The 12 √ 2 well tempered.
Playing the guitar can 'rise to the question (or not?) Because of' the frets are arranged in a way that not even on the neck. The frets are the metal arms that are perpendicular to the strings on the neck and demarcate "notes" of the guitar (as in instruments such as violins, violas and basses do not exist).
And then I wondered why ' their distance decreases in the guitar, and especially as decreases, because' I knew something interesting. And indeed it and '(or not?). And it goes far away, the trees of Western music as we know it. Despite being a total ignorant of music history, I venture to an account that will be 'full of historical errors (I hope not mathematicians).
Before Bach would make it officially recognized, the music (or rather the way of playing and writing) was different. The reason was its fundamental building blocks, notes. The notes were not those that we know. Every note and 'in truth' a frequency of an acoustic wave, then the frequencies used, say in the Middle Ages, were not those who use Max Pezzali nowadays (which begs the question whether there has been a real evolution).
The reason for this difference lies in the scales used. Before Bach (which I use as a reference I do not know how well) crazy Pythagorean scale or a relative call Natural , while Bach and Max Pezzali decided that this did not go too well together and coined the so-called Temperate . Then Bach wrote The Well-Tempered Clavier and Max Pezzali Myth . Both were quite successful, demonstrating the fact that the ear of the people is not 'out of tune and too sensitive to small deviations in frequency. Problems
Pythagoreans
Why 'discordant notes were there. For 883 and 'easy to imagine, but also for Bach non ci si puo' esimere dalla constatazione. Perche' stonature? La scala Pitagorica e' naturale in quanto si basa su un principio immediato: la divisione in due di un intervallo o di una corda di chitarra. Dividendo a meta' una corda (sempre alla stessa tensione) si ottiene il doppio della sua frequenza. Dal principio della divisione a meta' si possono generare tutte le note. E all'orecchio umano questo piace, l'armonia tra le frequenze generate pare naturale . Ma esistono problemi, legate alle terribili quinte del diavolo e al loro famigerato Circolo delle quinte , che mai puo' essere chiuso.
La quinta e' la nota che si ottiene moltiplicando una frequenza f o per 1.5, cioe' 3/2 (cioe' aggiungendole meta' di se stessa, cosa che Pitagora ama fare sempre). La quinta di Do e' Sol per esempio, ovvero f sol = 1.5 f do . Il problema con Pitagora e la sua scala e' che se parto da Do e salgo in frequenza sempre di quinta in quinta, prima o poi vorrei tornare a un Do . Di qualche ottava sopra, ma sempre un Do vorrei. E invece no.
Per ottenere la stessa nota ma all'ottava superiore devo raddoppiare la sua frequenza. I chitarristi sanno che bisogna dividere a meta' una corda per salire an octave, reduce it to one quarter to go up yet another octave, etc.. So if I start with a frequency f or, in the second octave avro ' 2f or, in the third octave frequency will' 4f or and so 'on. At n-th octave the frequency will be 'in general 2 n f or . So Pythagoras
with its scale that would, rising fifth in the fifth , a frequency f or we can get to some of the eighth f or . Go up fifth in the fifth by f or , say a number of times m, mean change from frequency f or frequency 1.5 m or f (following the same reasoning above for the "passing of eighth in eighth, but with 1.5 instead of 2). And here's the problem. Pythagoras then that would be true for some n eighth and fifth some m that 1.5 m or f n = 2 f or , or that 1.5 m n = 2 . But it is not 'possible' cause no two numbers n, m that makes this expression true. So
. Well 'the problem with the Pythagorean scale was therefore moving away from the initial note is not returned to precisely the same after a few octaves (n = 7 should be ). The thing that made it difficult to write and play music, especially when composers began to not just the usual few eighth in their compositions. It 's like saying that the Do the right of the keyboard of a piano was out of tune with what the left. Bach as unacceptable for Max Pezzali (even using him more than three notes, that was a matter of principle).
Mix via the linearity '
So how? On the one hand, the Pythagorean Scale was naturally harmonious because of divisions in the middle ', the other could not close the circle of fifths evil. The solution was to let small imperfections and approximate frequency to get a piano and a guitar can be tuned and played. Bach heads 'that was more' important to have the same Do across the keyboard of his piano, rather than preserve the harmony of course "linear". On the other hand, had to think, who in future would have appreciated those little imperfections Max Pezzali Pythagorean case would not have done too. And Myth pote' librarsi in cielo, e il Karaoke vedere la luce del giorno quando l'umanita' fu finalmente pronta.
Il problema era riuscire a dividere l'intervallo di un'ottava (cioe' da f o a 2f o ) in 12 parti (perche' compresi i # le note sono 12) in maniera speciale. Cioe' in maniera che le frequenze di due note successive stessero sempre nello stesso rapporto . Vediamo come e poi il perche' si capira'.
Se parto dalla frequenza f o , voglio che dopo 12 di questi intervalli la nota sia la stessa ma un'ottava sopra, cioe' 2f o .
Diciamo che f 1 and 'the frequency of the note next to f or in this new scale, and f the next 2 to 1 f , etc.. (So \u200b\u200bit would be for example, or f = Do , f 1 = C # , f 2 = Re , etc.).. Then we said that we should have the twelfth note of the scale equal to the first but an octave higher, ie 'f 12 = 2f or .
addition, however, 'we want two successive notes are always in the same report. We call this relationship x, and the problem becomes find it. So that has to happen: 1
f = f x or
f 2 = f x 1 = f or x 2
f 3 = f 2 x = f x or 3
(...)
11 f = f x = f 10 or x 11
12 f = f 11 x = f or x 12 = 2f or
And finally comes the last report find that the relationship between two successive frequencies that would solve all the problems and 'x = 12 √ 2 ( = 1.0594 ..., irrational number, and that 'an opinion on him but indicates that it is not' a number expressed by a fraction). Pezzali cheers, his records are safe. To pass from one note f n the next f n +1 we need to multiply the frequency of the first to x = 12 √ 2 . Here invented semitone. Indeed
see what happens with the fifth (the 'cause of this choice.) A fifth interval (to to Do Sol for example) contains 7 semitones, then the fifth or f and 'f
f = 7 or ( 12 √ 2) 7
ie 'the fifth report and' 7 f / f or = ( 12 √ 2 ) 7 = 1.4983 ... , instead of the Pythagorean 1.5. Minimum difference, the ear does not notice. In addition 'the important thing' that if by hours or f climb to fifth with 12 intervals (ie 12 times 7 semitones) you get:
f (7 x 12) = f or [(12 √ 2) 7] = f 12 2 or 7 ,
which means that after 12 fifths ( f (7 x 12) ) finally ritroviamo perfettamente un'ottava (la 7 ma ottava) della nota di partenza f o (cioe' f o 2 7 ). Il circolo delle quinte e' finalmente chiuso.
Capotasti
Tornando ai capotasti della chitarra, da cui tutto e' nato, la cosa e' esattamente la stessa, tranne per il fatto di ricordarsi che la frequenza di una corda e' inversamente proporzionale alla sua lunghezza l . La nota f o viene dalla corda intera l o , la nota al primo capotasto f 1 viene dalla lunghezza della corda l 1 , ... . Then the lengths of the rope to the various notes of the scale should be (it is always x = 12 √ 2): 1
the or = l / x
the 2 or = l / 2 x
(...)
the n = l or / x n
If we are to what the frets spaced together, the length of the keyboard between two frets higher (nth and (N +1)-th ) will '
c n = l n - l n +1 the or = (1 / x n-1 - 1 / x n )
So 'the first nut and' along the 5.61% of the or , the second 5.29%, 5.00% the third, ... The twelfth 2.80%. Anyways, 'who wants can' build a guitar now. Leave you a comment myth will 'appreciated but does not hold.
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