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.      

Musical Intervals

                As I mentioned last week I am writing a series of posts (accompanied by You Tube videos) geared toward musicians who desire to learn the basics of musical notation.  In last week's post one of the topics we discussed was the musical distance of a half step and a whole step.  In real music, the distance between pitches can extend beyond a whole step.  There is a musical unit of measure that we use to measure the distance between pitches.  That unit of measure is an interval, and it is the topic of today's discussion.

                As we mentioned last week, there are seven letters in the musical alphabet.  When measuring distances between these seven letters, we count the starting letter as one and then count our way to the destination letter.  So, the distance from A to C would be a third, because we count A as one, B as two and C as three.  This graphic helps to demonstrate all of the intervals between the seven letters of the alphabet.  

                We can count the same way on the musical staff.  Every line and space on the staff represents a letter.  We count the starting note as one, and then continue to count the lines and spaces up to (and including) the target note to determin the interval.  This graphic displays the musical intervals of one cycle of letters from C to C. 

                If you notice, that last graphic included an interval of an 8th.  Remember that the musical alphabet is a seven letter cycle that repeats back to A.  Since the letters repeat in a cycle, the interval of an 8th will always be a repeat of the letter you started on.  It is not an identical match of the starting pitch (which is called a unison).  It is the same letter in a higher or lower register.

                When two pitches are sounded at the same time they create harmony.  Harmonies that are pure and free from discord are called consonant.  There are two types of consonence in music, perfect and imperfect consonence.  The perfect consonent intervals in music are the unison, perfect fourth, perfect fifth and octive.  The imperfect consonences are thirds and sixths.  Harmonies that are discordant or clashing are called dissonant.  The dissonant intervals in music are seconds and sevenths. 

                The pairings of numbers mentioned in the last paragraph have an interesting relationship.  Each pair (2&7, 3&6, 4&5) are inversions of each other.  An inversion is the flipped version of an interval.  If a person plays a C and the E directly above it, this is the interval of a third.  If you then change the order of these two pitches by playing the E below the C (instead of the one above) you are playing the interval of a sixth.  You are still playing the same two pitches, but you have flipped the order of the pitches.  The same is true when flipping a second into a seventh, or a fourth into a fifth.

                Consonance and dissonance are used as tools by composers to create tension and release.  Any form of good entertainment (a book, a movie, a sporting event, a piece of music, ect.) will contain conflict that builds twoard a climatic moment and then resolves.  In music, this is accomplished by having dissonance resolve to consonance.  There are other tools used to compose great musical climaxes, but the most basic elements are dissonance and consonance. 

                Unfortunatly, musical intervals are not always just simple numbers.  Remember, there are actually 21 different pitches in music due to the sharps and flats.  This means that there are variations on each number when pitches are raised or lowered.  The unison and octive cannot be altered at all.  Once these distances are changed they are no longer a unison or octive.  The other perfect consonent intervals can be shortened (diminished) or lengthened (augmented). 

diminished 4th  / perfect 4th      / augmented 4th

diminished 5th  / perfect 5th      / augmented 5th

Imperfect consonent intervals and dissonant intervals have two versions, major and minor.  In addition, they can also be diminished or augmented.     

diminished 2nd / minor 2nd / major 2nd / augmented 2nd

diminished 3rd / minor 3rd / major 3rd / augmented 3rd

diminished 6th  / minor 6th / major 6th / augmented 6th

diminished 7th  / minor 7th / major 7th / augmented 7th

                All of these interval names can get confusing.  Remember the principal of enharmonic equavlince from last week's post.  Every sharp pitch can also be identified by a corrisponding flat (or in some cases natural) name.  This means that many of these interval names overlap each other.  A diminished 2nd, for example, is the same thing as a unison.  An augmented 4th is the same thing as a diminished 5th.  However, if you were measuring the distance from C to F#, you would call it an augmented 4th.  If you were measuring the distance from C to Gb (same pitch as F#), you would call it a diminished 5th. 

                Some of the other examples, like the diminished 2nd mentioned above, occure rarely.  For this to take place, you would have to be measuring the distance between a C and a D double flat.  The term double flat means that you have lowered the pitch two half steps instead of one.  These types of musical concepts are too complex for this basic discussion, so we will leave them for a future date.  The most common musical intervals are listed in this figure.

                This concludes our discussion on musical intervals.  Please refer to the attached video for further understanding.  I also provide private music instruction online.  Private lessons can be booked from this page on my website (http://www.raymelograne.com/private-lessons.html).  I will be sharing more music lessons combined with videos (like this one) in future posts.

The Musical Alphabet

                As mentioned in the description, the posts in this blog are intermediate to advanced level music theory and performance discussions.  However, the ratings of intermediate and advanced are relative to the situation being rated.  I would consider the Beatles to be advanced song writers, but they did not know how to read musical notation.  Therefore, I am going to write a series of posts geared toward musicians who desire to learn the basics of musical notation.  They will be accompanied by You Tube videos that provide further explanation and visual aid.  Readers at advanced music theory levels may even find this material useful in teaching their own students.  I only request that you refrain from creating/distributing illegal copies of the You Tube videos.  If you would like to purchase personal copy, they are available on my website.

                The musical alphabet is composed of the first seven letters of the English alphabet (A-B-C-D-E-F-G).  Once you get to G, the group repeats over again.  Each repeat is a higher register of the same pitches.  This cyclical nature of the musical alphabet is easer seen when the letters are drawn in the manor displayed here.

                Musical notation is written on a collection of horizontal lines (and the spaces between those lines) called a staff.  There are five lines and four spaces on a single staff, although there are actually an infinite number of lines and spaces in music.  The specific five lines and four spaces being listed on a particular staff are identified by a special label which is placed at the front of the staff.  This label is called a clef, and there are several different types of clefs (more on this later).        

                Most music teachers like to use catchy phrases to aid students in remembering the names of musical pitches.  They accomplish this by separating the pitches which fall on the lines of the staff from those that fall on the spaces.  The figure shown here is an example of this teaching method.

                However, separating the lines and spaces makes it difficult to see the alphabet within the pitches.  Listing the pitches on the staff in a consecutive "line - space" fashion reveals the alphabet.  This way of listing the pitches also displays the cyclical repetition of the alphabet across registers. 

                Although there are seven letters within the musical alphabet, there are more than seven pitches.  For every letter there is a natural (regular), sharp (slightly higher) and flat (slightly lower) version.  That makes for a total of 21 pitches in music. 

                                A#          B#           C#           D#          E#           F#           G#

                                A             B             C             D             E              F              G

                                Ab          Bb           Cb           Db          Eb           Fb           Gb

                The distance from one of these 21 pitches to the very next pitch is called a half step.  The distance of two half steps equals one whole step.  The distance from a letter to its corresponding sharp or flat is a half step.  However, the distance between the letters of the musical alphabet is not always a whole step.  There are two groups of letters (B & C / E & F) which are actually a half step apart. 

                The final topic of this discussion is enharmonic equivalence.  This topic was already discussed in a previous post, so I will quote that post here for your convenience. 

                When a sharp is applied to a pitch it raises the pitch by a half step.  When a flat is applied the pitch is lowered by a half step.  This creates an interesting dilemma as is seen by the graphic below.

In this example, the pitches C# and Db end up residing on the same key of the piano keyboard.  These two names actually lable the same pitch. 

                This concept can be confusing at first.  How can one pitch have two different names?  When I am teaching my public school students I describe it this way.  I also have two names (a first and last name).  At home, my wife calls me Ray, but at work my students call me Mr. Melograne.

                                                               

I am the same person, but it is more appropriate to use my first name in some settings and my last name in other settings. 

                The same is true of musical pitches.  If you are raising a C it is more appropriate to call that pitch a C#.  If you are lowering a D the resulting pitch is the same.  However, in this case it is more appropriate to call that pitch a Db. 

                This concludes our discussion on the musical alphabet.  Please refer to the attached video for further understanding.  I also provide private music instruction online.  Private lessons can be booked from this page on my website (http://www.raymelograne.com/private-lessons.html).  I will be sharing more music lessons combined with videos (like this one) in future posts. 

Long Tones

                Last week we discussed the topic of overtones, and we mentioned the use of long tones when practicing overtones.  This week I would like to spend some more time discussing the importance of overtones in an instrumentalist's practice routine.  This discussion may not be applicable to the study of instruments that are not capable of sustaining their tone (like many percussion instruments).  However, the study and use of long tones is a vitally important, and often overlooked topic in the study of instrumental performance.

                Long tones are exactly what the name suggests them to be.  They are tones that are sustained for a long period of time.  As I mentioned, they are often overlooked in instrumental studies.  This occurs because most students desire to obtain speed in their playing.  The reason we all strive to learn how to play an instrument (or sing) is to impress others.  Performing passages with fast, flashy and elaborate notes always seem to be the best way to impress others.  Long tones seem easy to achieve and boring to perform.  However, long tones that are executed correctly are both challenging and constructive tools for increasing one's performance level.

                Tone quality is just as important of a performance quality as flash and speed.  Without a desirable tone the fastest notes in the world will still be perceived as annoying instead of being entertaining.  Developing your tone as a performer should be accomplished in two stages.  First, one should look to develop a tone that is consistent, neutral and controlled.  This will allow for the performer to achieve changes in pitch, rhythm, dynamic, and articulation while maintaining a consistent tone.  Developing a consistent tone quality with long tones is the first step in this process.  Then a performer can gradually increase the fluctuation of musical elements within his/her playing while striving to maintain a consistent tone.

                Consistent long tones also help to improve intonation on instruments where intonation can fluctuate due to performance technique.  Many performers practice long tones in front of an electronic tuning device to have a visual display of their intonation.  Repeated practice in this manor causes the performer to remember what it feels like to play in tune.  Eventually this type of tone production become second nature.  

                The second stage of tone production is developing one's own unique sound.  Each performer's tone is unique, since it is shaped by specific details of his/her playing style.  The unique quality of a performer's tone is even more relevant with instruments which require breath, because the tone is shaped by both the instrument and the performer's body.  Finding and perfecting your unique tone is a performance quality that will distinguish you from other performers.  Being identifiable and unique is often a beneficial quality in musical performance. 

                One of the best ways to develop your unique and identifiable tone is through the use of long tones.  After a musician has developed a consistent and controlled tone, he/she can begin to listen to an examine the unique qualities of that tone.  While holding out long tones, desirable tone qualities can be accentuated through experimentation.  The result will be a signature tone that is unique to the performer.

                I hope that this discussion has helped you to recognize the importance of long tones in the study of musical performance.  A well rounded musician develops every aspect of their performance ability, including tone quality.  I still incorporate long tones into my practice routine regularly.  The results of long tone studies are noticeable and valuable.   

Overtones

                What are overtones?  Every musical pitch is actually composed of a mixture of many different pitches or frequencies.  To avoid confusion I will use the term pitch to refer to a melodic note and frequencies to refer to the pitch elements which compose that melodic note.  The lowest frequency within a pitch is called the fundamental.  The additional frequencies present within the pitch are called overtones.  The first overtone is always one octave above the fundamental.  The second overtone is a fifth higher than the first overtone, and the next one is a fourth higher than that.  The interval of each overtone gets closer as one travels up the series.  This mixture of tone colors gives each pitch it's unique tone color.

                This post applies most directly to wind players, although knowledge in this topic can be applicable to other areas of musical performance.  Wind players blow air into their instrument to cause the vibration that produces sound.  By making slight modifications in the way we blow, we can emphasize certain overtones more than others.  Varying the emphasis of different frequencies across the overtone spectrum can change the color of a wind performer's tone.  Skilled musicians use this technique to adapt their tone to the musical style that they are performing.

                The modifications in mentioned in the last paragraph are made through slight adjustments in a performer's throat, tongue, mouth and jaw.  Through experimentation with long tones, a performer can  discover how to produce the tone that they desire with accuracy and consistency.  Many wind players disagree over how many of the body parts listed above should be used in shaping ones tone and to what degree they should be manipulated.  However, most agree on the concept of hearing the desired tone in one's mind and then experimenting to achieve it.

                I cannot speak specifically on this topic for every wind instrument.  I can, however, add specifics about the study of overtones in saxophone performance.  Saxophone players often practice shifting between the overtones of a particular note while fingering the fundamental on a long tone.  In addition, we practice overtone scales by using the fingerings of lower fundamentals to produce the pitches of higher notes within the sequence of a scale.  The ability to eliminate lower frequencies within a pitch and bring out a certain overtone as if it is the fundamental is helpful.  A saxophone player can use this ability to gain more control over the shape of their tone.  We can also use this ability to increase the range of our instrument by playing harmonic pitches. 

                As a student, I always found it helpful to understand the application of the elements I was practicing.  Overtones are one of those elements that many people include in their practice regiment without understanding the application.  I hope that this post will help you to understand the significance and use of overtone studies in your practice time.