My first ever comma pump using tetrads

The tones used are exactly the same as in the triadic amity pump, except that they're in different octaves.

The interesting thing here is that it takes as much as 13 triads if we don't accept diminished triads, it takes 11 triads if we do, and it only takes 7 tetrads. So, no matter if we go for triads or tetrads, the 21-tone progression is still the same.

http://dl.dropbox.com/u/8497979/pp_amity_tetrads.ogg

Petr

Petr Pařízek <petrparizek2000@...> wrote:

> http://dl.dropbox.com/u/8497979/pp_amity_tetrads.ogg

Wow!!!

> The interesting thing here is that it takes as much as 13 triads
> if we don't accept diminished triads, it takes 11 triads if
> we do, and it only takes 7 tetrads. So, no matter if we go for
> triads or tetrads, the 21-tone progression is still the same.

Are you saying the scale needed to play this progression has
21 notes/oct in each of these three situations? That's quite
something.

> The tones used are exactly the same as in the triadic amity
> pump, except that they're in different octaves.

This I don't follow at all. Can you explain?

-Carl

Carl wrote:

> Are you saying the scale needed to play this progression has
> 21 notes/oct in each of these three situations? That's quite
> something.

Yes. At least that's what comes out as the shortest possible amity pump using major/minor triads.

> > The tones used are exactly the same as in the triadic amity
> > pump, except that they're in different octaves.
>
> This I don't follow at all. Can you explain?

I'll explain using meantone as an example. You can get a triadic meantone pump like "C major, A minor, D minor, G major, C major" but you can also get a meantone pump with tetrads like "C-major6, F-major6, G-major6, C-major" (triad). The former takes 4 chords, the latter takes 3 chords, both of them use "G E C A F D B G E C" as the parent sequence of tones (i.e. a 7-tone scale). But, if you don't stick to a single-octave range for all voices of the progression, you'll probably use different octave equivalents of the same scale in each rendition. That's what I meant - both progressions use the same notes if you omit octave equivalence, but not "the same" in terms of absolute pitches including octaves.
And now, as meantone does this with at least 7 tones per octave, amity does this with 21-tones per octave. Interestingly, a 7-tone meantone chain makes a MOS, while a 21-tone amity chain does not.
The answer is that it's not a matter of MOS within an octave, it's a matter of avoiding the chromatic step -- i.e. meantone maps the chromatic step to 7 generators, while amity maps it to 21 generators.

Petr

On Sun, May 22, 2011 at 4:25 PM, Petr Parízek <petrparizek2000@...> wrote:
>
> I'll explain using meantone as an example. You can get a triadic meantone
> pump like "C major, A minor, D minor, G major, C major" but you can also get
> a meantone pump with tetrads like "C-major6, F-major6, G-major6, C-major"
> (triad).

Does C-major6 not mean C minor with a major 6? The - symbol generally
means minor in usual lead sheet notation.

> The former takes 4 chords, the latter takes 3 chords, both of them
> use "G E C A F D B G E C" as the parent sequence of tones (i.e. a 7-tone
> scale). But, if you don't stick to a single-octave range for all voices of
> the progression, you'll probably use different octave equivalents of the
> same scale in each rendition. That's what I meant - both progressions use
> the same notes if you omit octave equivalence, but not "the same" in terms
> of absolute pitches including octaves.
> And now, as meantone does this with at least 7 tones per octave, amity does
> this with 21-tones per octave. Interestingly, a 7-tone meantone chain makes
> a MOS, while a 21-tone amity chain does not.
> The answer is that it's not a matter of MOS within an octave, it's a matter
> of avoiding the chromatic step -- i.e. meantone maps the chromatic step to 7
> generators, while amity maps it to 21 generators.

How do you define the "chromatic step?" 25/24?

-Mike

Mike wrote:

> Does C-major6 not mean C minor with a major 6? The - symbol generally
> means minor in usual lead sheet notation.

I'm not familiar with all the conventions. But to me personally, something like "C" usually means a note rather than a chord or a key signature. So I wanted to mean the thing which is often pronounced "C major 6" and I didn't know how to write it.

> How do you define the "chromatic step?" 25/24?

The one which spans the highest number of generators out of the three basic altering steps (10/9, 16/15, 25/24). For meantone or amity it's 25/24, for porcupine or hanson it's 16/15, etc.

Petr

Petr wrote:

> I'll explain using meantone as an example. You can get a
> triadic meantone pump like "C major, A minor, D minor, G major,
> C major" but you can also get a meantone pump with tetrads like
> "C-major6, F-major6, G-major6, C-major" (triad). The former
> takes 4 chords, the latter takes 3 chords, both of them use
> "G E C A F D B G E C" as the parent sequence of tones
> (i.e. a 7-tone scale). But, if you don't stick to a single-
> octave range for all voices of the progression, you'll
> probably use different octave equivalents of the same scale
> in each rendition. That's what I meant - both progressions use
> the same notes if you omit octave equivalence, but not
> "the same" in terms of absolute pitches including octaves.

Got it, thanks.

> And now, as meantone does this with at least 7 tones per
> octave, amity does this with 21-tones per octave.
> Interestingly, a 7-tone meantone chain makes a MOS, while
> a 21-tone amity chain does not. The answer is that it's
> not a matter of MOS within an octave, it's a matter of
> avoiding the chromatic step -- i.e. meantone maps the
> chromatic step to 7 generators, while amity maps it to
> 21 generators.

How do you define "chromatic step"? Or how do you know
what it is?

-Carl

Carl wrote:

> How do you define "chromatic step"? Or how do you know
> what it is?

If you have a 5-limit major/minor triad and you want to get another triad by changing one pitch and preserving the other two (i.e. a major triad turns into a minor triad or vice versa), you get three possible intervals for pitch alteration -- namely 10/9, 16/15, 25/24. The interval which is mapped to the smallest number of generator in a particular temperament works as the primary diatonic step, the one mapped to a higher number of generators works as the secondary diatonic step, and the one mapped to the highest number of generators works as the chromatic step. Interestingly enough, for meantone, tetracot or amity, they're the same (10/9, 16/15, 25/24), while for hanson it's different (25/24, 10/9, 16/15) and for porcupine it's still different (10/9, 25/24, 16/15). So a 25/24 in hanson has a completely different meaning, in the hanson harmonic context, than a 25/24 in the meantone harmonic context.

Petr