"Well-Tempered Clavier" and J.S. Bach

西方古典音樂  2017

 "Well-Tempered Clavier"

and J.S. Bach

Jack  是我的一位忘年好友，音樂造詣極高，日前給我們的社團做一個非常好的演講 J.S. Bach’s Harpsichord Concertos Part I. Slides 在
 https://1drv.ms/p/s!AhbiS47AEZXlojpufC5Vi6ktOiff （ Please note that  DO NOT play it online. there is an option to download it on the upper right hand corner with the "...."  icon. There will be a pull down menu from which you can select. 

會後，我問了幾個問題，下面是他的回答：

1. What is the difference between the Frequency and Pitch?  One answer I found is  "Frequency is cycles per second.(measured in terms of Hz). Pitch is something we perceive, whereas frequency is the physical measurement of vibration."  For example, frequencies of C4 and C5 are
respectively 261.63 Hz and 523.25 Hz, but we usually will refer these two notes have the same pitch.  Is this correct?
(Answer) Somewhat. It is true that pitch is a physchoacoustical term. It's related to frequency but not exactly. The layman's term for pitch is if a note sounds higher or lower. This also depends on the person. Trained musicians can tell the difference between two tones of frequencies less than 1 Hz. This also depends on frequency. The higher the frequency, the larger the difference before one can tell two notes sound "different".

For example, you can listen to a 200 Hz tone and a 204 Hz tone and can maybe tell that they are different. However, when you listen to a 4000 Hz and 4004 Hz tone, it may not be so obvious. We perceive the pitch to be the same, even though on the spectrum analyzer it's clearly different.

C4 and C5 are different notes and different pitch. But the ratio of 2:1, the octave, is so ingrained in our biology that even though they are different, they sound "similar". Apparently more primitive cultures that have no exposure to Western classical music still identify the octave.

2.  ~500 BC, Pythagoras discovered that between 261.63 Hz and 523.25 Hz, we can use "simple multiple principle" assigning 7 notes between these 2 frequencies, based on the fact that we, human beings, feel these pitches sounds pleasant.  Is this correct?
(Answer) Notes that have simple ratios between them are deemed to be pleasant. There is still some research and debate on why this is. Aside from the octave (2:1), the next is called the fifth (C to G / do to so)(ratio 3:2) and then the fourth (C to F / do to fa) (ratio 4:3). The simplest explanation is that notes at these ratios have many of their harmonics line up. Take the fifth.  Let's use the simple frequencies.  Call note1 N = 100 Hz and note2 M = 150 Hz (ratio 3:2). The harmonics of N will be 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz .. etc...   the harmonics of M will be 150 Hz, 300 Hz, 450 Hz, 600 Hz, 750 Hz, etc.  As you see, they have common harmonics of 300 Hz and 600Hz, 900 Hz, etc.  These harmonics are in the audible range, and so this interval is more pleasant. Now take two notes H = 400Hz and J = 410Hz.  The first common harmonic is 16400 Hz! Given that the normal human has an audible range of 20 - 20 kHz, this is close to the limit. Most people have problems hearing sounds higher than 16 kHz anyways. Therefore, when you play these two notes together, they can sound harsh, or dissonant. 

Of course, with the arrival of atonal music in the 20th century, people have gotten used to more complex intervals. But to the general audience, it's still dissonant. I cannot take this music in long doses.   

3.  Somehow, octave was expanded to 12 notes. Does anybody know the history?
(Answer)  This is mostly an artificial construct in western music. Other cultures divide the octave differently. There have also been microtunings (more than 12 notes / octave). The western diatonic scale was first developed (do re mi fa so la si do) in the church and then was slow augmented, I'm sure, due to progresses in music theory. There is a lot of literature online about music history. I don't know the specifics nor mislead you :)   

4.  In any case, Bach and his student has developed a way to assigning the 12 notes between these 2 frequencies. I guess Bach's contribution was composing 20 preludes and fugues.  Is this correct?
(Answer) It's actually 24. There are twelve notes in the octave. And then major and minor makes 12 x 2 = 24. Bach's contribution was demonstrating that you could compose in all the keys. Most composers did not, in principle, write in keys that had too many flat / sharps.  

One of the biggest reasons is because one tuning that works for one key does not work well for another key. There have been different tuning schemes to try to be able to play music that sounds good in all keys. 

The modern piano tuning is called equal temperament, dividing all the notes equally by taking the twelfth root. Note that this is a huge compromise, because other than the octave, no other interval is pure. When you play C and G on the piano, the ratio is 2^(7/12) = 1.49830707688...    This is not 3/2 = 1.5, but this is close enough that we just accept it to be good enough. Obviously string players will not have this problem, and they can play a perfect interval. 

So this might be a chicken-egg scenario. (Write music in all keys first or come up with a tuning so you can write music in all keys.) It's not exactly sure if Bach favored the equal temperament or if he preferred others, or even came up with his own. 

5. Scientifically, the way to divide it by 12 notes should be multiplying itself  by 2 to the (1/12).
中國明朝，有個名叫朱載堉（1536-1610）的宅男，他除了是天文學家、物理學家、數學家、音樂家、舞學家、作家、樂器製造師之外，他還是個「算盤神」。他竟然用「81
位數的算盤」，算出了 2 的 12 次方根到小數點後 25 位，也就是真正的「平均律常數」：
1.059463094359295264561825。
http://pansci.asia/archives/74969
這個 網站 非常有趣。

(Answer) Yes. I have read about this person. It was a great revelation to find out that he was the first person in history to come up with the concept of the equal temperament. Too bad it was more of a concept and experiment and was not readily adopted by the mainstream Chinese musicians of his time.

It's unsure whether or not his theories actually spread to Europe through the cultural exchanges.  

6. In practice, nowadays, how do music instruments get tuned?

(Answer)  Pianos are tuned by the equal temperament, as explained above. The violin family is tuned according to perfect fifths. Since A3 = 440 Hz is the accepted concert pitch, everything else can be tuned from this. The violin has the notes G3, D4, A4, E5. Since A4 is an octave above A3, so it can be tuned so. All the adjacent strings are a perfect fifth apart. The double bass and the guitar are tuned according to perfect fourths. Musicians and most people can identify when the interval is "pure" (exact), so they can tune by ear, once they have A3 =  440Hz.