# Pcs: how to retrieve inverted forms from this function ?

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Dear Friends,

I'm working with pcs now.

A lot of pcs have inverted forms.

For example:

Pcs 3-11 correspond to the minor triad (037) and also the major triad (047)

since both have the same interval vector.

But when I evaluate the expression below, I get only

the prime form (037) and not the IF (047).

Is it possible to retrieve also the inverted form, like

the example below ?

(setf pcset (pcs '3-11))

>> (0 4 7) ???

Thanks a lot

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```(pcs-analysis '(0 3 7))
? pcs-analysis
Sequence: (0 3 7)
Set: 3-11
Prime Form: (0 3 7)
Pitch: (c4 eb4 g4)
Normal Order: (0 3 7)
Complement: (1 2 4 5 6 8 9 10 11)
Inversion: (0 9 5)
Vector: (0 0 1 1 1 0)

(pcs-analysis '(0 4 7))
? pcs-analysis
Sequence: (0 4 7)
Set: 3-11
Prime Form: (0 3 7)
Pitch: (c4 eb4 g4)
Normal Order: (0 4 7)
Complement: (1 2 3 5 6 8 9 10 11)
Inversion: (0 8 5)
Vector: (0 0 1 1 1 0)```

Note: The pitch output form PCS-ANALYSIS is the pitch form prime form.

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Really Cool, Janusz !

But what I was refering would be getting the

inverted form from the Forte Number, i.e.,

from 3-11 and not the opposite, informing

the integer list and getting all the info, although

this can be very useful too.

In my book, I specify this by adbing a "b" after

the set:

3-11 = 037

3-11b = 047

like this, from

Best !

Julio

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To do this I would need to rewrite the pitch class set functions.

The functions where based on Allen Forte 'The Structure of Atonal Music' book.

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I will add the inverted form into to the system with the next update.

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Very  cool, Janusz!

Im doing my doctoral dissertation now  and I'm doing some thing interesting related to rhythm that certainly could be transformed in a function. I'll tell you soon. This Will be useful there.

Best

Julio

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It's a really  interesting issue to study the different approaches of the post tonal Theory. The more hardcore atonal side does not consider prime and inverted form as different things,  ontologically but in fact, our tonal oriented ear really can tell the difference between a major and a minor triad as a clear episthemological distinction.

Counting or not counting the inverted forms also have implications on the Problems of enumeration of the types of series,  as Frippertinger &  Lackner  recent studies related to group theory and a more modern approach to Hauer Tropes demonstrates.

Best

Julio

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3 hours ago, JulioHerrlein said:

It's a really  interesting issue to study the different approaches of the post tonal Theory. The more hardcore atonal side does not consider prime and inverted form as different things,  ontologically but in fact, our tonal oriented ear really can tell the difference between a major and a minor triad as a clear episthemological distinction.

Counting or not counting the inverted forms also have implications on the Problems of enumeration of the types of series,  as Frippertinger &  Lackner  recent studies related to group theory and a more modern approach to Hauer Tropes demonstrates.

Best

Julio

I agree totally !!!

S.

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Thanks, Stephane !

Nice to hear from you.

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The PCS-ANALYSIS function returns two more results: inverted-form  and interval-class:

```(pcs-analysis '(4 1 8 10 3 7))
=> Sequence: (4 1 8 10 3 7)
Set: 6-z29
Prime Form: (0 2 3 6 7 9)
Inverted Form: nil
Pitch: (c4 d4 eb4 fs4 g4 a4)
Normal Order: (1 3 4 7 8 10)
Complement: (0 2 5 6 9 11)
Inversion: (8 11 4 2 9 5)
Vector: (2 2 4 2 3 2)
Interval Class: (3 5 2 5 4)

(pcs-analysis '(0 3 7))
=> Sequence: (0 3 7)
Set: 3-11
Prime Form: (0 3 7)
Inverted Form: (0 4 7)
Pitch: (c4 eb4 g4)
Normal Order: (0 3 7)
Complement: (1 2 4 5 6 8 9 10 11)
Inversion: (0 9 5)
Vector: (0 0 1 1 1 0)
Interval Class: (3 4)

(pcs-analysis '(0 4 7))
=> Sequence: (0 4 7)
Set: 3-11b
Prime Form: (0 3 7)
Inverted Form: (0 4 7)
Pitch: (c4 e4 g4)
Normal Order: (0 4 7)
Complement: (1 2 3 5 6 8 9 10 11)
Inversion: (0 8 5)
Vector: (0 0 1 1 1 0)
Interval Class: (4 3)```

The new function PCS-FORMS takes care of both forms: prime-form and inverted-form.

The inverted form set ends with letter b:

```(pcs-forms '((0 4 7) (0 3 7)) :type :set)
=> (3-11b 3-11)```

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Really Cool, Janusz !!

Thanks a lot.

Can we do the reverse ?

Typing the Forte code 3-11b and getting the integer list (0 4 7) ?

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of course:

```(pcs '3-11b)
=> (0 4 7)```

INTERVAL-CLASS:

```(interval-class '(0 1 2 11 10 9 5 4 3 6 7 8))
=> (1 1 3 1 1 4 1 1 3 1 1)

(interval-class '(0 1 3 8 4 9 10 7 6 5 11 2))
=> (1 2 5 4 5 1 3 1 1 6 3)

(interval-class '(0 11 5 7 6 1 3 4 2 9 8 10))
=> (1 6 2 1 5 2 1 2 5 1 2)```

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WOW !!! GREAT !

Thanks !

It was really a lot of work done !

Lots of sets, but it's really useful, I think.

Best !

Julio

This will be in the next release, I think. Looking Forward !

If you like Set Theory and Guitar, you gonna like my book. Please, Check it out !

Best,

Julio

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Congratulation to your book, looks very good.

JP

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Thanks a lot, Janusz !

Best,

Julio

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On December 11, 2017 at 9:15 AM, JulioHerrlein said:

Best,

Julio

Going to, looks good...thanks for the notice ;-)

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Thanks a lot LOOPYC!  Hope you like it!  Best Julio

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Nice work and a good companion to Allan Forte.

/Lasse

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15 hours ago, JulioHerrlein said:

Thanks a lot LOOPYC!  Hope you like it!  Best Julio

I opted for a 'physical copy', so will have to wait until Dec. 21st...just in time for a Christmas gift to myself ;-)

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3 hours ago, Stephane Boussuge said:

Hey, Stephane!  Thank you!  Hope you like it. What a lovely combination!  I feel very honored with this picture!  Best! Julio

4 hours ago, loopyc said:

I opted for a 'physical copy', so will have to wait until Dec. 21st...just in time for a Christmas gift to myself ;-)

Thank you, LOOPYC!  It's really cool to have a physical book. Best regards and happy xmas to you!  Best Julio

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12 hours ago, lviklund said:

Nice work and a good companion to Allan Forte.

/Lasse

Thank you, Iviklund! Best Julio

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