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Chapter 6 discusses chords in harrowing detail. For now, just be aware that, if you confuse the meanings of. You may wonder who first figured out the relationship between lovely-sounding overtones, simple frequency ratios, and their application to scale building. People usually credit the Greek philosopher, mathematician, and comedian, Monty Pythagor. As you know, Mr.
Pythagor also formulated the Pythagorean Theorem about the square hide of the hippopotamus and the sum of the other square hides, which apparently revolutionized the footwear industry. Pythagor BC - BC may have figured out the mathematics of overtones and scales 2, years ago but he certainly was not the first to discover musically pleasing scales. As discussed in Chapter 1, Neanderthals had bone flutes with diatonic scale notes tens of thousands of years ago.
As for Mr. Pythagor, it seems he realized that if you kept adding tones in consecutive frequency ratios of perfect fifths , you would get a pleasing-sounding musical scale.
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The highest note, B, is almost three octaves above the C you started with. The next step is to play all six notes in the same octave , and in scale order. Then add another C to complete the scale.
Now you have the following seven-note scale:. There you go. And, as discussed earlier, when you plunk a bunch of these notes into the same octave, you end up with other simple frequency ratios within the scale as well, such as octave , perfect fourth , major third , and so on.
So, since Mr.
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Pythagor figured out the principle of creating scales derived from simple frequency ratios, such scales are called Pythagorean scales. So far, you've seen that if you use the strict Pythagorean method, you get these six different notes the octave note is repeated :. So, why not try to get that last note by playing the next note, a fifth interval seven semitones up from B, which was the last note you played in the series?
Worse still, suppose you go away from the piano and instead decide to derive the series of notes using a calculator.
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You start with the frequency What you discover is that all the theoretical notes you calculated are slightly but noticeably sharper than the notes on the piano! In any case, the fact that you can almost get a complete major diatonic scale simply by using notes derived from consecutive overtone frequencies with the single simple frequency ratio the perfect fifth illustrates the central role of simple frequency ratios in scale building.
Suppose you were to start with the frequency for Middle C and just keep on going, up and up in leaps of perfect fifth intervals, until you eventually reach the note C again, in a much higher octave. The first question is, would you ever get to C again, somewhere over the rainbow, way up high? Yes, indeed. Especially in Kansas.
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It takes 12 leaps of perfect fifths to get to another C. You end up seven octaves above the C that you started with. If you start from Middle C and use a calculator to multiply each successive frequency by a ratio of the simple frequency ratio of the perfect fifth interval , you get the data in Table Way over the rainbow.
C, seven octaves up from Middle C. Now, just for fun are you having fun? Start with Middle C at Table 13 shows what you get.
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Have a look at the last frequency in Table 12 and compare it with the last frequency in Table The ratio between them, 33, That ratio of 1. In music, a tiny interval is called a comma. The Pythagorean comma caused all sorts of havoc with instrument tuning for more than 2, years after Monty Pythagor died of laughter, without telling anybody how to fudge the Pythagorean comma and stay in tune.
Chapter 5 discusses some clever jiggery-pokery, called equal temperament , that gets around the Pythagorean comma and cures all problems with scales forever. Well, sort of. That means the brain has the ability to understand and appreciate simple ratios of frequencies, whatever form they take—overtones of a single tone, or scales consisting of notes in simple-frequency relationships.
So, whenever humans stumble upon a way of generating a series of notes in simple-frequency relationships, they find the notes pleasing and make music. The harmonic series is a phenomenon of nature that anybody anywhere can generate with nothing more than a string or a piece of catgut or sinew attached via some sort of bridge to a resonator. Easy to make. Pleasing, You get simple-frequency-ratio discrete notes. Humans everywhere prefer music made with tones in relationships of simple frequency ratios.
A certain amount of variability is possible even within CBR. This is done by means of a bit reservoir in frames. A bit reservoir is a created from bits that do not have to be used in a frame because the audio being encoded is relatively simple. The reservoir provides extra space in a subsequent frame that may be more complicated and requires more bits for encoding.
Field I in Table 5. MP3 allows for one mono channel, two independent mono channels, two stereo channels, and joint stereo mode. Joint stereo mode takes advantage of the fact that human hearing is less sensitive to the location of sounds that are the lowest and highest ends of the audible frequency ranges. In a stereo audio signal, low or high frequencies can be combined into a single mono channel without much perceptible difference to the listener.
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The joint stereo option can be specified by the user. A sketch of the steps in MP3 compression is given in Algorithm 5. This algorithm glosses over details that can vary by implementation but gives the basic concepts of the compression method. MP3 compression processes the original audio signal in frames of samples. Each frame is split into two granules of samples each. Frames are encoded in a number of bytes consistent with the bit rate set for the compression at hand.
In the example described above with sampling rate of Use the Fourier transform to transform the time domain data to the frequency domain, sending the results to the psychoacoustical analyzer. The fast Fourier transform changes the data to the frequency domain. The frequency domain data is then sent to a psychoacoustical analyzer. One purpose of this analysis is to identify masking tones and masked frequencies in a local neighborhood of frequencies over a small window of time.
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The psychoacoustical analyzer outputs a set of signal-to-mask ratios SMRs that can be used later in quantizing the data. The SMR is the ratio between the amplitude of a masking tone and the amplitude of the minimum masked frequency in the chosen vicinity. The compressor uses these values to choose scaling factors and quantization levels such that quantization error mostly falls below the masking threshold. Step 5 explains this process further. Another purpose of the psychoacoustical analysis is to identify the presence of transients and temporal masking.
When the MDCT is applied in a later step, transients have to be treated in smaller window sizes to achieve better time resolution in the encoding. If not, one transient sound can mask another that occurs close to it in time. Thus, in the presence of transients, windows are made one third their normal size in the MDCT. Steps 2 and 3 are independent and actually could be done in parallel. Dividing the frame into frequency bands is done with filter banks.