Scales and Triads Review (Gutwein)
This handout is a distillation of the material covered in class and in the on-line textbook.  For a complete reading list and links to the on-line textbook go to the Music 201 web page and use the ink for Materials and Assignments.

The Origin of Scales
The first six partials of the harmonic series are not only responsible for creating our perceptions of octave equivalency and pitch-class (or pitch-chroma) , but partials 1-6. "the chord of nature",  (C1, C2, G2, C3, E3, G3) are in large part responsible for creating our perception of harmonic consonance and stability.  (See the handout The Harmonic Series)  One way of prolonging this collection of consonant or stable intervals over time is to insert other pitches in between: (For example:  C, d, E, f, G, a, b, C.)  The intervening pitches (d, f, a, and b) are dissonances in relation to the consonances (c, e, g, c) that conform to the harmonic series.  This method of prolonging consonance is called embellishing, and was probably responsible for the origin of scales.  As the minds of composers evolved throughout history, composers began to apply these processes by logical extension to harmonies that did  not have their origins in the harmonic series.  We will spend a significant amount of time learning to control these processes.

Diatonic Collections and Scales
If you project seven P5ths above F, you obtain the following collection of pitches:  F, C, G, D, A, E, B.  (See the handout The Cycle of Perfect Fourths and Fifths) This is the collection of "white notes" on the piano keyboard.  We refer to any projection of seven P5ths as a diatonic collection.  When we arrange them in ascending order within one octave, we create a diatonic scale:   C, D, E, F, G, A, B.    We can also represent the diatonic scale as a series of whole-steps (W) and half-steps (H): WWHWWWH.  Of course, you can begin the projection from any one of the 12 pitches in the chromatic scale and thus obtain all  12 transpositions of the diatonic scale.

Modes
By rotating the order of the pitches (or whole-steps and half-steps) in a diatonic scale we can obtain seven different  modes.  Of  these seven modes, this course will focus exclusively on major and minor.
 
MODE NAME Rotation based on C  Each transposed to begin on C
(click scale to hear)
Major or Ionian
C D E F G A B
C D E F G A B
Dorian
D E F G A B C
C D E-flat G A B-flat
Phrygian
E F G A B C D
C D-flat E-flat F G A-flat B-flat
Lydian
F G A BC D E 
C D E F# G A B
Mixolydian
G A B C D E F
C D E F G A B-flat
Natural Minor or Aeolian
A B C D E F G
C D E-flat G A-flat B-flat
Locrian
B C D E F G A
C D-flat E-flat F G-flat A-flat B-flat

Scale Degree Names
1 2 3 4 5 6 7
Tonic Supertonic Mediant Subdominant Dominant Submediant Leading-tone

Relative Relation between Major and Minor
Using the major mode as our point of reference, and rotating it from the 6th to 6th step, we can obtain the relative minor scale. Scales that are rotations of each other are in relative relation.  Another way of understanding the relation is to observe that relative scales share the same key signature, in the table above, no sharps or flats. To obtain the relative minor scale when given the tonic pitch of any major scale, simply construct the major scale (using the WWHWWWH formula) and determine the name of the 6th scale degree of that major scale. (If C is the major tonic, A would be the 6th scale-degree and the tonic of the relative minor.).  To obtain the relative major scale if given the tonic of any minor scale, simply construct the minor scale (using WHWWHWW) and determine the name of the third scale degree.  (If A is the minor tonic, C would be the 3rd scale-degree and tonic of the relative major.)

Parallel Relation between Major and Minor
Major and minor scales that share the same tonic pitch are in parallel relation; however,  they will not share the same key signature. To obtain the key signature for the parallel minor scale when given the tonic pitch of any major scale,  you must calculate the interval of a minor 3rd above the given tonic, and that pitch will be the tonic of the parallel minor scale.  For example, if given C major, the parallel minor key signature would be that of the key of E-flat major.  To obtain the key signature for the parallel major scale when given the tonic pitch of any minor scale, simply construct the major scale starting on the given tonic or memorize the key signatures for your major scales.

Tonality and the Resolution of Dissonance by Semitone  (Click the notation to playback the examples)
Common Practice tonality (as opposed to modality), is produced primarily by the ascending half-step or semitonal resolution of the dissonant Leading-tone to the consonant Tonic in the major scale.    It is produced secondarily by the descending semitonal resolution of the dissonant Subdominant to the consonant  Mediant.  These resolutions comprise the only half-steps in the scale.  They may occur as temporally separate melodic events   or as simultaneous melodic/harmonic events.

Natural, Harmonic, and Melodic Minor(Click the notation to playback the examples)
Natural minor is synonymous with the Aeolian mode,  the rotation of major from 6-6.  As you can see, there are whole-steps rather than semitones between the 7th step and tonic, and the 4th step and the mediant.

In order to create tonality through the use of a "leading-tone" resolution to tonic, composers raise the 7th step by a semitone through the use on an accidental.  This procedure produces the harmonic minor scale.

If the 6th and raised.7th scale-degrees occur in succession (ascending or descending), they produce a melodic interval of an augment 2nd Composers throughout the history of the tonal tradition usually raise both the 6th and 7th degrees when melodically adjacent and ascending.  This is primarily because the augmented 2nd dramatically alters the diatonic (stepwise) nature of the scale, being larger than a whole-step.  In fact, the  leap sounds like the arpeggiation of an enharmonic minor 3rd,  suggesting the arpeggiation of any one of several triads (F A-flat C,  D-flat F A-flat, or D F A-flat), completely unrelated to the key.   Composers often lower both pitches when melodically adjacent and descending; however, if the supporting harmony dictates they may remain raised.  This procedure essentially replaces the upper portion (the upper tetrachord) of the minor scale with the upper tetrachord of the parallel major scale, thus  producing the melodic minor.

MELODIC RULES OF THUMB FOR MINOR
1.   Always raise the 7th degree when it ascends to tonic.
2.   Always raise both 6th and 7th degrees when they occur in succession and both are ascending.

Scale-degree Triads and their Qualities in Relative Major and Minor Modes
Triads are collections of three pitches.  The triads we will be studying are produced by choosing a pitch in the diatonic scale ( for example D), and "stacking up" every other pitch in the scale above it until we have obtained two more pitches (for example F and A).  This process produces a tertian triad, because it is composed of  two diatonic thirds (in this case, D-F, and F-A).   Harmony generated by superimposing thirds is called tertian harmony.  The triad-members are labeled from the bottom up: root, third,  and fifth. Since all the steps of the diatonic scale are not the same size, triads projected above each of the tones will contain thirds of different sizes; therefore, different triad qualities will be produced.  The graphic below shows the rotational relationship between the relative major and minor scales.  Even though their scale-degree numbers and names are not rotated (they are number from 1-7 and the first degree is always tonic),  their triad qualities are rotated and therefore, easier to remember.  The alterations of the 6th and 7th scale-degrees in the harmonic and melodic forms of minor will also result in changes to triad quality.  Chords affected by these changes are "shadowed" in gray, and gray circles are drawn around the accidentals.  (Remember, the accidentals in the graph only apply to the altered forms of minor, not to the major and natural minor scales).  The following is a list of the four types of triads found in these scales.  In parentheses are their interval constructions followed by the Roman numeral case used to label each.  Use the chart to memorize the Roman numerals and triad qualities for each of the major, natural, harmonic, and melodic minor scales in the chart.