Recordings of comma pump examples

Hi folks.

After spending a few months with comma pumps in 2D temperaments, I've come to an interesting idea. Some chord progressions are so specific to particular temperaments that you could actually, just by listening to it, tell which temperament it is -- as long as the first and the last chord of the progression is the same. That means that one particular chord progression doesn't say much about what EDOs the temperament combines but it can say, very clearly, which interval is tempered out -- and what characteristic harmonic possibilities that gives us. So, if you want to take a listen, here's what I've ended up with:

http://www.sendspace.com/file/8ky2ng

Petr

PS: Chris, if you wish, you can upload it somewhere, I probably won't save it this copy "permanently" anywhere out there in the near future.

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:

> http://www.sendspace.com/file/8ky2ng

Your supposed to say what comma is being tempered out, so interested microtonal composers can find a use for it.

2011/4/22 Petr Pařízek <petrparizek2000@...>
>
> Hi folks.
>
> After spending a few months with comma pumps in 2D temperaments, I've come
> to an interesting idea. Some chord progressions are so specific to
> particular temperaments that you could actually, just by listening to it,
> tell which temperament it is -- as long as the first and the last chord of
> the progression is the same. That means that one particular chord
> progression doesn't say much about what EDOs the temperament combines but it
> can say, very clearly, which interval is tempered out -- and what
> characteristic harmonic possibilities that gives us. So, if you want to take
> a listen, here's what I've ended up with:
>
> http://www.sendspace.com/file/8ky2ng

What's the first one? After listening to it a thousand times, and
trying to work out the math in my head, I think it's 2048/2025. I
didn't even realize that what it was was supported in 12-tet at first
though.

The second one to me sounds like porcupine.

Third one is unfamiliar to me, might be negri...?

Fourth one is spinning my head upside down, but I'm hearing lots of
movement by major thirds, as well as some motion by some really small
interval that seems to be functioning similarly to a chromatic
semitone, so I'm going to guess magic, since it equates 128/125 and
25/24.

Fifth one - I thought that this was Hanson until I heard #6. Can't
figure it out. What is it...?

Sixth one - I couldn't tell it was Hanson until you made it obvious
with the bassline moving in minor thirds near the end.

Seventh - a bunch of 10/9's turning into 3/2? What temperament is
this? Is this tetracot?

-Mike

PS, you might want to check out the following comma pump - in 12-equal
notation, try C-E-G -> E-G-C -> Eb-Ab-C -> Eb-G-Bb -> Db-Gb-Bb -> Db F
Ab -> Cb Fb Ab -> Cb Eb Gb. So you can see in meantone the progression
drops by a chromatic semitone, except in JI it actually drops by
250/243, so in porcupine you end up back at C again. The whole thing
sounds very "natural" to me, maybe you'll dig it.

Gene wrote:

> Your supposed to say what comma is being tempered out, so interested > microtonal composers can
> find a use for it.

Maybe, but I'll wait for at least a few hours.
To be honest with you, in the first "edition" of the archive, I had the examples named. But then I thought: "Why should they be named if the chord progressions themselves have been deliberately made in a way which demonstrates what's tempered out in the first place?"
And then, I've observed some people having strange "prejudices" against certain temperaments and this way they can (without being sure about the temperament before listening carefully) possibly hear if those temperaments are really as "weird" as they claim them to be.

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What's the first one? After listening to it a thousand times, and
> trying to work out the math in my head, I think it's 2048/2025. I
> didn't even realize that what it was was supported in 12-tet at
> first though.
>
> The second one to me sounds like porcupine.

I agree, I think these are diaschismic and porcupine.
I thought number 3 might be magic, but I'm not sure why.
Number 4 was awesome; I don't know what it is either.

-Carl

On Fri, Apr 22, 2011 at 12:57 AM, Mike Battaglia <battaglia01@...> wrote:
>
> Seventh - a bunch of 10/9's turning into 3/2? What temperament is
> this? Is this tetracot?

Another good trick with tetracot is to do the same thing as mentioned
above with Porcupine, but start by going up a 3/2 and moving down
10/9's instead. So in meantone notation, you would do

C-E-G -> G-C-E -> G-B-D -> F Bb D -> F A C -> Eb Ab C -> Eb G Bb -> Db
Gb Bb -> Db F Ab -> Cb Fb Ab -> Cb Eb Gb

But if you were in a tetracot temperament, like say 34-tet, you'd end
up back where you started instead of at Cb.

BTW, how are you making these examples? Are you just using 12-note
subsets of various temperaments?

-Mike

On Fri, Apr 22, 2011 at 1:08 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's the first one? After listening to it a thousand times, and
> > trying to work out the math in my head, I think it's 2048/2025. I
> > didn't even realize that what it was was supported in 12-tet at
> > first though.
> >
> > The second one to me sounds like porcupine.
>
> I agree, I think these are diaschismic and porcupine.
> I thought number 3 might be magic, but I'm not sure why.

I'm pretty sure #3 is Negri, listen to it - the basic root movement is
something like 1/1 -> 6/5 -> 15/16, so he's moving down a diatonic
semitone each time. Then after three of those, he moves back up by a
5/4 and ends up back where he started. At least that's what I think is
going on.

> Number 4 was awesome; I don't know what it is either.

What was your take on #5? That one messed my whole head up.

These are incredible though - this is basically the kind of music I
was searching for when I joined this list. This is a total trip. I now
see the problem is that I've been focusing too much on the MOS's
generated by these temperaments, and not enough on the actual harmonic
structure that they imply. At this point, I might as well just think
purely harmonically and allow my ear to suggest proper scale closures
for whatever chords it is I choose to play.

-Mike

Mike is amazing. ;-)

Yes, the first two are indeed diaschismatic and porcupine.

> Third one is unfamiliar to me, might be negri...?

And it is.

> Fourth one is spinning my head upside down, but I'm hearing lots of
> movement by major thirds, as well as some motion by some really small
> interval that seems to be functioning similarly to a chromatic
> semitone, so I'm going to guess magic, since it equates 128/125 and
> 25/24.

And it is.

> Fifth one - I thought that this was Hanson until I heard #6. Can't
> figure it out. What is it...?

Probably that was because of the poor bass line. It was semisixths. If you have my "Run Down The Whistle 3", there's a more understandable one if you "scroll down" to about 2:33.

> Sixth one - I couldn't tell it was Hanson until you made it obvious
> with the bassline moving in minor thirds near the end.

Intentionally.

> Seventh - a bunch of 10/9's turning into 3/2? What temperament is
> this? Is this tetracot?

Absolutely correct. :-)

> PS, you might want to check out the following comma pump - in 12-equal
> notation, try C-E-G -> E-G-C -> Eb-Ab-C -> Eb-G-Bb -> Db-Gb-Bb -> Db F
> Ab -> Cb Fb Ab -> Cb Eb Gb. So you can see in meantone the progression
> drops by a chromatic semitone, except in JI it actually drops by
> 250/243, so in porcupine you end up back at C again. The whole thing
> sounds very "natural" to me, maybe you'll dig it.

I've also come to a similar progression recently but from the opposite side -- as in "C-E-G C-E-A D-F#-A D-F#-B E-G#-B E-G#-C# F#-A#-C# [C#-]E#-G#-C#". But I haven't delved into these a great deal since Herman already has a fairly obvious pump at the end of his porcupine overture.

Petr

PS: The magic progression was made via a Scala sequence file. In other cases, I tune more MIDI channels to the different generators (for example, a semisixth pump requires at least 16 tones per octave which is more than 12). Or if it's enough, I just take a 12-tone chain, which allows me to play in real time.

--- Mike Battaglia <battaglia01@...> wrote:

> > I agree, I think these are diaschismic and porcupine.
> > I thought number 3 might be magic, but I'm not sure why.
>
> I'm pretty sure #3 is Negri, listen to it - the basic root
> movement is something like 1/1 -> 6/5 -> 15/16, so he's
> moving down a diatoni semitone each time.

Yeah, you're right, and Petr just confirmed.

> What was your take on #5? That one messed my whole head up.

It sounded familiar from Petr's other stuff, but I didn't
recall Run Down the Whistle specifically.

> These are incredible though - this is basically the kind
> of music I was searching for when I joined this list.

Yep, and all with plain triads!

> At this point, I might as well just think purely harmonically
> and allow my ear to suggest proper scale closures for
> whatever chords it is I choose to play.

I like what I've heard of yours so far, but yes, this is
how I operate in 12-ET and I reckon a natural way to
operate in other systems also.

-Carl

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
> But then I thought: "Why should they be named if the chord
> progressions themselves have been deliberately made in a way which
> demonstrates what's tempered out in the first place?"

Because it's supposed to be a resource to help composers, not a game of musical chairs.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> These are incredible though - this is basically the kind of music I
> was searching for when I joined this list. This is a total trip. I now
> see the problem is that I've been focusing too much on the MOS's
> generated by these temperaments, and not enough on the actual harmonic
> structure that they imply.

Watch out or you'll end up like me. :)

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:

I hope to hell if Chris puts these up he puts them up with attached labels, so it does some good.

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:

Which of these temperaments are the ones people have a strange prejudice against?

2011/4/22 Petr Pařízek <petrparizek2000@...>
>
> Mike is amazing. ;-)

Haha, I'm getting better at this!

> > Fifth one - I thought that this was Hanson until I heard #6. Can't
> > figure it out. What is it...?
>
> Probably that was because of the poor bass line. It was semisixths. If you
> have my "Run Down The Whistle 3", there's a more understandable one if you
> "scroll down" to about 2:33.

I'm not familiar with semisixths, but a search of the tuning-math
archives says it tempers out 78732/78125. The generator is a 9/7, but
if we're in the 5-limit, what is it then? An augmented third?

I can't find Run Down The Whistle 3, I'm searching the tuning archives
now and nothing comes up. Can you point me to the file?

> > Seventh - a bunch of 10/9's turning into 3/2? What temperament is
> > this? Is this tetracot?
>
> Absolutely correct. :-)

Nice. I've never actually played with tetracot before, but I heard 3/2
being divided by 4 parts, and since this whole "cot" thing seems to
have some mysterious thing to do with 3/2, I went for it.

Do you work with comma pumps that involve higher error tunings? For
example, I was trying to find a suitable 5-limit "tricot" temperament,
so I thought maybe if you temper 75/64 really flat that'd do the
trick, such that three 75/64's give you a 3/2. This vanishes
140625/131072 - the page for it can be found here

http://x31eq.com/cgi-bin/rt.cgi?ets=10_16&limit=5&key=3_-1_2_2_5&error=14.705

16 and 26 are some tunings that support this temperament, and 26 might
be ideal for it. One comma pump that works is (try this in 26-equal)

Cmaj -> Em -> Bmaj -> D#maj -> F##m -> C##maj -> E##maj -> G###m ->
D###maj -> F####maj -> B###maj

Where the B###maj at the end is actually Cmaj again. This progression
actually ends a major second up in 12-equal, but turns out to be a
unison in 16 or 26 equal. It only ends up one step off in 19-equal
though. But then again, maybe the error's a bit too high for you. And
maybe there's a better 5-limit "tricot" as well.

> I've also come to a similar progression recently but from the opposite
> side -- as in "C-E-G C-E-A D-F#-A D-F#-B E-G#-B E-G#-C# F#-A#-C#
> [C#-]E#-G#-C#". But I haven't delved into these a great deal since Herman
> already has a fairly obvious pump at the end of his porcupine overture.

Right. I think you can make the one you wrote above also sound a bit
more forceful by throwing in some V-I's and iv-I's

C-E-G -> C#-E-A -> D-F#-A -> D#-F#-B -> E-G#-B -> E#-G#-C# -> F#-A#-C#
-> F#-A-D# -> C#-G#-E#!!! (now you're back at C again in porcupine
temperament)

I'm trying to also explore the use of tiny pieces of "functional"
harmony in the middle of these comma pumps, as per the above. To my
ears when you do that, they start sounding really natural, and after
some repetition don't sound any stranger than comma pumps in meantone.

As a last note, I'm about to make a post on tuning-math that attempts
to figure out exactly what this "characteristic sound" of each comma
pump is, so maybe you'll find that interesting. I think Gene worked a
lot of it out before with something called an "involution map" that
allows you to turn one temperament into another.

-Mike

On Fri, Apr 22, 2011 at 1:44 AM, Carl Lumma <carl@...> wrote:
>
> I like what I've heard of yours so far, but yes, this is
> how I operate in 12-ET and I reckon a natural way to
> operate in other systems also.

So when all of this theory developed, how did people originally see
temperaments playing a role in music? As generating scales, as
generating comma pumps, as both?

That is, the first insight was, for every comma, there's a
temperament. And then the next insight, or at least the next insight
that I got from everything, was that for every temperament, there is a
generator and there are MOS scales (and MODMOS scales).

But, Petr's work implies a different insight, which is that for every
temperament, there's also a pun. Was this the original "point" of all
of these temperaments, with the focus on MOS's coming later?

-Mike

On Fri, Apr 22, 2011 at 2:01 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > These are incredible though - this is basically the kind of music I
> > was searching for when I joined this list. This is a total trip. I now
> > see the problem is that I've been focusing too much on the MOS's
> > generated by these temperaments, and not enough on the actual harmonic
> > structure that they imply.
>
> Watch out or you'll end up like me. :)

Do you generally make a lot of use of comma pumps in your music? I
haven't noticed any when I listen to stuff like Chromosounds.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I'm not familiar with semisixths, but a search of the tuning-math
> archives says it tempers out 78732/78125. The generator is a 9/7, but
> if we're in the 5-limit, what is it then? An augmented third?

It's a tempered 162/125, tempered to 443 cents where it sounds like a decidedly sharp 9/7.

> As a last note, I'm about to make a post on tuning-math that attempts
> to figure out exactly what this "characteristic sound" of each comma
> pump is, so maybe you'll find that interesting. I think Gene worked a
> lot of it out before with something called an "involution map" that
> allows you to turn one temperament into another.

I've spent some thought on turning one temperament into another, with interesting results, but don't recall calling that an involution.

On Fri, Apr 22, 2011 at 3:06 AM, genewardsmith
<genewardsmith@...> wrote:
>
> > As a last note, I'm about to make a post on tuning-math that attempts
> > to figure out exactly what this "characteristic sound" of each comma
> > pump is, so maybe you'll find that interesting. I think Gene worked a
> > lot of it out before with something called an "involution map" that
> > allows you to turn one temperament into another.
>
> I've spent some thought on turning one temperament into another, with interesting results, but don't recall calling that an involution.

Ahem

http://xenharmonic.wikispaces.com/Microtonal+Music+by+Gene+Ward+Smith

"If you exchange 3/2 with 5/3 and 7/4 with 10/7, leaving 2 and 5
fixed, the resulting involution map also interchanges 126/125 and
64/63."

-Mike

Mike Battaglia wrote:

> So when all of this theory developed, how did people
> originally see temperaments playing a role in music? As
> generating scales, as generating comma pumps, as both?
[snip]
> Was this the original "point" of all of these temperaments,
> with the focus on MOS's coming later?

It all happened together. A timeline is probably the best
format. Omissions/corrections welcome (Graham?).

1997

I join; Paul is talking a lot about 81:80 defining meantone and
the impossibility of performing common-practice music in JI.
This view of temperament as 'natural' raises a lot of hackles in
the JI crowd.

1998

Paul's 22-tone paper is published in Xenharmonikon 17. It
describes what will later be called the pajara and injera
temperaments, and gives optimal generator tunings for pajara.
Monz and Paul will later discover a similar optimization had
been done for meantone by Woolhouse in 1835.

Kraig launches the Wilson archives, and Wilson's work on linear
temperaments and MOS will soon attract wider discussion.

I run into 225:224 when making 7-limit scales for my piano, and
issue a challenge to come up with what we would now call Fokker
blocks for this comma.

In an offlist thread, Monz spurs the rediscovery of Fokker's
papers on periodicity blocks.

1999

Paul, Paul Hahn, Graham and others begin playing with Fokker's
matrix formalism.

Dave publishes his "Harmonic erors in single..." and "...double-
chain-of-fifths tunings" spreadsheets, systematically exploring
tuning error as a function of generator size and for the first
time ever (we think) including the case of half-octave periods.

Dave publishes his "Method for optimally distributing any comma",
which foreshadows TOP.

Paul posts 'diatonic and decatonic scales as periodicity blocks'
to the tuning list.

2000

Negri, Hanson, and Valentine are re/discovered by Dave, Robert
Valentine, myself, and others (they won't be named for another
year or two).

2001

Joseph Pehrson challenges the list to come up with an ideal
7-limit scale of around 19 notes/oct. This leads us back to
225:224 and a lot of scales that are obviously related.
MIRACLE temperament is christened.

I ask Paul if there's a connection between MOS and Fokker blocks.
He comes up with his "Hypothesis", later proven by Gene.

Tuning-math is started, to relieve the main list of all the talk
about MIRACLE.

Herman creates the first 2-D scale tree, showing the relationship
between rank 1 and rank 2 temperaments.

Gene joins; introduces wedge products and other algebraic tools.
The temperament explosion begins.

2002

Dave and George begin developing Sagittal notation.

2003

Gene and I start playing with tempered octaves.

2004

Paul demonstrates codimension-1 TOP tuning on tuning-math.

2005

Paul leaves the lists.

2006

A Middle Path is published in Xenharmonikon 18.

Graham releases primerr, his first of many PDFs that will
ultimately result in the adoption of the Tenney-Euclidean error
and complexity metrics.

2007

Gene leaves the lists.

2008 ... 2009 ...

2010

Gene rejoins, begins contributing to the xenharmonic wiki.

-Carl

"Carl Lumma" <carl@...> wrote:
> Mike Battaglia wrote:
>
> > So when all of this theory developed, how did people
> > originally see temperaments playing a role in music? As
> > generating scales, as generating comma pumps, as both?
> [snip]
> > Was this the original "point" of all of these
> > temperaments, with the focus on MOS's coming later?
>
> It all happened together. A timeline is probably the best
> format. Omissions/corrections welcome (Graham?).

I was taking "original" a different way, and looking at the
history of schismatic temperament. What I've discovered is
that Helmholtz did not use the term "schismatische
Verwechse?lung". You can find this by searching Google for
"schismatische Verwechslung" and reading the Google Books
preview of Liberty Manik's "Das arabische Tonsystem im
Mittelalter" (The Arabian tone system in the Middle Ages).
Unfortunately I exhausted my previews before I could track
down who did coin it. In case it helps another researcher,
here are the pages (according to the index) where
"schismatische Verwechslung" is covered: 77, 88, 91-96,
101, 108, 122, 127. From what I remember, Hugo Riemann was
ahead of Helmholtz, but I couldn't find the page he said
that.

Ellis was pretty early with "skhismic temperament" in 1885.
Shohé Tanaka had "schismatische Verwechselung" in 1890.

> 1997
>
> I join; Paul is talking a lot about 81:80 defining
> meantone and the impossibility of performing
> common-practice music in JI. This view of temperament as
> 'natural' raises a lot of hackles in the JI crowd.

I joined the list in 1996 but didn't post until early
1997. By then I'd also been in contact with Brian McLaren
about my matrices and at some point he sent me copies of a
load of papers, including Fokker's English one.

I don't have a clear idea of what I knew at this point and
what I learned from the list. I was much impressed by the
community in general and Paul Erlich in particular. The
main influence was that it got me interested in real-time
MIDI retuning and so I wrote MIDI Relay.

I have backups of my old messages somewhere, and I'll find
them one day I'm sure. I wrote an exposition of the
matrices, and sent it to (I think) Manuel Op De Coul (in the
belief that he was Paul Erlich, because Paul wasn't using
his own e-mail at the time) asking if the list would be
interested in it. I decided not to post it.

> 1998
>
> Paul's 22-tone paper is published in Xenharmonikon 17. It
> describes what will later be called the pajara and injera
> temperaments, and gives optimal generator tunings for
> pajara. Monz and Paul will later discover a similar
> optimization had been done for meantone by Woolhouse in
> 1835.

I'm surprised the 22-tone paper was so late. I had a
private conversation with Paul around that time, anyway.
The important thing, to me, was that I learned how to deal
with cases where the octave wasn't the period. Before
that, I had matrices to describe meantone and schismatic,
and I could invert to them to get between mappings (vals)
and unison vectors. That method gave a fractional matrix
for the diaschisma. I thought it meant the system was ill
formed, but Paul had already dealt with it, and showed the
octave was divided in two.

> Kraig launches the Wilson archives, and Wilson's work on
> linear temperaments and MOS will soon attract wider
> discussion.

Some of us had dead tree copies of a few papers, and I
think there was already discussion of them before the
archive went up.

> In an offlist thread, Monz spurs the rediscovery of
> Fokker's papers on periodicity blocks.

It's a shame that was offlist because I don't remember
anything about it.

> 2000
>
> Negri, Hanson, and Valentine are re/discovered by Dave,
> Robert Valentine, myself, and others (they won't be named
> for another year or two).

The thing is, I have no recollection of knowing about these
things until 2001. But this page, last time I could read
it, mentioned me as having been part of the discussion:

http://dkeenan.com/Music/ChainOfMinor3rds.htm

What was clear was that regular temperaments, MOS scales,
keyboard mappings, and notations were all tied up
together. There was talk about generalized diatonics, but
also MOS scales as keyboard tunings.

At this point I reckoned I had all the theory I needed, and
should concentrate on making music with it. I kept one eye
on the Tuning List in case something interesting came up.

> 2001
>
> Joseph Pehrson challenges the list to come up with an
> ideal 7-limit scale of around 19 notes/oct. This leads
> us back to 225:224 and a lot of scales that are obviously
> related. MIRACLE temperament is christened.

The 10 year anniversary's approaching. Are any celebrations
planned?

What I did fairly quickly was to work out a set of unison
vectors or commas to produce Miracle. That also entailed
recognizing generators other than a fifth or a simple
division of a fifth (Mohajira) which seems obvious but had
held me up until then.

Dave Keenan wrote a program (or spreadsheet) to search for
linear temperaments by choosing the size of the
generator. This already threw up Orwell as a standout,
but confirmed Miracle as the best if you wanted a lower
error. Then I wrote a program to search by pairing off
consistent equal temperaments, which was more general
because it could handle periods other than the octave. I
also wrote a program to take a set of unison vectors and
find the temperament they implied (given an optimization),
which meant hammering down the matrix inversion method to
deal with arbitrary periods and generators. By the end of
the year I had it working solidly and dealing with torsion
(which had come up in the Fokker discussion before).

Monz and I both came up with decimal notations at the same
time. They were fairly obvious as generalizations staff
notation applied to Miracle. A number of us were working
with Miracle. I produced something in Magic, with a keyboard
mapping and notation based on Miracle because I hadn't
worked out a native Magic notation then. I decided Miracle
was the future and concentrated on that. Other
temperaments were being cataloged and tuned up. There was
a huge amount of progress that year. Gene turned up in the
middle of it all.

Graham

Mike wrote:

> I'm not familiar with semisixths, but a search of the tuning-math
> archives says it tempers out 78732/78125. The generator is a 9/7, but
> if we're in the 5-limit, what is it then? An augmented third?

It's 162/125. Stacking two of these approximates 5/3.

> I can't find Run Down The Whistle 3, I'm searching the tuning archives
> now and nothing comes up. Can you point me to the file?

Well, for the time being, I've uploaded both of my semisixths pieces here (hope I can leave it there as long as needed):
http://dl.dropbox.com/u/8497979/StillDifferently.mp3
http://dl.dropbox.com/u/8497979/DownWhis.mp3

> http://x31eq.com/cgi-bin/rt.cgi?ets=10_16&limit=5&key=3_-1_2_2_5&error=14.705

Thanks. I haven't tried that one. But I know of a similar one which I think Herman could say the most about. An interesting view on that is here:
/tuning/topicId_53750.html#53750

> Where the B###maj at the end is actually Cmaj again. This progression
> actually ends a major second up in 12-equal, but turns out to be a
> unison in 16 or 26 equal. It only ends up one step off in 19-equal
> though. But then again, maybe the error's a bit too high for you. And
> maybe there's a better 5-limit "tricot" as well.

This reminds me very strongly of my recent ideas for the temperament which I finally decided to call "sixix" (i.e. 6 generators make a wider approximation of 16/5) and which tempers out 3125/2916. You can still find it in my folder in the Tuning Files here.

> C-E-G -> C#-E-A -> D-F#-A -> D#-F#-B -> E-G#-B -> E#-G#-C# -> F#-A#-C#
> -> F#-A-D# -> C#-G#-E#!!! (now you're back at C again in porcupine
> temperament)

Wow, what a "romanticizing" porcupine pump! :-D Well, I'll leave these to you and others, probably.

> I'm trying to also explore the use of tiny pieces of "functional"
> harmony in the middle of these comma pumps, as per the above. To my
> ears when you do that, they start sounding really natural, and after
> some repetition don't sound any stranger than comma pumps in meantone.

Agreed.

> As a last note, I'm about to make a post on tuning-math that attempts
> to figure out exactly what this "characteristic sound" of each comma
> pump is, so maybe you'll find that interesting. I think Gene worked a
> lot of it out before with something called an "involution map" that
> allows you to turn one temperament into another.

That sounds "like a challenge". I can see some interesting ideas and conclusions lurking there "deep in the ground" so maybe you'll eventually manage to dig it out one day.

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> So when all of this theory developed, how did people originally see
> temperaments playing a role in music? As generating scales, as
> generating comma pumps, as both?

Before the whole regular temperament thing, I saw it always in terms of comma pumps.

> But, Petr's work implies a different insight, which is that for every
> temperament, there's also a pun.

Juat to get a canonical list of puns you'd need to get a canonical list of commas first.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Do you generally make a lot of use of comma pumps in your music?

Often enough there are quite a lot of them.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> "If you exchange 3/2 with 5/3 and 7/4 with 10/7, leaving 2 and 5
> fixed, the resulting involution map also interchanges 126/125 and
> 64/63."

Right. That's an involution, but you don't always get involutions.

On Fri, Apr 22, 2011 at 10:02 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > "If you exchange 3/2 with 5/3 and 7/4 with 10/7, leaving 2 and 5
> > fixed, the resulting involution map also interchanges 126/125 and
> > 64/63."
>
> Right. That's an involution, but you don't always get involutions.

Is this different from what I posted on tuning-math? I'm trying to
see, if you remap the intervals from porcupine into 12-tet, where the
internal logic is inconsistent. Maybe a better way to start is by
remapping them from porcupine into meantone. Have you or has anyone
else worked with this sort of thing before? I think that by doing
this, we'll be able to understand better when porcupine comma pumps
make our perception "shift," which I guess really just means mapping
all of the places that 250/243 appears in meantone (and hence these
will be the places that porcupine comma pumps don't line up and sound
trippy).

-Mike

Hi Petr,

I've added it to your folder
http://micro.soonlabel.com/petr_parizek/
and your download page
http://micro.soonlabel.com/petr_parizek/~Petr_Parizek_Files.html

Have a great day!

Chris

2011/4/22 Petr Pařízek <petrparizek2000@...>

>
>
> Hi folks.
>
> After spending a few months with comma pumps in 2D temperaments, I've come
> to an interesting idea. Some chord progressions are so specific to
> particular temperaments that you could actually, just by listening to it,
> tell which temperament it is -- as long as the first and the last chord of
> the progression is the same. That means that one particular chord
> progression doesn't say much about what EDOs the temperament combines but
> it
> can say, very clearly, which interval is tempered out -- and what
> characteristic harmonic possibilities that gives us. So, if you want to
> take
> a listen, here's what I've ended up with:
>
> http://www.sendspace.com/file/8ky2ng
>
> Petr
>
> PS: Chris, if you wish, you can upload it somewhere, I probably won't save
> it this copy "permanently" anywhere out there in the near future.
>
>
>

Hi Graham,

> I have backups of my old messages somewhere, and I'll find
> them one day I'm sure. I wrote an exposition of the
> matrices, and sent it to (I think) Manuel Op De Coul (in the
> belief that he was Paul Erlich, because Paul wasn't using
> his own e-mail at the time) asking if the list would be
> interested in it. I decided not to post it.

I seem to remember seeing your page on matrices
http://x31eq.com/matrices.htm
pretty early on, but I searched all our offlist mail and your
first use of "matri" was in Jan 1999. I also searched your
list posts a bit, but I got the list in digest form at that
time and they're hard to search.

> > 1998
> > Paul's 22-tone paper is published in Xenharmonikon 17.
>
> I'm surprised the 22-tone paper was so late.

He had a draft of it when I joined, and originally wrote it
as a thesis (or perhaps extra credit) paper at Yale. He was
class of '94, so I'll guess '93 or '94. But it was under
revision until XH17 came out.

> Some of us had dead tree copies of a few [Wilson] papers,
> and I think there was already discussion of them before
> the archive went up.

Denny had given me a bunch of his stuff before I joined
the list. I visited him in summer '98 and subsequently
worked with him to dig up Boomsliter & Creel papers. We
were in regular correspondence until about 2001.

Of course his stuff was discussed (Warren Burt's articles
in 1/1 spurred some). But the Wilson archives really made
a difference in the level of attention.

> > In an offlist thread, Monz spurs the rediscovery of
> > Fokker's papers on periodicity blocks.
>
> It's a shame that was offlist because I don't remember
> anything about it.

I checked and the participants were Monz, Paul, Paul Hahn,
John Chalmers, John Starrett, John Haluska, David Canright,
Dave Hill, and myself.

> What was clear was that regular temperaments, MOS scales,
> keyboard mappings, and notations were all tied up
> together.

Yep.

> At this point I reckoned I had all the theory I needed, and
> should concentrate on making music with it. I kept one eye
> on the Tuning List in case something interesting came up.

I concluded I had all the theory I needed on a daily basis
from 1997 to 2010.

> > 2001
> > MIRACLE temperament is christened.
>
> The 10 year anniversary's approaching. Are any celebrations
> planned?

I think Mike is planning something...

> Then I wrote a program to search by pairing off
> consistent equal temperaments, which was more general
> because it could handle periods other than the octave. I
> also wrote a program to take a set of unison vectors and
> find the temperament they implied (given an optimization),
> which meant hammering down the matrix inversion method to
> deal with arbitrary periods and generators. By the end of
> the year I had it working solidly and dealing with torsion
> (which had come up in the Fokker discussion before).

Was that 2001? Were you using Java?

> Monz and I both came up with decimal notations at the same
> time. They were fairly obvious as generalizations staff
> notation applied to Miracle.

I believe I was the first to suggest a decatonic notation
on the 10-note MOS, but I didn't make any music or scores.
I was working with 8-nominal hanson notations in 1999, and
posted some mp3s at the time.

> I produced something in Magic,

I know MAGIC came soon after MIRACLE. Can you track it down?

I know I was looking at your "catalog of linear temperaments"
page in 2001 or 2002, and it may have been out in 2000
already. Can you track that down?

-Carl

On Fri, Apr 22, 2011 at 5:04 AM, Carl Lumma <carl@...> wrote:
>
> Mike Battaglia wrote:
>
> > So when all of this theory developed, how did people
> > originally see temperaments playing a role in music? As
> > generating scales, as generating comma pumps, as both?
> [snip]
> > Was this the original "point" of all of these temperaments,
> > with the focus on MOS's coming later?
>
> It all happened together. A timeline is probably the best
> format. Omissions/corrections welcome (Graham?).

Nice. Thanks for taking the time to lay that all out. Hopefully we'll
be able to add some stuff to that timeline this year.

-Mike

On 4/22/2011 2:53 AM, Mike Battaglia wrote:
> On Fri, Apr 22, 2011 at 1:44 AM, Carl Lumma<carl@...> wrote:
>>
>> I like what I've heard of yours so far, but yes, this is
>> how I operate in 12-ET and I reckon a natural way to
>> operate in other systems also.
>
> So when all of this theory developed, how did people originally see
> temperaments playing a role in music? As generating scales, as
> generating comma pumps, as both?
>
> That is, the first insight was, for every comma, there's a
> temperament. And then the next insight, or at least the next insight
> that I got from everything, was that for every temperament, there is a
> generator and there are MOS scales (and MODMOS scales).
>
> But, Petr's work implies a different insight, which is that for every
> temperament, there's also a pun. Was this the original "point" of all
> of these temperaments, with the focus on MOS's coming later?
>
> -Mike

In my case, the porcupine comma pump pre-dated the discovery of porcupine as a regular temperament. I also noticed that an interval I was using in a 5-limit JI scale was close to 7/4, and got the idea of tempering out 126/125 by using tempered major and minor thirds as generators (the rank-3 starling temperament). I don't recall the exact dates, but I believe this was around the same time that miracle and magic were being discussed on the tuning list. So the comma pumps and other sorts of "puns" have been around all the while.

On 4/22/2011 2:47 AM, Mike Battaglia wrote:
> 2011/4/22 Petr Pařízek<petrparizek2000@...>
>>
>> Mike is amazing. ;-)
>
> Haha, I'm getting better at this!
>
>>> Fifth one - I thought that this was Hanson until I heard #6. Can't
>>> figure it out. What is it...?
>>
>> Probably that was because of the poor bass line. It was semisixths. If you
>> have my "Run Down The Whistle 3", there's a more understandable one if you
>> "scroll down" to about 2:33.
>
> I'm not familiar with semisixths, but a search of the tuning-math
> archives says it tempers out 78732/78125. The generator is a 9/7, but
> if we're in the 5-limit, what is it then? An augmented third?
>
> I can't find Run Down The Whistle 3, I'm searching the tuning archives
> now and nothing comes up. Can you point me to the file?
>
>>> Seventh - a bunch of 10/9's turning into 3/2? What temperament is
>>> this? Is this tetracot?
>>
>> Absolutely correct. :-)
>
> Nice. I've never actually played with tetracot before, but I heard 3/2
> being divided by 4 parts, and since this whole "cot" thing seems to
> have some mysterious thing to do with 3/2, I went for it.
>
> Do you work with comma pumps that involve higher error tunings? For
> example, I was trying to find a suitable 5-limit "tricot" temperament,
> so I thought maybe if you temper 75/64 really flat that'd do the
> trick, such that three 75/64's give you a 3/2. This vanishes
> 140625/131072 - the page for it can be found here
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=10_16&limit=5&key=3_-1_2_2_5&error=14.705

As Petr has pointed out, this is a 5-limit version of lemba. I'm most familiar with the 7-limit version, and indeed 26 is a near optimal tuning for it.

> 16 and 26 are some tunings that support this temperament, and 26 might
> be ideal for it. One comma pump that works is (try this in 26-equal)
>
> Cmaj -> Em -> Bmaj -> D#maj -> F##m -> C##maj -> E##maj -> G###m ->
> D###maj -> F####maj -> B###maj
>
> Where the B###maj at the end is actually Cmaj again. This progression
> actually ends a major second up in 12-equal, but turns out to be a
> unison in 16 or 26 equal. It only ends up one step off in 19-equal
> though. But then again, maybe the error's a bit too high for you. And
> maybe there's a better 5-limit "tricot" as well.

If you mean dividing 3/2 into 3 parts, there's gorgo

[<1 1 1|, <0 3 7|]

or you could try rodan

[<1 1 -1|, <0 3 17|]

But like lemba, these are better as higher-limit temperaments. I've used 7-limit gorgo, and rodan tends to come up in higher limit temperament searches although I haven't used it for anything.

On 23 April 2011 00:28, Carl Lumma <carl@...> wrote:

> I seem to remember seeing your page on matrices
> http://x31eq.com/matrices.htm
> pretty early on, but I searched all our offlist mail and your
> first use of "matri" was in Jan 1999.  I also searched your
> list posts a bit, but I got the list in digest form at that
> time and they're hard to search.

The Wayback Machine has it as
http://www.cix.co.uk/~gbreed/matrices.htm back to October 1999, which
isn't much help. I think it was based on recycled Tuning List posts
anyway. Oh, wait, http://www.cix.co.uk/~gbreed/lintemp.htm is there
December 1998.

http://tinyurl.com/5vcxqze

>> > In an offlist thread, Monz spurs the rediscovery of
>> > Fokker's papers on periodicity blocks.
>
> I checked and the participants were Monz, Paul, Paul Hahn,
> John Chalmers, John Starrett, John Haluska, David Canright,
> Dave Hill, and myself.

There you go. I noticed people started talking about Fokker's work,
and assumed it was something everybody knew and had always known
about.

>> Then I wrote a program to search by pairing off
>> consistent equal temperaments, which was more general
>> because it could handle periods other than the octave.  I
>> also wrote a program to take a set of unison vectors and
>> find the temperament they implied (given an optimization),
>> which meant hammering down the matrix inversion method to
>> deal with arbitrary periods and generators.  By the end of
>> the year I had it working solidly and dealing with torsion
>> (which had come up in the Fokker discussion before).
>
> Was that 2001?  Were you using Java?

All of that was 2001. I kept working on it, and searched inconsistent
ETs better, thanks to feedback from tuning-math. It was originally in
Python 1.5.2 because that's all that worked on my Revo, and eventually
my website. The unison vector code needed Numeric Python until I
re-wrote it with wedgies.

> I know MAGIC came soon after MIRACLE.  Can you track it down?

I formally announced it in tuning-math message 586, July 28th 2001.

http://tinyurl.com/3epw4bh

From that, it's obvious that I was using it and had the name before,
and it was in the catalog with the expanded backronym. I checked the
Wayback Machine before for a copy of that catalog, but didn't come up
with anything. I've kept my eye out ever since for earlier references
to Magic. The only thing I found was George Secor looking at 22&19,
among other pairs, but not being interested because he was looking for
something in the 13-limit.

> I know I was looking at your "catalog of linear temperaments"
> page in 2001 or 2002, and it may have been out in 2000
> already.  Can you track that down?

Not before August 2001 in the Wayback Machine. I don't think it was
there before Miracle because there wasn't much to catalog . . . or, at
least, I wasn't keeping up with what there was.

Graham

Thanks. I don't know if the timeline is worth saving, but
I've saved a copy on my hard drive anyway. If you or anyone
else would like items added, please supply them! Put a month
on so I can get them in the right order under each year.

It looks like you were 3-6 months ahead of the rest of the
gang on some things but nobody was paying much attention.
I remember looking at lintemp.htm and thinking it was an
overkill version of something I already understood. To
quote James Burke, How wrong can you get?

-Carl

--- Graham Breed <gbreed@...> wrote:

> The Wayback Machine has it as
> http://www.cix.co.uk/~gbreed/matrices.htm back to October 1999,
> which isn't much help. I think it was based on recycled Tuning
> List posts anyway. Oh, wait,
> http://www.cix.co.uk/~gbreed/lintemp.htm is there
> December 1998.
> http://tinyurl.com/5vcxqze

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> It all happened together. A timeline is probably the best
> format. Omissions/corrections welcome (Graham?).
...
> 2001
...
> Gene joins; introduces wedge products and other algebraic tools.
> The temperament explosion begins.

Carl,

You need to include Graham Breed's contribution to the temperament explosion. If I remember rightly, Graham's temperament finder was first written without the benefit of the Grassmann Algebra that Gene introduced.

-- Dave Keenan

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> The thing is, I have no recollection of knowing about these
> things until 2001. But this page, last time I could read
> it, mentioned me as having been part of the discussion:
>
> http://dkeenan.com/Music/ChainOfMinor3rds.htm

Hi Graham,

Yes it does, in 1998. Its URL is now
http://dkeenan.com/Music/ChainOfMinor3rds.htm

Any other old links to my articles should be amenable to the same transformation.

-- Dave Keenan