifths on the Piano A state of minimum unpleasantness for the fifth is gotten when the tuning is around 1 penny higher than 3/2 * major. Beneath I have ascertained the fifth interim taking into account the center C of 261.63 Hz. The main fifth is 1.5*C 4 and the second one is the equivalent temp

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