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2. Scale Estimation (included in An_aadf_pitch_estimator).
The estimation of the pitches pertaining to a scale involves further choices:
1) A selection of the possible pitches expected to appear in the scale or set of scales chosen for our analysis. For instance the choice of the tempered chromatic scale reduce of possible pitches to 12 within an octave, the any measurement must be related of one of them. Only tempered notes can appear. However the frequency reference of all of them should be found because many instruments or voices doesn't take the A3 for 440 Hz. Therefore even if you have only this 12 pitches some estimation of the relative intervals must be made in order to find these general reference. The former of this considerations make us to understand that we need a finer scale for pitch calculation in order to be able to calculate the decided interval estimation.
For instance, if we want to find affine we must check all the possible proportions between all the found pitches and try to find a value of 3/2 or near and this nearness depends of our pitch discrimination the minimum pitches of two continuous calculations is the pitch resolution of determination, an d determines the exactness of the interval calculation.
So we need always to calculate intervals between the estimated pitches.
But are the estimated pitches? They are the pitches that happen frequently in a piece of music or in a song. And we say frequently because in actual music or song the pitch is always changing, because vibrato, glissandi and portamenti interpolate pitches between neighboring notes. Since we understand we hear notes because the spite these pitch changes the melody reposes frequently on the notes of the scale, our algorithm must make the same. We must accumulate the pitch estimations in a histogram of frequency. The frequency of each value is the time that this frequency of pitch have appeared in the piece (see a short reminder in Analysis modal automático or in Analisis de intervalos y Escalas Modales). This histogram extends over all possible pitches and will show peaks for the values of scaled pitches if the piece was long enough to show the appearances of all possible pitches and preponderates of the most important, usually the five dominant and the tonic of the scale. There are many problems derived of this technical and the surprising one that can be formulated like this: what is a peak? There is no perfect unique of perfect questions but an easy one is a peak is a value that is higher enough to its neighbors decided that we must know the closeness of this peak over the histogram variable; in our case we must expect almost 12 peaks within an octave.
This selection of peaks in the histogram to find the notes is already solve in our signal is a MIDI signal, because MIDI events include the note (key) as a parameter.
For instance if you find a peak in 220 Hz and another in 438 you will take this as an octave and you must call them with a name, A or A minus and to check if others peaks are in a simple or tempered relationship with this A. If you fail you can not consider tempered note but still you have musical notes.
In the scale estimation you have two problems: pitch estimation and after given a name for this estimation and for these scales, you must attribute this name to a specific pieces of the signal. You must call segments of speech with a code. Let's suppose we are trying to note an Arabic song, we need first to measure the pitches in the piece, to find the peaks in the histograms of the pitches and take a name of a scales, symbolic or related with the westerns, with modification like mayor or minus or flat, or something. Afterward believing that we found the scales of the song we must attribute voiced segments of the signal of the song to the former live found note of the scale. Now you can represent the melody with the name of the note and an estimation of the duration, another big problem.
If what you want is to recognize an scales, to decide that the scales that you have recently measured is the same that one in a set of previously save and coded scales you need now to establish a distance between scales.
Since scales really a set of intervals you can establish the distance between 2 scales as a mean or general value of the individual intervals of each one of them. For that you must compare both scales in all the possible relative positions in order to find the minimum distance. If these distances are smaller than a threshold of scale similarity you can decide that this scales are the same or you can recognize that is a new scales in your scales corpus.
All the algorithms that we have described must be trained with known signals and scales to find the thresholds that allow us to decide whether something is A or B, if Yes or No.
This kind of philosophy is based on the general idea of a metric spaced where points or events are more less closer, and this closeness correspond to our projective closeness. Thus metric space is a mapping of our perception in this matters, pitch in our case. There are other approaches to similarity decisions with statistics methods, neural nets or others.
If we are interested not in actual pitches, but only in pitches within an octave, such as in scale or modality estimation we solve some of the problems about encountered as the harmonic miss-pitched since all the pitches will be reduced to the same octave and in that way we behave the algorithm.
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