Lesson 8: Intervals - Part 3

Lesson Contents:

Nondiatonic Intervals

In Lesson 5, I presented 8 intervals diatonic to the key of C major. Here they are again in case you forgot:

Diagram 1
Diagram 1 - 8 Diatonic Intervals in the Key of C Major

Trick #1: The intervalic relationships expressed with these 8 intervals are the same for all major scales, not just C major. This means that if you compare all the notes from any major scale to its root you will always come up with:

  1. a unison - between the root and itself
  2. a major 2nd - between the root and the 2nd note of the scale
  3. a major 3rd - between the root and the 3rd note of the scale
  4. a perfect 4th - between the root and the 4th note of the scale
  5. a perfect 5th - between the root and the 5th note of the scale
  6. a major 6th - between the root and the 6th note of the scale
  7. a major 7th - between the root and the 7th note of the scale
  8. and an octave - between the root and the 8th note of the scale
Afterall, a major scale is still a major scale no matter which of the 12 notes you start it from. The scale will always contain its internal intervalic structure - the major scale formula; so it will always have these 8 diatonic intervals.

And here's a short-cut way of expressing this formula: R, 2, 3, 4, 5, 6, 7, R. The W,W,H,W,W,W,H formula is implied!!!

Why is this useful? Well, if you know the note names in all 15 theoretical major keys and if you also remember the 8 diatonic intervals of a major scale (P1, M2, M3, P4, P5, M6, M7, P8)...

15 (keys) x 8 (diatonic interval names) =
120 (interval names for all major keys)

...you have also memorized 120 interval names! So, be sure to memorize the key signatures for each major key. They are extremely important to know because not only do they tell us which notes are in any major scale, they also form the foundation upon which so many other concepts are built - like why the interval between A and C# is a major third... because C# is the 3rd note of an A major scale.

Trick #2: Now, using the 8 diatonic intervals from Diagram 1 as a foundation, we can create many nondiatonic intervals by altering the top notes of the intervals. Consult Diagram 2:

Diagram 2
Diagram 2 - 8 Nondiatonic Intervals to the Key of C Major

As you can see in Diagram 2, I have altered the top notes of the 8 diatonic intervals that were presented above in Diagram 1. The bottom note, the C note, stays unaltered. Afterall, we need to keep our frame of reference from shifting, otherwise we would get very confused and wouldn't learn anything. Now some interesting facts emerge as a result of these alterations. Most of these facts are review from Diagram 8 - The "Rules" of Intervals from Lesson 5 but some of facts involve new ideas - particularly, the idea of an enharmonic spelling for an interval name. Read on:

Trick #3: We can use the major scale formula to help us figure out nondiatonic intervals.

For example:

Say we have the interval G to Bb. What is it? Well, we know we have "some kind of 3rd" because G=1, A=2, Bb=3. If we know that G to B is a major third (using the G major scale as a frame of reference), then it must follow that G to Bb is a minor third.

Weird Intervals

Here's another example employing trick #3, a harder one: What interval is Db to G#? Well, we know we have a 4th because Db=1, E=2, F=3, G#=4. We also hopefully know that in the key of Db major the interval between Db and Gb is a perfect 4th. Now using this perfect 4th as a "frame of reference", if we compare Gb to G# we see that Gb got "altered" or raised by two half-steps.

Take this rationale one step further and we see that we raised the perfect 4th by two half-steps... and this leads to the correct answer, a doubly augmented 4th,. Weird? Yes, but this is a great way to think about how to solve these "weird" interval questions.

What's a more practical name for a doubly augmented 4th? That is, what's a better enharmonic spelling for that interval name? Well, turn G# into a diatonic note to the key of D flat major. Make it an A flat. Aha! Db to an Ab is a perfect 5th. So, Db to G# will sound like a perfect 5th. Case closed.

I know this way of thinking may seem very abstract and it is, but if you can think in this way, you will become a better musician. As you study more theory, you will find many little tricks to help you along the way. But, the only way to really discover these "shortcuts" or "tricks" is to play with this stuff. Try to find patterns. And above all, use the major scale as a frame of reference for your understanding of music theory! Why? Because most ideas and useful concepts in music theory will come from understanding the underpinnings of the major scale.

Enharmonic Spellings for Intervals - A Summary

In Lesson 5, I presented a table showing the relationship between interval names and half steps. I would now like to update this table by adding enharmonic spellings to the list. If this table makes sense, then you've "got it" when it comes to intervals.

Interval Names Commonly Used Symbols Number of Half Steps Examples Relative to C
Diminished 1st
[an eharmonic spelling for a m2]
dim. 1, °1 -1 C1 to Cb1
Perfect Prime (Unison) 1, P1 0 C1 to C1
Augmented 1st
[an eharmonic spelling for a m2]
aug. 1, +1 1 C1 to C#1
Diminished 2nd
[an eharmonic spelling for a P1]
dim. 2, °2 0 C to Dbb
Minor 2nd m2, -2, b2 1 C to Db
Major 2nd M2, 2 2 C to D
Augmented 2nd
[an eharmonic spelling for a m3]
aug. 2, +2 3 C to D#
Diminished 3rd
[an eharmonic spelling for a M2]
dim. 3, °3 2 C to Ebb
Minor 3rd m3, -3, b3 3 C to Eb
Major 3rd M3, 3 4 C to E
Augmented 3rd
[an eharmonic spelling for a P4]
aug. 3, +3 5 C to E#
Diminished 4th
[an eharmonic spelling for a M3]
dim. 4, °4 4 C to Fb
Perfect 4th P4 5 C to F
Augmented 4th (Tritone)
[an eharmonic spelling for a b5]
aug. 4, #4, +4, TT 6 C to F#
Diminished 5th (Tritone)
[an eharmonic spelling for a +4]
dim. 5, b5, °5, TT 6 C to Gb
Perfect 5th P5 7 C to G
Augmented 5th
[an eharmonic spelling for a m6]
aug. 5, #5, +5 8 C to G#
Diminished 6th
[an eharmonic spelling for a P5]
dim. 6, °6, bb6 7 C to Abb
Minor 6th m6, -6, b6 8 C to Ab
Major 6th M6, 6 9 C to A
Augmented 6th
[an eharmonic spelling for a m7]
aug. 6, +6 10 C to A#
Diminished 7th
[an eharmonic spelling for a M6]
dim. 7, °7, bb7 9 C to Bbb
Minor 7th m7, -7, b7 10 C to Bb
Major 7th M7, 7 11 C to B
Augmented 7th
[an eharmonic spelling for a P8]
aug. 7, +7 12 C to B#
Diminished 8th
[an eharmonic spelling for a M7]
dim. 8, °8 11 C1 to Cb2
Perfect 8th P8 12 C1 to C2
Augmented 8th
[an eharmonic spelling for a m9]
aug. 8, +8 13 C1 to C#2

And so ends yet another painful lesson in the land of intervals. If you've made it this far and understand this lesson and the other 2 lessons on intervals, you pretty much know what I know about them. Remember to play these suckers and invent lessons of your own. Draw diagrams. Make them colorful, make them pieces of art. Write notes out on the staff, learn this stuff in as many ways as you can think of - not just one way. Personalize this stuff. Make it fun and it will be a joy. Remember what they say, whoever "they" are, a journey of 1,000 miles is made by a series of small steps. (Or in a musician's case, a series of half-steps. I know this is bad, but I just couldn't resist.)

Don't give up. Don't get discouraged. You're not here for that. Be patient. If things seem overwhelming, put things into perspective and start over - fall back to what you already know and build from there. Add a little bit of knowledge to your understanding each time you study and practice. You'll get there. Perseverance furthers.

Proceed to Lesson 9 or go back to the main menu.