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Chapter 1
Notation
      The idea of representing notes of different pitches on a staff of horizontal lines is an old one. A note placed on a line, or on a space between lines, is intended to represent a pitch that is higher (of greater frequency) than one placed on a space or line vertically below it on the page. Any given note is placed centered either on a line or in a space. A succession of notes are written from left to right, representing a succession of sounds proceeding through time. Notes have different shapes to represent different durations. Such an arraignment can encode a scale, a melody, or multiple simultaneous melodies.

      Regardless of the notation method, a piece of written music intended to be performed is called a score. The notation method described in the previous paragraph requires that each line and each space be assigned a specific pitch before the score is performed.

      Traditionally, a staff has been comprised of five lines. The lines and spaces were given letter names. In ascending order of pitch, the lines and spaces on the staff have been named in alphabetical order. Here is an example:





      Notice that the letter names are seven in number. They include A through G. This order of letter names repeats as the pitches of the staff increase and as pitches are placed on the staff in successively higher positions.

      The pitches were assigned to lines and spaces in this way over two centuries ago. This decision is responsible for the complexity of the traditional music notation system. The thinking that accounts for these choices also introduced unnecessary complexity into the design of the harpsichord, the piano, and the majority of keyboard instruments existing today.

      The development of the keyboard and the notation system were intimately linked. For this reason, the traditional keyboard is used here to explain points affecting notation.



      The keyboard shown above has a layout of white and black keys that is typical of present-day pianos and electronic keyboards. Different layouts exist, but this is the traditional design. Every black key has a white key immediately to its left and also immediately to its right. White keys are adjacent at the front edge of the keyboard, but there is sometimes a black key between two adjacent white keys back away from the front edge. Groups of two black keys and three black keys alternate along the length of the keyboard. These groups are each separated by a pair of white keys having no black key between them.

      If two different keys are pressed, the key that is furthest to the right produces the higher pitch.

      The white keys have been given the letter names A, B, C, D, E, F, G. The leftmost key on the keyboard shown above is an A. These names are repeated as the white keys proceed to the right. If a scale is played on the white keys from C to the next C right of that key, the eight pitches produced are called do, re, me, fa, so, la, ti, do.

      A great many songs can be played on the white keys without playing any black keys. There are many pleasant harmonies that can be played by striking more than one white key at the same time. Reasons of this sort led to viewing the pitches of the black keys as exceptional or advanced. This view is long obsolete, but the influence (and in my view the damage) is still with us.

      Fitting a maximum number of keys within the reach of one hand from the thumb to the little finger was another reason for the arraignment of keys in the traditional design. The black keys being interspersed between white keys augments the number of pitches that can be straddled by the hand; however, this problem was solved more effectively in 1882 by Paul von Jank√≥:





      The pitches of the scale do, re, me, fa, so, la, ti, do are not equally spaced in frequency. There is a black key between C and D, but there is no black key between E and F. In this context, the interval between two adjacent keys (regardless of color) is called a half step. An interval spanning two keys is called a whole step.

      The Interval from B to C is a half step. The interval from E to F is a half step. No black key separates them. A-B, C-D, D-E, F-G, and G-A are whole steps. Each of these pairs of keys has a black key between them.

      The pattern of whole steps and half steps in the ascending scale C, D, E, F, G, A, B, C (all white keys) is exactly the pattern that we recognize as do, re, me, fa, so, la, ti, do. The half steps are me-fa and ti-do.

      The peculiar pattern of whole steps and half steps in this most famous musical scale of the western countries comes from the way it was formed. The frequencies of the pitches form simple ratios with each other. In this scale, widely learned by ear, the whole steps are not exactly the same as each other and the half steps are not exactly the same as each other, but they are extremely close to being so. The most extensively used tuning system for musical instruments is called even temperament. It simply ignores these tiny pitch differences to make all of the half steps identical to each other. Consequently, the whole steps are also identical to each other. An interval of n half steps is the same interval (frequency ratio) as an interval of n half steps anywhere else on the keyboard.

      There are exceptions to the use of even temperament. Singers who are not accompanied by even-tempered instruments tend toward the original ratios of pitch. Barber Shop Quartets make a point of it.

      The traditional notation system (which is currently almost universal) uses the seven letter names for lines and spaces, while there are twelve pitches in any octave (the distance between any pitch and the next pitch, lower or higher, having the same name). When the black key pitches are included, an octave contains 12 different pitches - not seven. This leads to a complication. If one wishes to produce the famous scale starting on some pitch other than C, the pattern of half steps and whole steps is not the same. To correct this, some notes of the scale drawn on the staff must be modified to indicate that the pitch is to be played either a half step higher or a half step lower than the letter name pitch of the line or space.

      This is done by placing a sharp sign (#) or a flat sign () next to the notation for the pitch. Here is an example:





      Both of the scales shown above are intended to sound like do, re, me, fa, so, la, ti, do. They are each placed on a five-line staff, except that the first note of the top scale starts on a ledger line, which extends the staff in such cases. The second scale has sharp signs before some of the notes. The first scale does not. This is because the first scale goes from C to C and has pitches named C, D, E, F, G, A, B, C, while the second scale goes from E to E and would not have the same pattern of whole steps and half steps without adjusting some of the pitches a half step upward.

      Here is an example of a scale that requires one flat:





      The notes in the picture above are called whole notes to distinguish them from notes having shorter or longer durations. They have a different appearance from half notes, which have half the duration in time as whole notes. There are many kinds of notes having various durations.

      The shape of a note, independent of any accidental that may be place ahead of it, determines the duration of the note:





      From left to right, the five notes shown in the figure above are named whole note, half note, quarter note, eighth note, and sixteenth note. A half note has half the duration of a whole note. Each note that is to the right of another note in the figure is sounded for half of the duration of the note to its left. Beyond the quarter note, time durations are divided in half by adding a flag to the vertical stem of the note. This is sometimes, but rarely, extended as far as the 128th note. Notes having vertical stems may be drawn so that the stem extends downward instead of upward from the round part of the note, but in either case, the flags are aways drawn extending to the right.

      In traditional music notation, the duration of a note can be augmented by half by placing a dot just after it. This can be done with numbers just as well, such as 2., which has the duration of a quarter note tied to a following eighth note. More rarely, a double dot is used to extend the duration further. The note 2.. is equal to a quarter note plus an eighth note plus a sixteenth note.

      Now that composers can write music on computers and have them performed automatically, the notation system could be extended to accommodate any duration without regard to the difficulties instrumentalists might encounter. The exact specification of phrase length and emphasis would no longer require the conductor's interpretation. The drama of rhetoric could be accurately created and precisely controlled.

      For composers who like to jot down ideas on a sheet of staff paper, numbers would be a lot faster and easier to draw than these detailed shapes. The numbers 0 through 7 would suffice. Nowadays, graphic software could translate a score from one notation system to another; but changing the appearance of notes is far less important than addressing the complications posed by the five-line staff which does not represent all of the pitches within an octave.

      Defining the staff with seven letter names corresponding to the white keys of the keyboard, without assigning places for the pitches that are between some pairs of white keys (the black keys), not only necessitates modifying the appearance of some notes with sharps and flats, it also creates an ambiguity. The white key B, for example has a black key immediately to its left. In the context of the scale shown above, this black key is called B flat. In another context, this same black key would be called A sharp. If names are both unique and unchanging, learning the notation system is less context dependent and therefore less complicated and confusing.

      By convention, when a given collection of sharps or flats are needed throughout a section of written music, they are placed on the left end of each staff or at the beginning of the section it applies to in the piece of music. This is called a key signature. A figure is also included to indicate the pitch of some place on the staff. Here is an example:





      The stylized figure shown above is known as a clef. In this case it is a G cleff. Although it is not very apparent, it indicates which line of the staff is the G line. A staff having a G clef is also called a treble clef. Consistent with the placement of the G line, the key signature includes F sharp, C sharp, G, sharp, and D sharp. This is the key signature for the key of E Major. E Minor is different. Its key signature is the same as that of G Major. The difference between major and minor keys does not affect how the performer decides on a pitch on the basis of the notation. It is a substantial matter that is independent of the notation issues being raised here.

      This traditional system results in fifteen different key signatures. Each key signature doubles as the key signature for a major key and for a minor key. Sharps and flats modify the meaning of a line or space as needed to preserve the pattern of the scale do, re, me, fa, so, la, ti, do (which is the pattern of whole steps and half steps that exists in the scale C, D, E, F, G, A, B, C, but not in the scale E, F, G, A, B, C, D, E, or in any other eight unmodified consecutive notes on the staff, or any eight consecutive notes of the white keys).

      To negate the effect of a sharp or flat in a subsequent note, there is another sign that is used called a natural sign which looks like this:





      Collectively, sharps, flats, and naturals are called accidentals.

      Another aspect of time in music is meter. This is a regular interval of time delineated by vertical bar lines drawn across the notated music:





      The vertical bar lines in the staff above are to separate notes that add up to the same duration between bar lines. The time signature indicates how many beats at equal time intervals are between each adjacent pairs of bars. It also indicates which kind of note gets one beat. This particular time signature indicates (with the upper 4) that there are four beats to each measure and that a quarter note (the lower 4) gets one beat.

      The area between bars and the area between the first bar and the time signature are to contain note durations that add up to four beats where the quarter note is the note duration that gets one beat. Such an area is called a measure. They may contain four quarter notes, or eight eighth notes instead. Any combinations of notes equal in duration to four quarter notes can be placed between adjacent bars.

      Most composition software assumes that the time signature will not change frequently. This seems to prompt programmers of composition software to make changing the time distance between bars inconvenient and time consuming. Announcing an enduring time signature explicitly may not be as useful as letting the note durations speak for themselves, with bars or other marks used merely to group notes of not-necessarily-uniform duration. A march would not be confused by this.

      A staff that provides places for all of the pitches on the keyboard avoids many complications and is very much easier to learn. Consider the solution below this paragraph. Eight octaves are notated (enough for the possible pitches of an entire orchestra). Every space is a half step different in pitch from the lines adjacent to it. Every line is a half step different in pitch from the spaces adjacent to it. It can be referred to as a twelve-tone staff (meaning twelve pitches per octave). There is no need to use accidentals on a twelve-tone staff. It is not necessary to use the full eight-octave staff when the written music is to be performed by an instrument of more limited range.





      This is one of the solutions that is especially convenient for composers. It is unbiased as to whether the music uses one scale or another or no particular scale (as in atonal music). When the feeling of using one key (scale) changes to another, there is no need to change a key signature. In the case of temporary or fleeting changes in key feeling, there is no need to write sharps of flats in front of notes. A scale or key can be defined as starting on any pitch (not just those of the white keyboard keys). Keep in mind that nowadays the composer's score can be automatically changed to traditional notation for the benefit of present-day performers. Some decades from now, we might even see bilingual performers. Certainly, children taught with a 12-tone staff would be accomplished sooner than usual. For composers, The benefits outweigh the difficulty. Writing music is not a real-time operation.

      Chords are collections of notes that occur at the same time in a score, or are sounded simultaneously in a performance. A twelve-tone staff makes recognizing written chords less complicated. Without accidentals to interpret, chords can be distinguished on the basis of the number of staff places that separate the vertically aligned notes. They form purely geometric patterns that are not complicated by the non-spacial symbology of the existing system.


Chapter  2 - Sound