Are the musical intervals the basis from which we can gain a sense of the true character of our artistic selves and advance a potentially limited number of pitch possibilities toward an infinite number of combinations? In the tonal scheme of things, does each of the intervals create it's own unique emotional quality? And as we progress in our art, do we each find for ourselves our favorite ones, those intervals whose quality we seem to innately know, attracted towards and perfectly express our emotional artistic ideas?
For some it may very well be the blue notes, whose roots might run the deepest in the American sounds. For some it is the 5th and it's power of suggestion and dominance in structured tonality. Many players dig the quality of the major 3rd and the spiritual joy and brightness it creates. Some dig the pentatonic magic of continual float. To each their own yes? Coolness emerges in knowing a bit of the theory so as to better understand an interval and it's sound when found within the music we love.
Why cool? Well, in a number of ways. First, that knowledge of the intervals allows us to become better at analytical listening, so that when we hear a musical idea that we dig, we can better understand it's theoretical structure, thus recreate it. Second, that a theoretical knowledge of the musical intervals connects us with a couple of hundred years of existing music, it's research and development by having a working vocabulary to understand and discuss specific pieces of music, musical topics etc. Third and perhaps most important initially for the creative artist, that musical interval studies and exercises are a sure way to learn one's chosen instrument and strengthen one's abilities in articulating their own musical ideas.
From one theoretical perspective, musical intervals are simply numerical labels to measure the distance between two pitches. This measuring of intervalic distance is made up of musical half and whole steps. Another perspective is to search to discover the strengths and emotional relationships between one pitch and another, or many pitches when working with harmony. And to be aware, (thus sensitive?), of the forces of tonal gravity, and learn how to begin to enjoy this natural force in our own artistic creations. Learning the numbers part of musical intervals might take a couple of weeks of diligent study. Finding the ones you like and getting them under your fingers takes awhile. Understanding the tonal gravity aspect of musical intervals is perhaps the easiest, because we physically and emotionally feel and respond to its forces. Being able to successfully recreate this tonal gravity on our chosen instruments and in our musical creations is another matter entirely.
Numerically, the numbers top off at 15 or so, so no need for a digitron. We can use fingers and toes and cover them all. The number nine eventually plays a unique role. Lets start with the C major scale. Know this sound? Example 1.
Here are the 7 different pitches of the major scale with their interval names and above them. All of the following intervals are measured from the root C, our key or tonal center. Try to sing along with the following musical idea. Example 2.
|prime or unison||major 2nd||major 3rd||perfect 4th||perfect 5th||major 6th||major 7th||perfect octave|
Pretty basic huh? Intervals within the span of one octave are termed simple intervals, so we can simply countthe lines and spaces between pitches if we get confused right?
|The pitch D is a major second above the root C and is also the second scale degree of the C major scale.|
|The pitch E is a major third above the root and the third scale degree of the C major scale etc.|
Who said we are not scientists? Are you hip to the musical terms prime, unison, major, perfect from example 2? Lets define them in terms of the theory of the overtone series from whence they occur.
Prime, unison and perfect intervals are more closely in tune with the fundamental pitch, which in these examples is the pitch C. The mathematical ratios of the prime, unison and perfect intervals / pitches are equally divisible by the number of cycles per second created by the fundamental. Hear these intervals extracted from the chart above. Example 2a.
|prime / unison||perfect fourth||perfect fifth||perfect octave|
Major intervals and soon to be encountered minor intervals, do not have this perfection of ratios of numbers, they are not quite so in tune in the naturally occurring overtone series and thus have to be more "tempered" to take their place in our system of modern music theory. Hear these intervals extracted from the chart 2 above. Example 2b.
|major 2nd||major 3rd||major 6th||major 7th|
Cool with these intervals of the major scale? The ability to hear these intervals and identify them by their proper numerical label can open up some initial doors into the world of music theory. How about the major scale's relative, the natural minor scale? Does this group of pitches sound familiar? Example 3.
Here are the 8 pitches of the natural minor scale with their interval names above them. All of the following intervals are measured from the root A, our key or tonal center. Try and sing along with the music of example 4.
|prime or unison||major 2nd||minor 3rd||perfect 4th||perfect 5th||minor 6th||minor 7th||perfect octave|
Can you vocalize the pitches? Are the wider interval pitches a bit more tricky to find than the smaller ones? If so, your not alone. And although theorists call the intervals within a one octave span "simple", the wider intervals in this range generally present a greater challenge to accurately sing and theorists call them "leaps." To create this natural minor color, we use the minor 3rd, 6th and 7th degrees. Lets create a chart to compare the musical intervals of the major and natural minor groups of pitches using the common root C. Example 5.
Look familiar? What potentially comes right off the chart is that these two important musical colors contain 4 common pitches. Also that by lowering the major intervals by half step, they become minor, i.e., major third E to minor third Eb etc. Are there other musical intervals besides perfect, major and minor? Yes there are. Lets default back to the chromatic scale and explore musical intervals a bit deeper.
Can you recognize the sound of the chromatic scale? It has an easy intervalic formula, simply consecutive half steps. Here we create the chromatic scale using C as the root. Example 6.
Lets measure each pitch from the root C and determine it's interval. Example 7.
|C to Db||C to D||C to Eb||C to E||C to F||C to Gb||C to G||C to Ab||C to A||C to Bb||C to B||C to C|
|min 2nd||maj 2nd||min third||maj 3rd||perf 4th||dim 5th||per 5th||min 6th||maj 6th||min 7th||maj 7th||per 8th|
Cool with this? For the melodic portion of the music, the above music and chart cover about 90 percent of the music intervals that we hear when we turn on the radio and surf through the presets. The chromatic scale is cool in that it contains all of the 12 pitches within the octave as prescribed by the equal temperament system, so all of our scales we can create from the chromatic grouping of pitches? Yes. Blues scale too? Well, yes and no. The pitches are available but the blues color is a bit more complex and elusive to really capture so simply. Anyway, right in the middle of example seven above is a new term for this page. See it? The diminished fifth interval is created by decreasing the size of the perfect fifth by half step. Like changing a major interval into a minor one. Lets respell the chromatic scale with its enharmonic equivalents and see what happens to the naming of the musical intervals. Example 8.
|C to C#||C to D||C to D#||C to E||C to F||C to F#||C to G||C to G#||C to A||C to A#||C to B||C to C|
|aug unison||maj 2nd||aug 2nd||maj 3rd||per 4th||aug 4th||per 5th||aug 5th||maj 6th||aug 6th||maj 7th||per 8th octave|
Holy cow, only one new term emerges, "augmented" ( aug ) , from respelling the chromatic scale from examples 6 and 7 to it's enharmonic equivalents. Any guesses as to its effect on a musical interval? To musically "augment", is to simply enlarge a musical interval by half step. Comparing examples 7 and 8, all of our minor intervals in example 7 have become augmented intervals in example 8. Thus, does a minor second interval from C to Db equal the augmented unison from C to C#? Does the augmented fourth interval from C to F# equal the diminished fifth interval from C to Gb? Why choose one over another? Is one used in one situation and the other in another? Yes, pretty much. Theoretically, a particular musical element in a particular environment is called a particular name. Change the placement of the element into a new environment and it's name could change. A #4 or b5 interval are for most people the same thing, but not everyone. The awareness of the possibilities is the key thing here, call it whatever you want, the color of the musical interval between the root and #4 / b5 still sounds most like a tritone, regardless of how we label it.
What determines how a pitch is labeled? Basically it's relationship to the tonic pitch at any given moment within a piece of music. This labeling system is more about the theoretical analysis of music than its emotional statement. Also the direction of the melodic line that contains the pitches under scrutiny is a factor. Like most aspects of our modern world, we oftentimes want to dissect something into all of its component parts, then label each of the parts we find, leaving no stone unturned. By this process, we can hopefully begin to understand the beauty of the creative process. So in this sense, perhaps the theoretical analysis of music is like any other science, with the labeling of the musical intervals being part of the fundamental structure. Who said we are not scientists!
Lets take a quick look at the number of half steps of each of the intervals created in the chromatic scale. As a guitar player where each fret is a half step up or down, this viewing of the intervals by number of half steps they each contain added a helpful perspective of my big picture. Example 9, viewing the chromatic scale by number of half steps followed by a picture of a 2 octave chromatic scale on the bignote keyboard.
|C to Db||C to D||C to Eb||C to E||C to F||C to Gb||C to G||C to Ab||C to A||C to Bb||C to B||C to C|
|min 2nd||maj 2nd||min third||maj 3rd||per 4th||dim 5th||per 5th||min 6th||maj 6th||min 7th||maj 7th||per 8th|
|1 half step||2 half steps||3 half steps||4 half steps||5 half steps||
6 half steps
|7 half steps||8 half steps||9 half steps||
10 half steps
11 half steps
|12 half steps|
So how many half steps in a whole step? Does this way of viewing the material help to understand the concept of musical intervals? Try getting a one octave chromatic scale under your fingers on your chosen instrument. Perhaps try and sing this scale. Lets do a quick review the types of "simple" musical intervals, those contained within the span of an octave, limiting the scope of this review to the major / relative minor scales used above. Example 10.
|major intervals||minor intervals||augmented||diminished||perfect|
|found on the 2nd, 3rd, 5th and 6th degrees||found on the 2nd, 3rd, 5th and 6th degrees||to augment an interval increases its size by half step||to diminish an interval is to reduce its size by half step||found on the root or tonic, fourth, fifth and octave degrees|
Cool so far? It is pretty straightforward but perhaps some new terms eh? Lets create a chart for the compound intervals, those that exceed the span of one octave. First the music then the chart. Example 11.
|C up an octave to C||C up 1 octave + a major 2nd to D||C up 1 octave + a major 3rd to E||C up 1 octave + perfect 4th to F||C up 1 octave + a perfect 5th to G||C up 1 octave + a major 6th to A||C up 1 octave + a major 7th to B||C up 2 octaves to C|
|octave||major 9th||major 10th||perfect 11th||perfect 12th||major 13th||major 14th||perfect 15|
|12 half steps / 6 whole steps||14 half steps / 7 whole steps||16 half steps / 8 whole steps||17 half steps / 8.5 whole steps||19 half steps / 9.5 whole steps||21 half steps / 10.5 whole steps||23 half steps / 11.5 whole steps||24 half steps / 12 whole steps|
Beyond 15? For sure, the pitches do continue but what is practical in terms of musical styles comes into play. What is interesting beyond 15 is in the major / minor third arpeggio where the "loop" runs through all twelve of the major and minor keys. Really? Yep. Although the major is Lydian major and the minor is Dorian minor, two of the original ancient Greek modes. See overtone series experiments.
Inverting. Up to this point in our look at the musical intervals, all of our measurements of the intervals have been from the root upward. Can we measure the intervalic distance from one pitch down to another? Of course we can, but you knew that right? Theorists oftentimes refer to this process as inverting musical intervals. The "inverting" of musical intervals is potentially an important aspect of understanding interval theory. For what goes up must come down eh? For example, the distance between the pitch C up to the pitch F is not the same interval as C down to F. C up to F is a perfect 4th. C down to F is a perfect 5th. Really? Yep. Example 12.
To arrive at the same pitch from the same place but from a different direction, we must use different intervals. Here is the music inverting simple pitch intervals and a chart which examines the distances. Example 13.
|C up to D||C down to D||C up to E||C down to E||C up to F||C down to F|
|major 2nd||minor 7th||major 3rd||minor 6th||perfect 4th||perfect 5th|
|1 whole step||5 whole steps||2 whole steps||4 whole steps||2.5 whole steps||3.5 whole steps|
|2 half steps||10 half steps||4 half steps||8 half steps||5 half steps||7 half steps|
|C up to G||C down to G||C up to A||C down to A||C up to B||C down to B|
|perfect 5th||perfect 4th||major 6th||minor 3rd||major 7th||minor 2nd|
|3.5 whole step||2.5 whole steps||4.5 whole steps||1.5 whole steps||5.5 whole steps||.5 whole steps|
|7 half steps||5 half steps||9 half steps||3 half steps||11 half steps||1 half step|
So, why does a the intervalic name of a simple interval and it's inversion add up to nine? Such as with the perfect fifth of C up to G and the perfect fourth of C down to G. Wish I knew. Any ideas on this phenomena? Perhaps a more important and stable idea is that when perfect intervals are inverted, they stay perfect. That when major intervals are inverted they become minor and minor intervals become major when inverted. Cool with this? Pretty much the same with an inverted augmented interval becoming diminished and visa versa.
Well, enough of the theory for now eh? Where can we begin to put these ideas to work in our music? A common pedagogical use for the creative artist with musical intervals is in the creation of what are commonly termed the interval studies. All we are going to do here is simply use the pitches of the C major scale to create as series of performance exercises to strengthen ones playing ability with the various intervals associated with the major scale. If a particular interval strikes your fancy or reminds you of a favorite melody, perhaps explore that interval and learn that tune. Getting the following music under your fingers will definitely get you in the direction of becoming "one" with your chosen instrument. Lets start with the stepwise major and minor seconds of the C major scale. Example 14.
This next idea is based on the interval of ascending diatonic major and minor thirds. Example 15.
This next idea is based on the interval of ascending diatonic fourths. Example 16.
This next idea is based on the interval of ascending diatonic fifths. Example 17.
This next idea is based on the interval of ascending diatonic sixths. Example 18.
This next idea is based on the interval of ascending diatonic sevenths. Example 19.
This next idea is based on the interval of ascending diatonic octaves. Example 20.
Well? Challenging? Fun? Work? These past examples are a simple introduction into the world of interval studies, melodic permutation and basic manipulation of the resource. My college professor Dr. Miller would call the above interval studies "grist for the mill", his way of suggesting the idea of combing through the resources to look for new ideas to create melodies and sharpen ones melodic prowess on their chosen instrument. Try running each of the examples 14 through 20 around the cycle of fifths, to encompass the 12 different major keys. These above examples constitute the beginning of one's interval studies. For a more thorough study of musical intervals click interval studies.
Here is a chart to review the musical intervals encountered thus far.
|pitches||interval||# of half steps||# of whole steps||commonly called|
|C to C||unison / prime||0||0||unison / prime|
|C up to C#||augmented unison||1||.5||sharp one|
|C up to Db||minor second||1||.5||flat two|
|C up to D||major second||2||1||two|
|C up to D#||augmented 2nd||3||1.5||sharp two|
|C up to Eb||minor third||3||1.5||flat 3 / blue 3rd|
|C up to E||major third||4||2||major third|
|C up to F||perfect fourth||5||2.5||fourth|
|C up to F#||augmented 4th||6||3||sharp four / tritone / blue 4th|
|C up to Gb||diminished fifth||6||3||flat five / tritone / blue 5th|
|C up to G||perfect fifth||7||3.5||fifth / dominant|
|C up to G#||augmented fifth||8||4||sharp five|
|C up to Ab||minor sixth||8||4||flat six|
|C up to A||major sixth||9||4.5||six|
|C up to A#||augmented sixth||10||5||sharp six|
|C up to Bb||minor seventh||10||5||flat seven / blue 7th / dominant 7th|
|C up to B||major seventh||11||5.5||major seventh leading tone|
|C up octave to C||octave||12||6||octave|
|C up octave to C#||augmented octave||13||6.5||sharp octave?|
|C up octave to Db||minor ninth||13||6.5||flat nine|
|C up octave to D||major ninth||14||7||ninth|
|C up octave to D#||augmented ninth||15||7.5||sharp nine|
|C up octave to Eb||minor tenth||15||7.5||minor tenth|
|C up octave to E||major tenth||16||8||tenth|
|C up octave to F||perfect eleventh||17||8.5||eleventh|
|C up octave to F#||augmented eleventh||18||9||sharp eleven|
|C up octave to Gb||diminished twelfth||18||9||diminished twelfth|
|C up octave to G||perfect twelfth||19||9.5||twelfth|
|C up octave to G#||augmented twelfth||20||10||sharp twelve|
|C up octave to Ab||minor thirteenth||20||10||flat thirteen|
|C up octave to A||major thirteenth||21||10.5||thirteenth|
|C up octave to A#||augmented thirteenth||22||11||sharp thirteen|
|C up octave to Bb||minor fourteenth||22||11||flat seven|
|C up octave to B||major fourteenth||23||11.5||leading tone|
|C up 2 octaves to C||major fifteenth||24||12||octave|
|C up 2 octaves + 1/2 step to C#||augmented fifteenth||25||12.5||sharp fifteen|
|C up 2 octaves + 1/2 step to Db||minor sixteenth||25||12.5||flat nine|
|C up 2 octaves + whole step to D||major seventeenth||26||13||ninth|
|C up 2 octaves + 3 1/2 steps to D#||augmented seventeenth||27||13.5||sharp nine|
|C up 2 octaves + 3 1/2 steps to Eb||minor eighteenth||27||13.5||minor third|
|C up 2 octaves + 2 whole steps to E||major eighteenth||28||14||8 av. tenth?|
The cool thing is that this chart does continue and although it oftentimes repeats itself, there is a consistent major third / minor third intervalic repetition which creates a gradual modulation by half steps of the 12 major and 12 minor tonal centers of the equal temperament system. How those cats figured this out 500 years ago is way beyond me. Needless to say, very advanced! Click overtone series experiments to explore these possibilities.
Are the intervals past fifteen practical? Ever hear of tonality without a tritone? Knowing that the arpeggio continues and evolves into succeeding keys is potentially important consideration for the creative musician. Why? Well, we each must answer that for ourselves as we progress in our art and continue to search for new elements to express our ideas and own spiritual evolution. Please bare in mind one cool thing about studying music theory, that once the principles are committed to memory, the knowledge is ours forever and free to share with those around us.
Click workbook to exercise your new knowledge with the theory of the musical intervals.
|Where to next?|
This will be our own reply to violence; to make music more intensely, more beautifully, more devotedly than before. Leonard Bernstein