View Full Version : Some Thoughts on Intervals

07-13-2002, 02:12 AM
Using Interval Knowledge

The formula for a DROP 2 Voicing is: Note Skip Note Note Skip

For instance:

R [SKIP 3] 5 7 [SKIP R] 3

Lets look at the intervals of a Min7 chord:

ROOT [minor 3rd] THIRD [Major 3rd] FIFTH [minor 3rd] SEVENTH

In terms of half steps:


Lets say we want to make DROP 2 voicings for MIN7th Chords.

First lets extend this interval pattern to include the octave.


Now we and extend this indefinitely.

3 4 3 2 3 4 3 2 3 4 3 2 3 4 3 2 3 4 3 2 3 4 3 2

So to form a Drop 2 like:


We need this section of the series:
3 4 3 2 3


But we are skipping the first 3rd and the second Root so:


To find the new interval we need to add the numbers on each side of the skipped notes:

(3 4) 3 (2 3)
(7) 3 (5)

We can use the same approach to find the next inversion's Drop 2:



(4 3) 2 (3 4)
(7) 2 (7)

The last two inversions are:


I have included a diagram to play these on first 4 strings DGBE.

07-14-2002, 12:15 PM
It is useful to notice the Octave as a Big Circle, since intervals can be thought of as ascending or descending.
The ascending interval is the one for which it is named Major 3rd is 4/2 Steps ascending, however it is 8/2 Steps if the notes are inverted (Since the Octave is a Big Circle with 12 Unique notes
(12 - 4= 8) Octave - Interval = inverted interval.
This leads to all the conclusions about:

inverted m2 = M7
inverted M2 = m7
inverted M3 = m6
inverted m3 = M6
inverted P4 = P5
inverted b5 = #4

So triads on stacked thirds (Tertiary) when truly inverted become stacked 6ths (Hexal ? what is the proper term here?).

Triads on stacked 4ths (Quartal) become stacked 5ths (Quintal).

07-15-2002, 07:11 AM
Yea, szulc,

I noticed that one will mix up inverted intervals at times as they really do have the same flavour. When I was ear training using Guni's software which was available online at the dawn of iBreahte I thought it a good idea to improve the feeling of inverted interval. It does make sense because the intervals often go topsy-turvy in the chord inversions still maintaining their distinctive sound.


07-15-2002, 07:33 PM
Hi boys and girls!

This thread drove me at thinking over the complementary intervals again and here's what I've
come up with. Have you ever taken a close look at the intervalic structure of the scale you
have to deal with to improvise or harmonize? Why some scales are natural while the others
sound weird and artificial or even awkward to the point you can hardly get an ear friendly
melody out of it?
Let's assume that complementary intervals are interchangable within a certain scale (to some
extent at least). So that if we have M6 we can substitute it for m3 (m6 <-> M3, P4 <-> P5 etc.)
as these intervals sound very similar especially when picked rather wide - over an octave for
Therefore we now can implement a definition of "intervalically full" scale:

Def.: "Intervalically full" scale is the one containing all possible intervals (except for tritone
interval!) or their complementary substitutes with relation to the root note.

I exclude the tritone interval here as the most dissonant and "out of scale". Well you might ask -
hey what's the reason for such a prejudice against the tritone interval? Music theory says that
it tends to resolve to P5 or P4 as in V7 -> I progression. It is still not good an explanation
sounding a bit dogmatic. It would be called a "devil interval" through the centuries. Hell of an
interval :) What we can notice about it practically is that it hasn't its complementary substitute
or it is a tritone again - vicious circle with no way out - stuck there forever - HELP! :)
Anyways we can discuss the mistery about it later - let's just take the tritone being no good
interval for granted now. Forget about the blues for a moment :)

Ex.1: Let's consider the C major scale:


See? We have F# dividing the scale in two making up c-f# & f#-c equal tritone intervals.
Let's figure out if C major scale is "intervalically full". For this purpose we'll reflect the notes
on the right of F# onto the left part making up some kind of "tritone symmetric scale". Here follows
the result of this perversion:

(c)-(db complementary to b)-(d)-(eb complementary to a)-(e)-(f) - NO GAPS! Intervals go half step all
through this tritone symmetric scale!

YES! So now the C major scale has proven to be "intervalically full" which means we can construct all
kinds of intervals using the notes out from C major - just perfect!

Ex.2: Now we're gonna take the A minor:


(D# is a tritone divider here)

This scale is destined to undergo the same painful procedure ;)
Here's the tritone symmetric scale out of A minor:

(a)-GAP HERE!-(b)-(c)-(c# complementary to f)-(d)

OK! It's not complete! But nothing is perfect in this life. So why not add g# to fill in the gap?:

(a)-(bb complementary to g#)-(b)-(c)--(d)-(D#)-(e)-(f)--(g#)--(a)

Right! This is A harmonic minor which is used a lot especially in good old V7->I cadence.
There is an option to add bb as well. But now we're getting deprived of our lovely V7-> :(

Now you will ask - what's good about this lunatic ideas? How can I apply it to my playing and
composition? Here's the answer: when constructing your own scales you can always test it for being
"intervalically full" which would mean it at least is capable of providing you with the full range of
complementary intervals.

Thus the "poorest" scale from this perspective is whole tone scale :):

(c)--(d)--(e)--(f#)--(g#)--(a#)--(c) wich goes reduced to (c)--(d)--(e)--(f#) notes set!

And it's time to mention about our lil' poppet - the tritone note. Of course we haven't forgotten about
it. Just think of it as of a spice note to make your life better and tasteful!

Should I see a doctor? ;P


Ultra mega best regards,

07-16-2002, 04:00 AM
Did you ever notice thast inverting the Major Scale results in the Phrygian Mode?

01-02-2003, 12:46 AM

01-02-2003, 05:03 PM
Hi szulc
and everyone who hopefully will find this of a slightest interest...

First of all - szulc - yea you are right about Phrygian! A very interesting fact adding to the beauty of symmetry in music.

As the interval is an essential building block of music it's a great idea to revive this thread. So where were we? (it's whole lot easier to type this sentence than pronounce it :) ) Now I felt like I would add even more crazy stuff here to scare off those who are reading this thread again after its resurrection ;)

I'd like to mention complementary intervals and some ideas about them once again. What I'm going to share with you is based on what I wrote above.

One of the favourite chord substition methods is tritone substitution. I'm not going to dwell upon it here and jump directly to the symmetric substitution which I dreamt last night of :)


Let's have a look at Cmaj7 again. Ionian scale. Everyone feels home here. The tritone symmetric scale (see above) out of Ionian would be as follows

[ C-Db(B)-D-Eb(A)-E(Ab)-F(G) ]-(F#)-G--A--B-C (*)

Now let's see what we've done to Cmaj7 to fit into tritone symmetric scale based on Ionian:

x - x - * - * - x - x - |

Looks like it's symmetrical in itself. Now we substitute B & E for their symmetrical brothers from (*) diagram:

x - x - * - * - x - x - |
C Db Ab G

What chord can we possibly construct using these notes? The very first one that came to my mind is Dbmaj7 (db f ab c). The idea is to maintain the common note C which never comes substituted. So let's call Dbmaj7 a symmetric substitutions for Cmaj7.

You can experiment with the other scales ruled by this logic. Let's take Am7 chord to play with Aeolian:

[ A-Bb(G#)-B(G)-C(F#)--D(E) ]-(D#)-E-F--G (**)

Am7 goes like

x - * - x - x - * - x - |

Applying tritone symmetry mutation to Am7:

x - * - x - x - * - x - |
A B F# D

The output is (b d f# a) = Bm7 chord - a symmetrical substitution for Am7.
Actually we can come up with different symmetry mutations and get required substitutions. Again - what is this good for? Although this method may interfere with your primary voice leading it provides room for:

* smooth bass movement due to bass note change
* modulations

while preserving the acoustic symmetry which gives a chance the new chord sounds somewhat similar (quality of the chord stays mostly the same) but fresh. I'm going to stop here cos I'll have to experiment with these ideas myself. It's way too raw by far y'know...
Sorry for my english.

Oooof. Anyone still here? :)
Just sharing thoughts.


01-02-2003, 05:05 PM
Oops :( my drawings crashed...

01-02-2003, 05:40 PM
Zatz - you really scare the hell out of me !!! :D

Ok, lemme go back and read that again ....

01-02-2003, 11:20 PM

please erase my last bit the hell out of this thread. When I got back home and read it I though I wished it was not me who wrote that :D

01-02-2003, 11:34 PM
all I wanted to say is - given a chord substitute all or some of the the intervals between the root and the other notes for their complementary ones.

Cmaj7 has this set of intervals:

* C - E
* C - G
* C - B

When I call Dbmaj7 is symmetric substitution I mean I change the intervals in the following way:

* C - E - > C - Ab
* C - B - > C - Db

and finally I get Dbmaj7 which contains 2 complementary intervals out of Cmaj7 and fit it into the progression with a new bass if the stressed melody note is C here (a common note for these 2 chords).

Forget it people! Never mind! Maybe it's hangovers that tells so awfully after New Year celebration...


01-02-2003, 11:35 PM
Originally posted by Zatz
please erase my last bit the hell out of this thread.Ya serious??? Maybe we wait until szulc read it :)


01-03-2003, 02:09 AM
This looks to me like you have inverted all of the intervals using C as the Pitch axis.
This would be exactly like playing some arpeggio on one string and having someone turn the guitar around at one of the notes.

01-03-2003, 04:11 PM

yes you are right! This may be either root or tritone axis - in both cases we get the same intervals. The thing is you don't have to invert all the intervals - just pick any depending on your intention.

I'll just add a couple examples.

Assume we have the simple progression:

Cmaj7 -> Fmaj7 -> G7 -> Cmaj7


Gbmaj7 is "symmetric substitution" of Fmaj7. Still it sounds different in a harmonic context. If such chords follow each other we feel the movement along with the acoustic similarity. Here's what I mean:

... -> Fmaj7 -> [Gbmaj7 -> G7] -> Cmaj7

The bass approach sounds good here anticipating the final tension to resolve.


Let's take G7. It's Mixolydian:

G - - A(F) - - B - C(D) - | - D - - E - F

For instance we've come up with such substitution:

Am9 (no 5) - (a c g b) for G7

Then we can construct something like:

... -> Am9 -> Abo -> G7 -> Cmaj7 to prolong tension and add to the acoustic variety.

Not too scary indeed :)


01-03-2003, 05:44 PM
Technically a complimentary interval is where you subtract the interval from the octave, so P5 becomes P4, in general these are just normally inverted intervals. WHat you are doing is actually mirroring the intervals about a center point. This is similar to pitch axis but different in the sense that pitch axis has one note stay and everything else moves around it (not necessarily in equal intervals up and down).

01-03-2003, 06:03 PM
yes - I agree - these are inverted not complementary intervals. Though they may appear the same - as in your ex. with P4 & P5:
But sometimes they differ:

Bb - C | C - D - inverted
Bb - C, C - Bb - complementary

In this approach the root stays where it is and becomes a part of the substitutive chord. Some of the intervals (R-3, R-5, R-7...) may get inverted.


01-03-2003, 10:32 PM
When you post your well formed artice or thread on this topic why don't you consider using powertab to put it into staff?
This will make it easier on ther reat of us who are trying to understand this.

01-03-2003, 10:47 PM
I'll try to take my time to wrap it up into friendly readable form. Always on the run is a bad life style... Shaving foam on the face, unhappy girlfriend, fast food, tangleminded and raw posts... Powertab would be a good start to change things for better :)


09-15-2003, 08:03 PM
Hi guys! I'm back to this thread again.

Well, where was I?..
There are several ways to harmonize a melody. The most common one is to do it in fifths, it sounds as well if you happen to attach fourth interval, 3rds will do the trick too. Still I decided to share with you yet another idea on how to give your solo/melody/riff an intervalic support. Ages since I posted my mental ramblings re "symmetric" intervals in this thread and I decided to let them one more chance.

What is "symmetric" harmonization? The idea is borrowed from atonal music. Suppose we've got all the chromatic notes numerated, 0 being the root.

For ex.:

c(0), c#(1), d(2), d#(3), e(4), f(5), f#(6), g(7), g#(8), a(9), a#(10), b(11), c(12).

The numbers in brackets represent the distance in semitones between the root and a given tone. So, the different tones are symmetric (related to the root) if they are equally distant from the root, i.e. the sum of their numbers is 12. In our case the following intervals are symmetric in C major:

* c# - b (1+11=12)
* d - a# (2+10=12)
* eb - a (3+9=12)

The main assumption is that the symmetric interval sounds good and even contributes to establishing the tonality due to its symmetric nature ;) Still there is an exception - tritone is no good for harmonization.

A curious fact about this symmetric procedure is that the church modes turn out to be in the following symmetric relation to each other:

* Ionian and Phrygian (with the same root) are symmetric modes (thanks szulc - you pointed this out for me):

C Ionian:

c(0), d(2), e(4), f(5), g(7), a(9), b(11), c(12).

C Phrygian:

c(12), bb(10), ab(8), g(7), f(5), eb(3), db(1), c(0) (reverse order)

Note that the sum of numbers remains 12 as we move from left to the right.

(see the picture)

* Dorian is symmetric to itself!
* Lydian is symmetric to Locrian (opposites attract)
* Mixolydian and Aeolian are symmetric too.

To be continued...

09-15-2003, 08:14 PM
In the below examples I harmonized simple descending melodies lines in all the church modes with the symmetric intervals. This resulted in superimposing modes which may be used in improvization.

In this post you can see the descending C Ionian (major) melody harmonized with C Phrygian bottom line. Note that I avoided harmonizing with tritones and instead used #4 as an approach or passing tone:

09-15-2003, 08:15 PM
Isn't a better term for this complementary(from geometry/trigonometry) or reciprocal(from fractions) intervals?

09-15-2003, 08:23 PM

well, whatever the term... But it's not quite a complementary interval in a common sense - it's rather 2 complementary intervals put together.

09-15-2003, 08:27 PM
This ptb contains harmonization experiments with 7 modes. Feel the crazy sound of them.

09-15-2003, 08:29 PM
I guess my point was you could look at the octave as 180 degrees and the tritone as 90 degrees 90 + 90 = 180 so a tritone is complementary to itself. Half a tritone(m3) is 45 degrees do its complementary angle is 135 (three minor thirds) once again complementary since they add up to an octave or 180 degrees.
When I think of symmetry I think of things sharing some similar geometry, to me this idea is one of retrograde interval structure.

09-15-2003, 08:44 PM
graphically I could depict it as follows. It could be called retrograde too maybe