LCC Interval Categories Chart aka Tonal Gravity Chart

[chespernevins]

Thanks ML!

MusicXML seems to be the recommended format for cross application sharing.

The filename should end in .xml

[chespernevins]

From my first post:

I thought at first that the intervals might have been chosen by the consonance of the tonic of each of the intervals within the LCS. For example:

The tonic of a stand-alone min 2nd interval is the top note. Thus the tonic of a min 2nd on +5 in C would be A, the tonic of the b3 interval would be E, the tonic on 5 would be Ab and the tonic on D would be Eb.

If we grade the tonics of these intervals: A, E, Ab, Eb in order of consonance, we get the order given on the chart.

I found in the 1959 version of the book on page xxviii where it says "The concept of interval tonics has been applied in the grading of the intervals of each category".

Given this idea, and the ideas that ML and Ben have shared, I’m coming up with a little different method of generating the chart. Let's use the minor 2nd interval as an example.

1) use the prime order (1 5 2 6 3 7 +4 +5 b3 b7 4 b2 ) as the order of most ingoing to most outgoing, which ML and Ben both mention.

2) find the intersection of a given interval (in this case, a minor second), built on each of the primes, with the 7 tone order.

3) find the tonic of each of those intervals (in this case, minor seconds).

4) put the tonic of each interval in the order of the primes to find the most ingoing to outgoing (minor 2nd) intervals in the 7 tone order.

5) repeat with the 9 tone order, the 10 tone order, the 11 tone order, and 12 tone order.

Here are the steps in detail:

1) Take your primes:

b2 <-- most outgoing in the LCS
4
b7
b3
+5
+4
7
3
6
2
5
1 <-- most ingoing in the LCS


2) find the intersection of minor seconds built on each of the primes with the 7 tone order ( 1 5 2 6 3 7 +4). In the key of C, the 7 tone order is C G D A E B F#.


b2
4
b7
b3
+5
+4 <-- F# and G make a min 2.
7 <-- B and C make a min 2.
3
6
2 etc…
5 <-- (G and Ab NOT in the 7 TO)
1 <-- (C and Db NOT in the 7 TO)

3) The tonic of the interval of a minor 2nd is the top note. (See page 6 of new book or page xxvii of 1959 book).

The 2 intervals we have in the 7 tone order, and their tonics are:

The interval of F# and G, whose tonic = G
The interval of B and C, whose tonic = C


4) Put these tonics in order of ingoing to outgoing based on the order of the primes (with 1 being the most ingoing and b2 the most outgoing):

b2
4
b7
b3
+5
+4
7
3
6
2
5 <-- the note G (tonic of the F# G interval) is the second most ingoing note in the C LCS
1 <-- the note C (tonic of the B C interval) is the most ingoing note in the C LCS

So the ingoing to outgoing order of the minor seconds in the 7 tone order is:

7 (the B C interval)
+4 (the F# G interval)

5) Go back to step 2 and repeat the process with the 9 Tone Order

2) the 9 tone order is (1 5 2 6 3 7 +4 +5 b3). In C, the 9 tone order is C G D A E B F# G# Eb.

b2
4
b7
b3 <-- Eb and E make a min 2
+5 <-- G# and A make a min 2
+4 <-- this interval is already accounted for in the 7TO
7 <-- this interval is already accounted for in the 7TO
3
6
2 <-- D and Eb make a min 2
5 <-- G and Ab make a min 2
1

3) The tonic of the interval of a minor 2nd is the top note. We have these intervals and tonics:

G and Ab, whose tonic = Ab
D and Eb, tonic = Eb
G# and A, tonic = A
Eb and E, tonic = E

4) Put these tonics in order of ingoing to outgoing based on the order of the primes:

tonics, in order of ingoing to outgoing: A, E, Ab, Eb

this gives us the minor 2nds in this order:

G# A (+5)
Eb E (b3)
G Ab (5)
D Eb (2)

5) If we keep going, we get:


10 tone order:
b7
6

(This is different than the 1990s chart)

11 tone order:
4
3

12 tone order:
b2
1

Now we have the order of min 2nds:

7
+4
+5
b3
5
2
b7
6
4
3
b2
1


The major seconds will be a little different because the tonic of a major 2nd is the lower tone of the interval.


1) the prime order (1 5 2 6 3 7 +4 +5 b3 b7 4 b2 )

2) find the major seconds in the 7 tone order

3) find the tonic of each of those intervals

4) order the tonics from ingoing to outgoing

5) repeat with the 9 tone order, the 10 tone order, the 11 tone order, and 12 tone order.

7 tone order:

b2
4
b7
b3
+5
+4
7
3 <-- maj 2
6 <-- maj 2
2 <-- maj 2
5 <-- maj 2
1 <-- maj 2


Spell out the intervals:


C D
G A
D E
A B
E F#

Get the tonics:

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

Order the tonics:

C, G, D, A, E

Get the order of major 2nds, ingoing to outgoing:

1
5
2
6
3


9 tone order (C G D A E B F# G# Eb):

b2
4
b7
b3
+5
+4 <-- F# G#
7
3 <-- already in 7TO
6 <-- already in 7TO
2 <-- already in 7TO
5 <-- already in 7TO
1 <-- already in 7TO

There’s only one major 2 in the 9 TO:

+4


10 tone order (C G D A E B F# G# Eb Bb):

b2
4
b7 <-- Bb C
b3
+5 <-- G# Bb
+4 <-- already taken in 9TO
7
3 <-- already in 7TO
6 <-- already in 7TO
2 <-- already in 7TO
5 <-- already in 7TO
1 <-- already in 7TO

Find the tonics of the intervals:

Bb C = Bb
G# Bb = G#

G# comes before Bb in the order of primes so our ingoing to outgoing order of major 2nds in the 10 TO is:

+5
b7


11 tone order (C G D A E B F# G# Eb Bb F):

b2
4 <-- F G
b7 <-- 10TO
b3 <-- Eb F
+5 <-- 10TO
+4 <-- 9TO
7
3 <-- 7TO
6 <-- 7TO
2 <-- 7TO
5 <-- 7TO
1 <-- 7TO

Intervals:

Eb F tonic = Eb
F G tonic = F

The order of tonics, ingoing to outgoing, is Eb, F, thus:

b3
4

12 tone order (C G D A E B F# G# Eb Bb F Db):

b2 <-- Db Eb
4 <-- 11 TO
b7 <-- 10TO
b3 <-- 11 TO
+5 <-- 10TO
+4 <-- 9TO
7 <-- B Db
3 <-- 7TO
6 <-- 7TO
2 <-- 7TO
5 <-- 7TO
1 <-- 7TO


Intervals:
Db Eb = tonic of Db
B Db = tonic of B

B comes before Db in tonal order so our order is:

7
b2

So the order of major 2nds is:

1
5
2
6
3
+4
+5
b7
b3
4
7
b2

[chespernevins]

Thanks for your help, ML.

[strachs]

The additional point about interval tonics was the missing puzzle piece for me.

Without the benefit of either of the older books, your step-by-step, visual description helped me to make sense of this chart. Until now, I was stumped.


Thanks

[chespernevins]

Tonal Gravity 2009

http://www.4shared.com/file/136247311/6 ... _2009.html


Interval inversions juxtaposed

http://www.4shared.com/file/136247432/c ... sions.html


TG arranged in cycle of fifths

http://www.4shared.com/file/136247606/e ... _5ths.html

[chespernevins]

I have a few exercises that we used to play with in regards to TG. I like to think of them as a 'wash of sound'.

ML, if you have them handy and get a chance, I would be VERY interested to see what exercises you have!

Thanks.