Help identifying pumped commas

Hi All,

Since this is my first post, I'd like to start with a big thank-you to all
the amazing people who have posted here over the years. I've learned a
colossal amount with still loads to learn but loving every minute of it.

I constructed a sequence of 4:5:6:7 chords in a Blackwood temperament (in
55deo which I know isn't optimal, just what I'm working in at the moment)
where the roots don't cycle in 720 cent "fifths" but in descending 240 cent
"tones". In each step of the sequence, there are 2 common tones where the
"7/6" of the previous chord becomes the "8/7" of the new chord.

What comma is this pumping? It's not 256/243. (7/8)^5 * 2/1? (7/6)^5 * 1/2?
49/48 each step?

Here's what it sounds like:
https://soundcloud.com/james-fenn-theoretical/blackwood-comma-pump-55edo

Cheers,
James

Awesome first post! Each chord change does temper out

7/6 - 8/7 = 49/48

where "-" is understood to subtract log-frequencies. Working
on the ratios, it gives

7/6 * 7/8 = 49/48

The reinterpretation of this interval between adjacent
chords in the progression (7/6 = 8/7) is called a "pun".

But the "pump" of the entire progression depends on the root
motion. Each root motion is 8/7 in the 'reference frame' of
the current chord. So the pump is

(8/7)^5 = 2

2/((8/7)^5) = 16807/16384

Cheers,

-Carl

James Fenn <james@...> wrote:

> I constructed a sequence of 4:5:6:7 chords in a Blackwood
> temperament (in 55deo which I know isn't optimal, just what
> I'm working in at the moment) where the roots don't cycle in
> 720 cent "fifths" but in descending 240 cent "tones". In each
> step of the sequence, there are 2 common tones where the "7/6"
> of the previous chord becomes the "8/7" of the new chord.
>
> What comma is this pumping? It's not 256/243. (7/8)^5 * 2/1?
> (7/6)^5 * 1/2? 49/48 each step?
[snip]

Hello James,

On Thu, Feb 21, 2013 at 8:49 AM, James Fenn <james@...> wrote:
>
> Hi All,
>
> Since this is my first post, I'd like to start with a big thank-you to all
> the amazing people who have posted here over the years. I've learned a
> colossal amount with still loads to learn but loving every minute of it.

Glad you enjoy it, welcome to the forum! Be sure to check out the
Xenharmonic Alliance Facebook group as well
(http://www.facebook.com/groups/xenharmonic2), along with the
Xenharmonic Wiki (http://xenharmonic.wikispaces.com/), where there's
even more people and information for you.

> I constructed a sequence of 4:5:6:7 chords in a Blackwood temperament (in
> 55deo which I know isn't optimal, just what I'm working in at the moment)
> where the roots don't cycle in 720 cent "fifths" but in descending 240 cent
> "tones". In each step of the sequence, there are 2 common tones where the
> "7/6" of the previous chord becomes the "8/7" of the new chord.
>
> What comma is this pumping? It's not 256/243. (7/8)^5 * 2/1? (7/6)^5 *
> 1/2? 49/48 each step?

The answer is, all of the above. You're using a 7-limit extension of
Blackwood called "Blacksmith" temperament, which tempers out 28/27 as
well as 256/243. Since both of those commas vanish, all products and
quotients of the commas vanish as well, such as (28/27) / (256/243) =
64/63, and (28/27) / (64/63) = 49/48, and (64/63) / (49/48) =
1029/1024, and (49/48) * (1029/1024) = 16807/16384.

So you're actually pumping through the whole set of commas at once, in
your case. When you have a temperament where more than one comma
vanishes (a so-called "codimension > 1 temperament"), sometimes you
end up in situations where you can interpret the choice of comma pump
in more than one ways, or treat it as though it tempers an entire set
of commas out at once. For instance, you're also moving by three 8/7's
to get to a 3/2, meaning that 1029/1024 vanishes as well.

Thanks and welcome to the forum!

-Mike

Why is there a distinction between Blacksmith and Blackwood? As I think I
understand, any tuning in Blackwood has 2 cycles of 5edo separated by a
major-3rd-like interval and in Blacksmith, the 7/4 interval is the interval
implied by 4 steps of 5edo which any tuning of Blackwood would also
contain. Does it mean that if you don't use that interval in your piece of
music then it isn't in Blacksmith temperament?

Is there a name for an 11-limit extension of Blackwood? I made another
comma-pump, this time it goes around a cycle of 720 cent 5ths and uses
inversions of 4:7:11 chords, tempered to 55edo. If there isn't a name, I'd
enjoy the poetic injustice of it being called Blackbird:
https://soundcloud.com/james-fenn-theoretical/comma-pump-4-7-11-inversions

On 22 February 2013 21:52, Mike Battaglia <battaglia01@gmail.com> wrote:

> **
>
>
> Hello James,
>
>
> On Thu, Feb 21, 2013 at 8:49 AM, James Fenn james@...> wrote:
> >
> > Hi All,
> >
> > Since this is my first post, I'd like to start with a big thank-you to
> all
> > the amazing people who have posted here over the years. I've learned a
> > colossal amount with still loads to learn but loving every minute of it.
>
> Glad you enjoy it, welcome to the forum! Be sure to check out the
> Xenharmonic Alliance Facebook group as well
> (http://www.facebook.com/groups/xenharmonic2), along with the
> Xenharmonic Wiki (http://xenharmonic.wikispaces.com/), where there's
> even more people and information for you.
>
>
> > I constructed a sequence of 4:5:6:7 chords in a Blackwood temperament (in
> > 55deo which I know isn't optimal, just what I'm working in at the moment)
> > where the roots don't cycle in 720 cent "fifths" but in descending 240
> cent
> > "tones". In each step of the sequence, there are 2 common tones where the
> > "7/6" of the previous chord becomes the "8/7" of the new chord.
> >
> > What comma is this pumping? It's not 256/243. (7/8)^5 * 2/1? (7/6)^5 *
> > 1/2? 49/48 each step?
>
> The answer is, all of the above. You're using a 7-limit extension of
> Blackwood called "Blacksmith" temperament, which tempers out 28/27 as
> well as 256/243. Since both of those commas vanish, all products and
> quotients of the commas vanish as well, such as (28/27) / (256/243) =
> 64/63, and (28/27) / (64/63) = 49/48, and (64/63) / (49/48) =
> 1029/1024, and (49/48) * (1029/1024) = 16807/16384.
>
> So you're actually pumping through the whole set of commas at once, in
> your case. When you have a temperament where more than one comma
> vanishes (a so-called "codimension > 1 temperament"), sometimes you
> end up in situations where you can interpret the choice of comma pump
> in more than one ways, or treat it as though it tempers an entire set
> of commas out at once. For instance, you're also moving by three 8/7's
> to get to a 3/2, meaning that 1029/1024 vanishes as well.
>
> Thanks and welcome to the forum!
>
> -Mike
>
>
>