Music : Patterns in Time and Frequency

 I don't think anybody can claim to be totally indifferent to music. At least I have not met such a person as yet. Tastes may vary, and so may predilection for certain forms and genres. But it can be confidently said that almost everybody among st us does like music of some kind or other. Being an engineer, I have experience in voice/music processing, I have always found an immense fascination in music and possible patterns in it. Though most music lovers may just feel it a mere digression to classify and identify patterns in music, my take would be that isn't music after all only a set of patterns in time and frequency. I can already hear the cry of pain from the reader, so trying not to inflict anymore pain; let us continue in our quest of patterns. It can be safely said that each piece of musical composition is unique in itself. In fact the rendition of same piece of music performed at different times even by the same artist tends to have uniqueness, which even the artist, the original creator, cannot replicate. This is especially true for music involving some kind of improvisation, like solos in Rock or Jazz music. But even for musical genres with static compositions, like western classical, each performance is unique.

So that leaves us with a bewildering variety of musical styles and compositions, each unique, and almost intellectually unanalyzable and need for classification. But is there an existing structure to all music which can help us classify, categorize and identify different types of music and hence makes the patterns intellectually visible? Yes indeed, a zillion types of classifications are possible, but the classification used has to satisfy the criteria of helping us in identification the essence (for want of better word) of a musical piece, matching our innate verbal descriptions, if possible. There are many types of classifications, modality, rhythm, genre, origin, etc. I plan to write down about modes, scales and the related though distinct concept of Raga in. I plan to discuss Indian musical traditions and their relation with Western Classical and related traditions, partly because I have some familiarity with those forms and partly because those are probably some of the most well defined forms which uses these set of classifications extensively. Later I plan to also add articles about specific Ragas and related Scales.

Notes: Natural and Sharps/Flats

Firstly let us review the well-known basics. Let us start from notes. As is well known, there are a total of seven natural notes in the scale. These are corresponding thswara (or sur) in the Indian system. The usual denotation of these notes in western musical system is C D E F G A B. Now that is a pretty colorless nomenclature, so the southern Europeans also tend to use the solfege system of denotation. Now the notes become Do Re Me Fa So La Ti. The solfege is pretty uncommon for Indian teachers of western music. We tend to follow mostly the alphabetical notation, probably again borrowing from the British as in other things. The corresponding swaras in Indian system are 'Sa Re Ga Ma Pa Da Ni Sa'. Each of these is a short form for a Sanskrit word i.e. 'Shadaja Rishab etc'. As is evident, the Ancient Indians were quite imaginative when it came to naming. Most people translating Indian texts (especially the Chinese and Japanese Buddhists); find Indians writers too abstract and symbolic to their liking. But that is how things are. How to explain it is beyond my ken. Anyway coming back to our story, the above 7 list isn't the complete list of notes. In fact there are notes lying in-between the natural notes. These are denoted by adding a Sharp or Flat before the corresponding natural note names, depending on whether it comes before or after the natural note. For example the note lying between "C" and "D" can be called a "C Sharp" or a "D Flat" note. That is, "C Sharp" and "D Flat" denote the same note, the one lying between C and D. So we should have a total of 14 notes, including Sharps (or Flats). But that is not the case. There are no notes between B and C and also between E and F. So we have only 12 possible notes. It should be notices that I have used the term "natural notes" and "notes" for 7 cardinality set and 12 cardinality set respectively. The natural notes correspond to the "White Keys" on a Piano or Synthesizer or Harmonium. The "Black Keys" are the Sharp (or flat) notes. So there are always a total of 5 "black keys" for each set of 7 "white keys". In Standard notation C Sharp may be denoted as C# and C flat as Cb. The corresponding terms for sharp and flat in Hindustani Classical system are Tivra and Komal (literally sharp and soft).

Property of nodes
The nodes of the octave are plotted on a logarithmic frequency scale. we will get a plot of notes which are at equal distances to each other. Hence this kind of scale is called an equitempered scale. To summarize the notes are equidistant on a logarithmic scale. Notice that we are talking about the 12 numbered set of notes and not only the natural notes. Only by including the sharps and flats can we get an equal-tempered scale.

Octaves
Now the set of 12 notes form an octave. If we go further up or down the frequency scale we will find the same set of notes in the higher or lower octaves. Depending on the instrument, we have 3 to 4 possible octaves. Each of these octaves has a unique set of 7 notes plus the sharp and flat notes. for example if a piano has 4 complete octaves, then we have total of 4x12=48 keys. The 6 string guitars have around 4 octaves. The human voice is considered have 2 to 3 octaves. Females usually have high pitched voices compared to Males so they tend to use higher octaves.

Another aside on the physical properties of octaves
The Frequency of one octave to next is related by a factor of 2. Hence if the first octave is considered to be at 440 Hz, the second octave will be at 880 Hz, the third octave will be at 1760 Hz and so on. Notice the logarithmic progression. the Third octave is not 3 times the first octave. It is 4 times. So the next octave will be 8 times the first one.

Scales
Scales can be considered a method of enforcing discipline on which notes to play for a particular composition. Simplistically, scale is the subset of the 12 possible notes. This subset can have any number of notes, though those having 7 or 5 are the most common. Though some compositions may change scales midway, or use multiplicity of scales, again for simplicity we may consider that a scale remains unchanged in a composition. By sticking to a scale a musical composer can give a free rein to his/her imagination and still stick to a theme or pattern he/she chose, or a performer can improvise during performance and still reduce possibility of moving into to jarring musical territory. A particular scale (and mode) tends to signify a particular human feeling in all music traditions. Though it may sound amazing, the effect of a scale considered and defined in one culture tends to be the same across all cultures. So a scale may be linked to cheerfulness, another to seriousness, and yet another to gloom. Again Ancient Indians, bent upon their classifying and enumerating ways, defined 9 rasas each corresponding to one human emotion. All ragas (Indian classical scales), correspond to a particular rasa. Even in Western tradition, different scales are generally ascribed to different human feelings. So Major scale is usually considered for cheerful songs, while Minor scale is used for songs in more somber mood. Let us take an example of "C Major" scale. For this scale we will use 7 out of the 12 available notes. These notes can be played in any order on on any octave (remembering that the same notes will also be available on many octaves). For "C Major" scale these 7 notes happen to be the natural notes as discussed above namely C D E F G A B C. Notice that we start from the note C for describing the scale. This note "C" is the root note for the "C Major scale. Now predictably "C Major" is almost the first scale taught in music. The raga corresponding to the same scale in Hindustani Classical is Bilawal. This raga is also very commonly taught as the first raga, along with Yamen (another popular raga). Taking another example the "C minor" scale corresponds to the notes "C D D# F G G# A# B". Here the # sign has been used to denote the sharp notes as is the common in notation. The main thing to notice is that even though here too we have 7 notes only, the set of notes is different. The root note is still the same "C" note, and the importance of root note will become clear as we move further into modal territory.

Modes

As evident from the examples above, a scale is fully described by a two parametrizes. The first is the root note as discussed. The second, still unexplored, is the mode. First two new terms: tone and semitone. A tone is defined as a distance of two in an equitempered 12-note system. A semitone is defined as distance of one in this equitempered metric. So next semitone after 'C' is 'C#', but the next tone after 'C' is 'D'. Again, next semitone after 'B' is 'C', but next tone of 'B' is 'C#' (because B doesn't have a sharp note). Now considering the same example of "C Major" scale again, we see that the set of notes "C D E F G A B" which constitute the scale are a distance of "T T S T T T S", if we denote tone by T and semitone by S. This set of distances is in fact the essence of the Major Mode. So if we select a different root node, instead of 'C' selected above, and keep the same tone/semitone distances, we will have another scale of Major Mode. Now considering the same example of "C Major" scale again, we see that the set of notes "C D E F G A B" which constitute the scale are a distance of "T T S T T T S", if we denote tone by T and semitone by S. This set of distances is in fact the essence of the Major Mode. So if we select a different root note, instead of 'C' selected above, and keep the same tone/semitone distances, we will have another scale of Major Mode. So if we select 'D' as the root note and try to find the other notes in "D Major", we just write down the 12 notes and selected distances for Major Mode, starting from 'D'. So the set of notes is
A A# B C C# D D# E F F# G G#
and writing the distance from D
7 9 11 0 2 4 5
We can remember that "T T S T T T S" turns out to be "0 2 4 5 7 9 11" if T is replaced by "2" and S by "1" and series is cumulatively added. Hence using this distance property scale for any root note can be created only by knowing the modal properties.
There are many modes in Western Music, the most popular ones being Major (or Ionian), Minor Aeolian) modes.

Composition and Structure
Besides the tonal disciplines, musical forms can be classified based on the polyphonic structure. Polyphony is when multiple melodies can be played together at the same time. In contrast monophonic music incorporates only one melody at a time. Polyphonic music cannot be performed by a lone musician, and is usually part of elaborate orchestras. Some forms of music incorporate some polyphonic characteristics while being inherently monophonic. Most rock and western popular music is mostly monophonic but incorporates polyphonic characteristics by invoking independent chord accompaniment to the main melody being played. Similarly Hindustani Classical forms are mostly monophonic but have drone accompaniment in form of strings instrument like tanpura playing independently. Even in the polyphonic structure case, the scale and mode characteristics are usually stuck to.