Half-Steps/Whole-Steps With sharps and flats (accidentals), it is possible to notate all twelve distinct half-step increments within the octave. To do this however, it is necessary to know that basic notes on the 5-line staff are not all equi-distant from one to the next. If there are only 7 basic notes, it is obvious that the distance between some of these adjacent pitches must be more than a half-step. The dis- tance from one frequency (pitch) to another is known as an inter- val. The half-step is the smallest interval in twelve-tone equal temperament. By custom, only two adjacent pairs of pitches are separated by an interval of a half-step. These are the intervals E to F, and B to C, in whatever octave range they appear: Two half-steps comprise a whole step. All of the basic-note inter- vals of adjacent pitches, except those mentioned above (E to F and B to C) are whole-steps. The way a note appears on the staff ("basic," with sharp, flat, natural, etc) is its spelling. Given twelve pitches within the octave, and the flexibility of applying accidentals to basic notes, it's readily apparent that the same pitch can be spelled in differ- ent ways. For instance, D-sharp and E-flat are the same frequency, but are spelled differently. The reason for these alternative spellings will become clear in later discussion of pitch patterns. Tones which sound the same but which are spelled differently are enharmonic equivalents. The Chromatic Scale When all 12 notes within an octave are sounded as a scale, the distance between each adjacent note is a half-step. This scale has the name chromatic, in analogy to the word chroma, meaning color. In spelling an ascending chromatic scale, it is customary to spell the appropriate half-steps with sharps: Conversely, a descending chromatic scale is spelled with flats: Half-steps which result from accidentalizing a basic note are chromatic half-steps. Do not confuse this term with chromatic scale. Chromatic half-steps may occur within a melody without being part of a chromatic scale. Half-steps that are chromatic involve no change of line or space. If adjacent half-steps are from a line to a space,or from a space to a line, they are diatonic: EXERCISE 1 The screen below displays the basic note pattern for the major scale. The pattern is WWHWWWH. Your assignment is to construct this same pattern beginning at each successively higher basic note. N = next pattern to enter. Beginning pitch is shown. P = prior pattern. R = ready to enter. At prompt, enter each note name, with necessary # or b (lower case B). Leave no spaces and enter in either lower or upper case. 3 chances allowed on each pattern. H = hear notes. S = see answer. PITCH PATTERNS: SCALES The previous exercise introduced you to a pattern of pitches referred to as a scale (from scala, "ladder"). We've all heard scales practiced by budding musicians and know how tedious, boring and unmusical they are. While practicing scales on an instrument is a useful means of acquiring technical proficiency, we rarely hear scales as componants of actual pieces of music. From the point of view of analysing music, scales are abstracted "pitch sets" which help us to better understand pitch organization. For the past 400 or so years, composers working within traditions developed in Western Europe have focused upon a means of organizing pitch materials known as "tonality." The fundamental notion of tonality is that within any given piece, there is a single tone which functions as a kind of center of gravity. This single tone is called the TONIC. The other pitches (within the framework of the 12 pitches of equal temperament) all have a tendency to sound unstable (up in the air?) in relation to this tone. Tonality is an issue we will continue to examine. For the moment, however, we'll look at it in the context of some particular scalar patterns. Look at and listen to the following tune, the first segment of the familiar "classic," Somewhere Over the Rainbow. Given the conditioning we've experienced by way of hearing tunes in the tonal system, you should have no difficulty hearing the beginning pitch, C, and especially the same pitch used at the end, as the "center of gravity" among all the other pitches. This is the tonic note, one which I sometimes refer to as the generating tone. If we write this pitch down on a staff and then deploy the other pitches in ascending order 'til we reach the octave, we have the pitch set from which this piece is made. This is, as you know, a scale. This particular pitch set should sound (and look) familiar. It is one of the most common pitch sets of tonal music, known as a MAJOR SCALE. The whole-step, half-step pattern, in ascending order, is: WWHWWWH. EXERCISE 2 In the first window, you can cycle through 5 examples of sections from common tunes which use major pitch sets. You can hear these tunes by hitting "H", cycle through them by hitting "N" (for next) and "P" (for prior). Use the Up/Dn Arrows to set the cursor on the line or space (in the second window) of the pitch you deduce is the tonic note in the given example. Hit when ready... If you are right, the rest of the pitches will appear in ascending order to form a scale. Examine this scale to determine indeed, consists of the correct pattern for MAJOR. If you're not right, you get one more chance before being given the answer. PREVIEW: INTERVALS Understanding scale patterns can be enhanced by a preliminary expansion of knowledge of intervals. A musical interval is the frequency difference between one pitch and another. In traditional terms, this difference is expressed as the numerical count from one letter-name to another, including the starting and ending letter-names in the count. Thus, the interval from C to D is a 2nd, C to E a third, and so on. And since the same frequency may be spelled in more than one way (enharmonic equivalents), there re- sults different varieties of the same numerically-named interval. The interval C to E is, numerically, a 3rd. So is the interval C to E-flat. The former contains 4 half-steps, while the latter consists of only 3 half-steps. C to E is a MAJOR 3RD. C to E-flat is a MINOR 3RD. The interval C to D is a 2nd. So is the interval C to D-sharp. C to D contains 2 half-steps (thus a whole-step); C to D-sharp is 3 half-steps. C to D is a MAJOR 2ND. C to D-sharp is an AUGMENTED 2ND. This latter interval sounds exactly the same as a minor 3rd, but is spelled differently. Thus these two intervals are enharmonically equivalent. A musical interval has two designations. One is numerical. The other is qualitative. We will discuss these designations in greater detail later. For now, there is one interval in particular which is most helpful in further understanding scales. It is the PERFECT 5TH. The interval from the first scale degree in the major scale we have been considering (the TONIC, or generating tone) and the 5th scale degree (called the DOMINANT), is, numerically, a 5th. This can be readily deduced in the C-Major scale, for instance, by counting C-D-E-F-G, a count of 5. IN ANY MAJOR SCALE, THE QUALITY OF THE INTERVAL OF A 5TH FROM THE TONIC TO THE DOMINANT IS PERFECT. The perfect 5th contains 7 half-steps. The perfect 5th is a useful interval for examining certain relation- ships among scale patterns. These relationships are the fundamental building-blocks of tonality: KEYS. Any tonal piece of music is said to be in a particular key. The key is the name of the generating tone. For example, if a tune's tonic note is C, and the pitch pattern is Major, the key is C-Major. As we have seen, the key of C-Major needs neither sharps nor flats. It is the "basic-note" pattern for the major scale. In EXERCISE 1 you "built" a major scale beginning on the basic note G. You would have determined that this scale necessitated raising the F a half-step (F#) to replicate the major-scale pattern. You also made a major scale beginning on D. The D-Major scale uses both an F# and a C#. Now you can begin seeing the pattern in which sharps are added. If you "project" upwards by the interval of a perfect 5th you arrive at the next "sharped" key. Each successively arrived at key adds a sharp, and that sharp is added to the 7th scale degree. This pattern forms the basis for what is known as the CIRCLE OF 5THS, a commonly-used device for learning key relationships. We do run into a problem in this circle of 5ths. Any keys beyond 7 sharps need double-sharps. This is very cumbersome. The remedy is to respell a key using flats. This usually is done when the use of 6 sharps is reached (F#-Major) because the respelling produces a key with 6 flats (Gb-Major): Now, if we continue on with the circle of 5ths, spelling scales with flats, we note that adding a sharp to the 7th scale degree is equivalent to elliminating a flat. The circle is completed when all flats are gone, and the key of C-Major is reached. Another way to approach the keys which employ flats is to project downwards by perfect 5ths. From the key of C-Major, the next key, with one flat, is F-Major. You spelled this as the only scale in EXERCISE 2 with a flat. To make the correct whole-step, half-step pattern for major beginning on F, the 4th scale degree has to be flatted. F-Major therefore has one flat, Bb. A perfect 5th down from F is Bb. The new flat added is Eb. The counterclockwise series in the circle of 5ths adds flats: MAJOR KEY SIGNATURES If a piece of music is tonal and therefore cast within any one given key, it will consistently use a given number of sharps or flats. Rather than employing accidentals, having to write the sharp or flat every time it is called for, the sharps or flats are written at the beginning of each staff line as a key signature. The placement of the sharps or flats follows a consistent pattern: EXERCISE 3 Enter signature for given major key. ARROW KEYS = move cursor to proper location # (or 3)=SHARP b (or B)=FLAT E=ERASE (reverse order) ENTER = when you think you have proper signature M=more (continue as long as you wish) Alt/F = to store score to disk +,- = next or prior page MINOR SCALES Examine the following tune to determine its key tone: All of the clues discussed with regard to the major scale lead to the conclusion that this tune's tonic note is A. Deployment of the other pitches to generate a scale produces a basic-note pattern with a different configuration from major: WHWWHWW. This pattern is, in the system of tonality, the complement to major: MINOR. The most crucial difference between the two patterns is the quality of the 3rd scale degree. The interval between the tonic note and the 3rd in major contains 4 half-steps and is referred to as a Major 3rd. The distance from the tonic to the 3rd scale degree in minor consists of only 3 half-steps and its quality is labelled minor. There are traditional learned associations with the difference in quality between these two patterns. Major scales, keys, and the major 3rd are "bright," "cheerful," "open," "up-lifting." Minor scales, keys, and the minor 3rd are "sad," "poignant," "doleful," "closed." Whatever associations you may have, the difference in sound is distinct. If you perceive sounds in terms of colors (syn- esthesia: one type of stimulus producing a secondary sensation), you may even "see" different colors in association with major versus minor. Transposition of the minor scale pattern via the circle of 5ths reveals the respective key signatures for minor. At the same time, another method for deriving minor signatures is exhibited. If C major has no sharps or flats in its signature and A minor is similarly disposed, C major and A minor have the same signature. Since A is the 6th scale degree of C major, you can deduce that every major key has a related minor which shares its signature, found on its 6th scale degree. This is a common way for learning minor keys, in association with major. Cycling through the circle of 5ths corroborates this relationship. Note that it is now the 2nd scale degree which is raised in each successive scale, rather than the 7th. Again, we encounter the problem of needing to use double sharps if we were to continue with the above process. Conversion to flats when 6 sharps are reached will result in the enharmonically equivalent scale of Eb minor. MINOR KEY SIGNATURES Minor key signatures have the same patern as major. Hit "B" (b) to see the flat keys, "#" to toggle to sharp keys. EXERCISE 4 Enter signature for given minor key. ARROW KEYS = move cursor to proper location #(or 3)=SHARP b(or B)=FLAT E=ERASE (reverse order) ENTER = when you think you have proper signature M = more (continue as long as you wish) Alt/F = to store score to disk +,- = next or prior page MINOR VARIATIONS Here's an English folk song which is clearly in minor. Note, however, the use of accidentals. For reasons having to do with the sense of direction of a melodic line and underlying harmonic implications, strict adherence to the notes called for by the signature of a minor key is rare. There are two common variants of the minor scale pattern. The example above refers to both. If a tune is in minor and the melodic contour contains a segment which ascends up through the 6th and 7th scale degrees to the tonic note, it is common for these pitches to be raised a half step. If the melodic contour, however, is descending through the 7th to the 6th and on downwards, it is standard practice for these pitches to be in their "key signature" positions. "The Three Ravens" has a measure demonstrating the latter, and the last measure exhibits the trait of the 7th raised a half-step to lead to the tonic. The scale abstracted from this pattern is called the melodic minor. As a scale pattern, it raises the 6th and 7th a half-step from "key signature" position when ascending, while lowering them back down to "correct" spelling when descending. Very few melodies actually appear having the exact pattern of ascending 6th-7th-tonic, then tonic-7th-6th, but a variety of uses could be construed as fitting the basic framework of melodic minor. Even without this specific pattern, however, unless the melody is a direct linear descent through the 7th to the 6th, the 7th scale degree is nearly always raised, especially if this note is going to be harmonized by the Dominant triad (the chord constructed from the 5th scale degree of the given key) because this chord sounds best (in the system of tonality) if it is Major. In a minor key, the dominant triad would "naturally" be minor, so its "third" needs to be raised a half-step to make the major chord. The "third" of the Dominant triad is the 7th scale degree of the given key. Abstracting from a minor key the pitch set which uses a raised 7th results in the scale pattern called harmonic minor. This pattern is the same ascending as descending, with a 7th scale degree raised a half-step. As you can hear, this scale has a distinctly Eastern European "flavor," which results from the augmented 2nd interval between the 6th and 7th scale degrees. As a strict melodic contour, this is not common to most traditional minor tunes. In this instance, one can understand directly how it is that a scale pattern is an abstraction. There are, then, three common variants of minor pitch sets. The one based upon the key signature, without accidentals, is called natural minor. It is the "model" scale from which the melodic and harmonic minor are derived. EXERCISE 5 "P"= play scale. "M"= more. Alt/F= save score. "R"= ready to identify quality. At prompt, enter MAJOR, NATURAL, MELODIC, or HARMONIC (upper or lower case). At 2nd prompt, spell scale in ascending and descending order, USING CAPS with no spaces (e.g. ABCDEF#G#A-AGnFnEDCBA for "A Melodic Minor" - note hyphen between up/down and lower-case N for "natural"). "##" for double sharp. "M"= more. Alt/F= save score.