Ascending substitute dominants

[motherlode]

Here's a little problem for the the idea of Forte primes as significant. The prime for the dominant 7 is [0,2,5,8]. No rotation of that makes a 7th.

Rotating the prime is simply revoicing the notes. If you would like for the prime to spell a dom7 you'd have to invert it. Let's look at it as notes:

Prime (0258): C .. D .. F .. Ab
Inverted form: E .. G .. Bb .. C, there's your dom7.

[nuffink]

Exactly. Looking forward to hearing the results of you explorations.

[motherlode]

I found an interesting comment regarding Allen Forte and PCS; I didn't realize that a controversy existed.

[nuffink]

I found this comment regarding Allen Forte and PCS as rather striking. I didn't realize that a controversy even existed.

Mathematicians like to squabble as much as anyone else. The most commonly cited "error" is in the calculation of primes. This was pointed out by John Rahn in his book "The Integer Model Of Pitch". Forte's method is complex, inelegant and wrong. Rahn's is simple, elegant and right.
However the methods only disagree on the primes of the following 6 sets...

Forte, Rahn.

{0,1,3,7,8}, {0,1,5,6,8}
{0,1,3,5,8,9}, {0,1,4,5,7,9}
{0,1,3,6,8,9}, {0,2,3,6,7,9}
{0,1,2,4,7,8,9}, {0,1,2,5,6,7,9}
{0,1,2,3,5,8,9}, {0,1,4,5,6,7,9}
{0,1,2,4,5,7,9,10}, {0,1,3,4,5,7,8,10}

[motherlode]

I don't know whose method I've learned to determine the primes, but so far it's been flawless.

[nuffink]

I don't know whose method I've learned to determine the primes, but so far it's been flawless.

Here's mine...

The octave can be directly mapped to a 12bit binary word. Ionian is 101011010101, where 1 = a scale note.
To convert it to a direct binary equivalent you must invert it... 101010110101 which gives a decimal value of 2471
Go through the other 6 rotations...

Dorian 011010101101 = 1709
Phrygian 010110101011 = 1451
Lydian 101011010101 = 2773
Mixolydian 011010110101 = 1717
Aeolian 010110101101 = 1453
Locrian 010101101011 = 1387

Locrian is the lowest, hence the most compact and should be the prime form. I believe Rahn's method concurs with this.

This method is blindingly fast for a computer.

[motherlode]

Ah Yes, the computer! And how does the computer compare?

The chord structure was produced from SonicMigration.
Arturo ex.png

Chord tones: Eb E Gb Ab B Db ---> Transposed to the fixed 'do': C Db Eb F Ab Bb ---> 0 1 3 5 8 10

PCS:
0 1 3 5 8 10
1 3 5 8 10 12
3 5 8 10 12 13
5 8 10 12 13 15
8 10 12 13 15 17 ---> Prime form: 6-32(024579) Interval vector: <143250>
10 12 13 15 17 20

[nuffink]

Ok, but how do you know that's the most compact rotation?

[motherlode]

By subtracting the 1st number from the last of each PCS. The one with the lowest number is the choice (barring a tie).
Then that number is reduced to the fixed 'do' ...from there the interval vector is calculated.

[nuffink]

Ok, that's effectively Forte's method, i think.

[nuffink]

Motherload (and anyone else who may be following this),

I discovered an interesting snippet on the early development of Forte's work and some of the controversy it generated. It's buried in a highly academic paper, the rest of which may not be of so much interest... http://chordspace.com/Goyette_rochester_0188E_10265.pdf

The pages from 18 - 26 detail the changes Forte's work went through in response to criticism from John Clough (a serious player in the world of music theory). It also explains the reason that Forte's numbering is so screwed up (pages 24 - 25).

[nuffink]

D'you know what's cute about this dialog between Forte and Clough? (About which I knew nothing before this thread started).
My map orders Forte's sets by Clough's algorithm for evenness. I feel like I've reconciled two old sparring partners.

[motherlode]

@nuffink

Briefly, I'm choosing my writing material based on the interval vector such as ...

<200022> (6) 4-9: (0,1,6,7) ... <644464> (6) 8-9: (0,1,2,3,6,7,8,9)

[nuffink]

Did you choose 4-9 and 8-9 specifically because they're symmetrical? Modes of limited transposition in Messiaen's term?

[motherlode]

On interval content alone ...