Major Scales

                 A scale is a consecutive listing of the 7 pitches within a diatonic key.  In our previous lesson on key signatures, we discussed that a diatonic key is a selection of 7 of the 21 possible pitches in music.  The selection is limited to 7 because each letter in a diatonic key is limited to one type (either flat, natural or sharp).  The listing of pitches within a scale always starts and ends with the tonic pitch.  The purpose for practicing scales on our instrument is to familiarize ourselves with the 7 diatonic pitches of a given key.

                Every diatonic key signature results in one major and one minor tonic.  Major scales are the consecutive listing of the 7 pitches within a major key.  These scales are classified as major because of the intervals formed between the tonic and each of the other pitches.  Both major and minor scales contain a perfect 4^th and 5^th.  However, all the other intervals (2^nd, 3^rd, 6^th and 7^th) within a major scale contain major qualities.  

           This sequence of major 2^nd, major 3^rd, perfect 4^th, perfect 5^th, major 6^th and major 7^th from the tonic can be used to determine the pitches of any major scale.  Another way to determine the pitches of a major scale would be to use one’s knowledge of the circle of fifths.  The circle of fifths lays out the keys signatures of every major tonic by either increasing sharps or flats.  There are many diagrams of the sharp and flat key signatures available in literature and on the internet.  A musician can determine the pitches of a major scale by starting on the tonic and applying the given key signature of that key while ascending or descending in stepwise motion until he/she reaches the repeat of the tonic in the next register.

           A third way to determine the pitches of a major scale is to examine the whole and half step relations from pitch to pitch within the scale.  Major scales are composed of two major tetrachords that are connected by a whole step.  A major tetrachord is a group of four pitches in which the distance between pitches follows the sequence of whole step – whole step – half step.  The entire sequence of a major scale is shown in this included figures.  The first figure labels the tetrachords and the connecting whole step between them.  The second figure displays the two tetrachords within a C major scale.  The scale is displayed both in musical notation and across a piano keyboard.

           Musicians can employ many methods when practicing scales.  We should realize the specific focus we wish to instill in our playing and use a method that supports that skill.  For example, practicing scales in order through the circle of fifths is an effective method for memorizing the key signatures.  Since key signatures increase by one sharp or flat as we progress through the circle of fifths, we are provided with a helpful reminder of each tonic and its corresponding key signature.  Practicing by ascending or descending chromatic tonics is an effective method for quizzing ourselves on rapid recall of the key signatures.  Finally, practicing varied patterns can help to solidify the pitches of the key signature in our minds and fingers.  Traveling up and down a scale the same way every time can become a repetitive and thoughtless activity.  This can be avoided by varying the sequence in which the pitches of the scale are played.  We can play extended "scale like" lines that change direction at destinations other than the tonic pitch.  In addition, we can create various repeating diatonic patterns such as 123-234-345-456... or 13-24-35-46-57-68-79-8.  These types of patterns help to test our ability to apply the key signature to musical content with varied melodic motion.    

Triads

                When more than two pitches are stacked together the resulting harmony is called a chord.  The simplest type of chord is a triad.  The prefix "tri" means three, so a triad is a chord composed of three pitches.  The pitches that compose a triad are the 1st, 3rd and 5th intervals in relation to the root.

                A root is the primary pitch of a triad.  The name of this pitch becomes the name of the triad, so if you want to construct a triad centered around the pitch C you would be making a C triad.  In this example C would be the 1st interval and the other two pitches would be the 3rd and 5th of C (which are E and G). 

     

           The three pitches of a triad can be stacked in different orders.  When the root is on the bottom the triad is said to be in root position.  When the third is on the bottom the triad is in 1st inversion.  When the fifth is on the bottom the triad is in second inversion. 

               

            The intervals between the three pitches of a triad determine the quality of the chord.  These quality defining intervals are measured with the root position version of the triad.  Triads can be labeled as having major, minor, diminished or augmented qualities. 

             Triads in which the root and fifth are a perfect fifth apart are either major or minor.  With these two cases a major triads possess a major third between the root and third.  Minor triads possess a minor third between the root and third. 

               

                   

             In root position, a diminished triad is composed of a diminished fifth (between the root and fifth) and a minor third (between the root and third).  This results in a stack of minor thirds.  An augmented triad is composed of an augmented fifth (between the root and fifth) and a major third (between the root and third).  This results in a stack of major thirds.  

               

Key Signatures & The Circle of Fifths

                A key signature is a collection of seven pitches that a diatonic piece of music is composed of.  One could think of it as the pitch ingredients within the recipe of a song.  As mentioned in my Musical Alphabet and Musical Intervals posts, there are 21 pitches in music.  This is a result of a seven letter musical alphabet in which each letter can be either sharp, flat or natural. 
A#     B#     C#     D#     E#     F#     G#
A       B       C       D       E       F       G
Ab     Bb     Cb     Db     Eb     Fb    Gb

                However, diatonic music selects seven of these 21 pitches as the pitch framework for a particular song.  A nonmusical illustration of this would be a person selecting four crayons from a box to make a drawing.  


The box may contain many colors, but the drawing will only contain the four colors that the person selected.  

Diatonic music is limited to one type of each letter in the musical alphabet.  There is only one type of A, B, C, etc. in a diatonic piece of music.  If a song contains a second version of a particular letter (example: A and A#) the extra pitch is chromatic and outside of the key.  Modern forms of a-tonal music are based off of other pitch systems that differ from the customary diatonic keys.  However, in this discussion we will focus on customary diatonic music.

                The key signature of a piece of music is usually indicated at the beginning of each staff (just after the clef) by listing the flat or sharp letters included within the given collection of seven.  Sharp or flat signs are placed on the line or space that illustrates the appropriate letter.  This one label signifies that every instance of that letter (regardless of register) will be a sharp or flat version.  Letters that are not mentioned within the key signature are assumed to be natural.  
           

   
               The key signature displayed here lists the sharps F# and C#.  A piece of music containing this key signature would be composed of the pitches circled in the following illustration.  

               Every key signature can result in either a major or minor key depending on the tonic that is established by the music.  The tonic is the primary pitch or harmony that the music is both centered around and resolves to.  The letter of the tonic pitch also acts as the name of the key.

        

        
               It is important to note that the existence of a sharp or flat within a key signature does not guarantee the occurrence of that pitch within the song.  Some simple songs contain less than seven different pitches within their structure.  However, the song is still based off of a particular collection of seven.  For example, a song in the key of G major (with an F# in the key signature) may not contain any F's.  However, if it did, the F would be sharp. 

                Due to enharmonic equivalence, every sharp can also be called by a flat name and every flat can be called by a flat name.  To avoid confusion, key signatures do not mix sharps with flats.  They are either a list of sharps or a list of flats.  The order in which sharps or flats are added to the list is determined by a mathematical component of music called the circle of fifths. 

      

            The key of C major or A minor contains no sharps or flats, so this key signature acts as the starting point of the circle of fifths.  If we travel up a fifth from there (counting C or A as one) we reach the key that contain one sharp (G major or E minor).  Continuing up in fifths will reveal the tonic for two sharps, three and so on.  Traveling down a fifth from C or A will reveal the tonic of the key with one flat (F major or D minor).  Continuing down in fifths reveals the other flatted tonics.  This circle of fifths diagram graphically depicts the order of the cycle.  Major keys are listed in the outside circle and minor keys are listed in the inside one.  At the bottom of the circle, we reach a point where additional sharps or flats would be inefficient.  Once the number of flats or sharps in a key signature exceeds six the resulting key overlaps the opposite side of the circle.  For example, the key of C# major which contains 7 sharps is the enharmonic equivalent to Db major which contains only 5 flats.  For this reason, the circle of fifths diagram usually only depicts the simplest form of each key signature.  For a more in-depth discussion on the circle of fifths, please refer to my blog posts from 5/28/14 and 6/4/14.  Please also refer to the included video for more insight into this discussion on key signatures.     

  

Equal Temperament

                I recently had a discussion with someone about the use of the term “perfect” when labeling the intervals of a fourth and a fifth.   As mentioned in my last post, these intervals are perfect consonances.  However, unlike octaves and unisons, fourth and fifths can be altered.  The label “perfect” is used to distinguish the consonant form of these intervals from the diminished or augmented forms. 

                This discussion caused me to think about the nature of perfect fifths and fourths in today’s music compared to music of the past.  Technically, today’s fifths and fourths are not exactly perfect.  To understand why, we need to have a brief discussion on tuning systems.

                A tuning system is a method or formula for obtaining the correct distances between musical intervals on an instrument.  The Pythagorean tuning system (created by the mathematician Pythagoras) was used till the beginning of the 16th century.  The system was based on a scale that was composed of actual perfect fifths which measure to be 702 cents in distance.  Unfortunately, this system results in uneven interval distances for across the pitch spectrum.  Unisons and octaves are perfect, but there is one fifths (the wolf fifth) within the sequence that is a different size.  This causes the other intervals within the sequence to have two different sizes throughout the series. 

                People experimented with other tuning systems throughout the years in order to have more consistent intervals.  Eventually the system of equal temperament was accepted as the dominant tuning system.   In this system octaves are subdivide into halve steps of equal distance.  This results in fifths that are slightly flat when compared to a pure perfect fifth.  However, the mathematical inconsistencies that resulted from Pythagorean tuning (and other systems) are eliminated.  The distances between intervals are equal across every key and register of the pitch spectrum. 

                This equal temperament system is the one that we are used to hearing now when we listen to music.  Most people do not even realize that the fifth they hear on an equal tempered piano is not pure.  Piano tuners, however,  are very aware of this.  They are trained to hear the proper beating sound of an equal tempered fifth. 

                A classic example of the impact of tuning systems is Bach's Well-Tempered Clavier.  This is a collection of preludes and fugues written in all 24 major and minor keys for solo keyboard.  This collection was composed before equal temperament, and was originally played on instruments that were tuned with other systems.  The mathematical inconsistencies of these systems caused the different keys to poses different sonic qualities and characters.  This sonic variety is lost when the pieces are performed on keyboard instruments tuned with an equal temperament.  So, we have gained symmetry in our intervals by sacrificing the individual character of the different key signatures and the pure fifth.