Some rather unfamiliar 5-limit temperaments

Hi again,

recently, I've suggested a 5-limit tuning based on tempering out 1953125/1889568. Some time later, I typed these numbers into Google and it found a message from the Tuning Math list where this particular temperament was called "shibboleth". I wonder how it happened overtime that temperaments like porcupine or blackwood are of much greater interest to some musicians than this one is (I haven't even found it in those 5-limit temperament lists to which some of us have already posted links a few times).
The listing in question is here in message # 5605:
www.robertinventor.com/tuning-math/s___6/msg_5600-5624.html

Petr

Petr Pařízek wrote:

> recently, I've suggested a 5-limit tuning based on tempering out > 1953125/1889568. Some time later, I typed these numbers into Google and it > found a message from the Tuning Math list where this particular temperament > was called "shibboleth". I wonder how it happened overtime that temperaments > like porcupine or blackwood are of much greater interest to some musicians > than this one is (I haven't even found it in those 5-limit temperament lists > to which some of us have already posted links a few times).
> The listing in question is here in message # 5605:
> www.robertinventor.com/tuning-math/s___6/msg_5600-5624.html

It immediately dropped off the lists because it has a high complexity for a modest accuracy. Perhaps over time it'll be recognized for other virtues. If you think it's neglected you can begin the rehabilitation.

Graham

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>
Hi Petr,
>
> recently, I've suggested a 5-limit tuning based on tempering out
> 1953125/1889568 .....

http://www.robertinventor.com/tuning-math/s___6/msg_5600-5624.html#udpm4
baptizes that by labeling it as

"shibboleth 1953125/1889568 ..." = 5^9 / (3^10 * 2^5)

1200 * ln (1 953 125 / 1 889 568)) / ln(2) = ~57.2734161...Cents

I really see nothing special in that 'rather-unfamiliar' ratio,
except that both nominator and demominator consist all two of
the usual
http://www.rcbj.net/Articles/HammingNumbers.html.html
http://en.wikipedia.org/wiki/Hamming_numbers
property.

Quest:
What's concrete the practical use of the so called "shibboleth"?

bye
A.S.

Graham wrote:

> It immediately dropped off the lists because it has a high
> complexity for a modest accuracy.

For one thing, I think there are other temperaments which are less accurate and are included there, like gorgo, for example. Maybe I should try that one to see if it should be more prospective than „shibboleth“.

For another thing, what about the one which tempers out 3125/2916? If gorgo can be of any use, why should this not be then?

Petr

Petr Pařízek wrote:

> For one thing, I think there are other temperaments which are less accurate and > are included there, like gorgo, for example. Maybe I should try that one to see > if it should be more prospective than „shibboleth“.

Included where? I have gorgo on a 7-limit list. Its 5-limit comma's [4 -7 3> for anybody following along at home. It looks pretty useless to me but it's simpler than shibboleth (4 major triads in the 11 note MOS compared to 1).

> For another thing, what about the one which tempers out 3125/2916? If gorgo can > be of any use, why should this not be then?

It's better than gorgo by the odd-limit scores, so sure, why not? Maybe it drops off some lists because the TOP complexity comes out worse. There are more generators to a 3:2.

Graham

Petr Pařízek wrote:
> Hi again,
> > recently, I've suggested a 5-limit tuning based on tempering out > 1953125/1889568. Some time later, I typed these numbers into Google and it > found a message from the Tuning Math list where this particular temperament > was called "shibboleth". I wonder how it happened overtime that temperaments > like porcupine or blackwood are of much greater interest to some musicians > than this one is (I haven't even found it in those 5-limit temperament lists > to which some of us have already posted links a few times).
> The listing in question is here in message # 5605:
> www.robertinventor.com/tuning-math/s___6/msg_5600-5624.html
> > Petr

It could be in this case that the 7-limit version "superkleismic" has drawn more attention.

[<1, 4, 5, 2], <0, -9, -10, 3]]

Extended versions of this temperament show up on my "best rank 2 temperaments" list for 7-, 11-, 13-, 17-, 19-, 23-, 29-, and 31-limit but not 5-limit. Here are the two 31-limit mappings that show up on my list:

[<1, 4, 5, 2, 4, 8, 10, 8, 8, 7, 13], <0, -9, -10, 3, -2, -16, -22, -14, -13, -8, -30]>

[<1, 4, 5, 2, 4, 8, 10, 8, 4, 7, 13], <0, -9, -10, 3, -2, -16, -22, -14, 2, -8, -30]>

Maybe one of these could keep the name "superkleismic" and the other one could revive the old "shibboleth" name.

Graham Breed wrote:
> Petr Pařízek wrote:
> >> For one thing, I think there are other temperaments which are less accurate and >> are included there, like gorgo, for example. Maybe I should try that one to see >> if it should be more prospective than „shibboleth“.
> > Included where? I have gorgo on a 7-limit list. Its > 5-limit comma's [4 -7 3> for anybody following along at > home. It looks pretty useless to me but it's simpler than > shibboleth (4 major triads in the 11 note MOS compared to 1).
> >> For another thing, what about the one which tempers out 3125/2916? If gorgo can >> be of any use, why should this not be then?
> > It's better than gorgo by the odd-limit scores, so sure, why > not? Maybe it drops off some lists because the TOP > complexity comes out worse. There are more generators to a 3:2.

[<1, 3, 4], <0, -5, -6]>

if that's the one, it's around the same error and complexity as (only slightly worse than)

[<7, 11, 16], <0, 0, 1]>

the 2187/2048 temperament I've mentioned on a couple of occasions in the tuning math list, which barely makes my 5-limit "best temperaments" list. (That's according to the particular measures of error and complexity I used to make that list, which IIRC was the same as in Paul Erlich's _Middle Path_ paper.)

The thing is, there's just so many temperaments, many of them that could be musically useful are buried in these huge lists. There could still be some gems waiting to be discovered that just haven't attracted much attention yet, and I've found musical uses for even some of the more "questionable" temperaments.

Herman wrote:

> The thing is, there's just so many temperaments, many of them that could
> be musically useful are buried in these huge lists. There could still be
> some gems waiting to be discovered that just haven't attracted much
> attention yet, and I've found musical uses for even some of the more
> "questionable" temperaments.

For me personally, the 3-limit approximations in gorgo are almost unusable. BTW: This is what the comma pump for the 3125/2916 temperament sounds like: www.sendspace.com/file/s5t9bz

Petr

PS: It’s interesting that if you try to play this in 12-equal, you end up two steps higher, not one.

Petr Pařízek wrote:
> 
> > Herman wrote:
> >> The thing is, there's just so many temperaments, many of them that could
>> be musically useful are buried in these huge lists. There could still be
>> some gems waiting to be discovered that just haven't attracted much
>> attention yet, and I've found musical uses for even some of the more
>> "questionable" temperaments.
> > For me personally, the 3-limit approximations in gorgo are almost > unusable. BTW: This is what the comma pump for the 3125/2916 temperament > sounds like: www.sendspace.com/file/s5t9bz

That has an interesting effect almost like an endlessly rising staircase. It does seem that it could be musically useful. I don't see this on the chart at http://tonalsoft.com/enc/e/equal-temperament.aspx but it would be just to the right of porcupine, passing through 7, 18, 25, and 32.

I had an example on my old Comcast web page that used gorgo temperament. I chose it as much for its melodic properties as anything else, but it also illustrates some of the potential for 7-limit harmony.

/tuning/files/hm_examples/gorgo-example.mp3

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>
> BTW: This is what the comma pump for the 3125/2916 temperament
> sounds like: www.sendspace.com/file/s5t9bz

Wow, that's one of the coolest things I've heard in a while.
It really proves that puns are real illusions: it sounds like
the pitch is rising.

It's one point that's getting lost in this whole JI vs. meantone
thread. Not only is there a wealth of existing music which
assumes the disappearance of the syntonic comma (and thus can't
be adequately performed in strict JI), there is a like quantity
of music that assumes the disappearance of other commas, which
has yet to be written.

-Carl

Herman wrote:

> it would be just to the right of porcupine, passing through 7, 18,
> 25, and 32.

Okay, now the remaining question is if this temperament is „musically fruitful“ enough to be worth finding a name for. But I’m not sure if I can comment on this because there were other people who have, unlike me, found something of use in temperaments like father or bug.

> I had an example on my old Comcast web page that used gorgo temperament.
> I chose it as much for its melodic properties as anything else, but it
> also illustrates some of the potential for 7-limit harmony.

Well, you seem to be more successful than I was. I think you would also be better at choosing the „right“ amount of tempering for dicot (another temperament that I was never satisfied with). Maybe my problem there was that I find the two intervals of 6/5 and 5/4 to be of such a different nature that I simply don’t know how to „effectively“ treat a tuning where they turn into one thing. Perhaps the main purpose is to use neutral thirds? Another possibility would be to use a generator equal to the cube root of either 15/8 (which would make 3/2 and 5/4 mistuned by the same amount)or 9/5 (then this would go for 3/2 and 6/5). But this would mean a clear preference for major thirds over minor thirds or vice versa. Okay, the cbrt(15/8) choice sounds acceptable to me, maybe I’ll try it one day too - and hear what it sounds like.

Petr

Carl wrote:

> Not only is there a wealth of existing music which
> assumes the disappearance of the syntonic comma (and thus can't
> be adequately performed in strict JI), there is a like quantity
> of music that assumes the disappearance of other commas, which
> has yet to be written.

Exactly. This is what I've been thinking about for some time. IIRC, the first time I used some sort of "mathematical models" for comma pumps in actual music was in my "Run Down The Whistle 3", where I calculated the version which used 9 chords in the 16-tone semisixthscale (which eventually turned out to be the least number of chords and the least number of tones in the scale with which this was possible). And last week, I did something similar for porcupine when I made "Among Other Things". Details here: /tuning/topicId_81664.html#81696

Petr

>
> It's one point that's getting lost in this whole JI vs. meantone
> thread. Not only is there a wealth of existing music which
> assumes the disappearance of the syntonic comma (and thus can't
> be adequately performed in strict JI), there is a like quantity
> of music that assumes the disappearance of other commas, which
> has yet to be written.
>

I must disagree with this. All music can be performed in strict JI.
Just await the Lasso example I'll post later today and I'll think even you
will be convinced :)

Marcel

Marcel wrote:

> I must disagree with this. All music can be performed in strict JI.
> Just await the Lasso example I'll post later today and I'll think even you
will be convinced :)

Have you heard the comma pump demo which I posted yesterday? If you have,
then I wonder what you're still trying to convince us about.

Petr

I looked for this, but I didn't see it?

Can you post the link? I'd like to hear it.

On Mar 1, 2009, at 9:22 AM, Petr Pařízek wrote:

> Marcel wrote:
>
>> I must disagree with this. All music can be performed in strict JI.
>> Just await the Lasso example I'll post later today and I'll think >> even you
> will be convinced :)
>
> Have you heard the comma pump demo which I posted yesterday? If you
> have,
> then I wonder what you're still trying to convince us about.
>
> Petr
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

>
> Have you heard the comma pump demo which I posted yesterday? If you have,
> then I wonder what you're still trying to convince us about.
>

Yes I have.
And nothing proves it can't be played in pure JI.
Not saying I'm going to attempt the interpretation but you can't say it
can't be played in pure JI either.

Marcel

Caleb wrote:

> I looked for this, but I didn't see it?
> Can you post the link? I'd like to hear it.

It should still be there: www.sendspace.com/file/s5t9bz

Petr

> IIRC, the first time I used some sort of "mathematical models"
> for comma pumps in actual music was in my "Run Down The Whistle 3",
> where I calculated the version which used 9 chords in the 16-tone
> semisixthscale (which eventually turned out to be the least number
> of chords and the least number of tones in the scale with which
> this was possible). And last week, I did something similar for
> porcupine when I made "Among Other Things". Details here:
> /tuning/topicId_81664.html#81696
>
> Petr

I keep about 1/3 of the files you post in a folder on my hard drive
with your name on it, and these two are in it. Can you show your
work or explain more about your method?

[Actually, I have Run Down the Whistle ... I don't know which
version.]

-Carl

Petr Pařízek wrote:
> 
> > Herman wrote:
> >> it would be just to the right of porcupine, passing through 7, 18,
>> 25, and 32.
> > Okay, now the remaining question is if this temperament is „musically > fruitful“ enough to be worth finding a name for. But I’m not sure if I > can comment on this because there were other people who have, unlike me, > found something of use in temperaments like father or bug.

Well, it doesn't rank very favorably on the "badness" scale, but that really only measures two properties that are easily quantifiable: complexity and error. Many other temperaments of the same or lower complexity are more accurate. But there are other considerations. It has a 7-note MOS, which is usually a good sign that it might have one or more musically useful modes. In this case, it has two modes with a tonic chord that has a perfect fifth (using A for the large step and B for the smaller step): A A B A B A B and A B A A B A B. For any other consonant fifths you have to use notes outside the basic MOS. One thing that's apparent is how little difference there is between major and minor. The 1/4 keyboard in Erv Wilson's generalized keyboard classification system (http://www.anaphoria.com/key.PDF) is a good match for this temperament. One interesting thing is that it splits the major third into two equal steps without being a meantone.

>> I had an example on my old Comcast web page that used gorgo temperament.
>> I chose it as much for its melodic properties as anything else, but it
>> also illustrates some of the potential for 7-limit harmony.
> > Well, you seem to be more successful than I was. I think you would also > be better at choosing the „right“ amount of tempering for dicot (another > temperament that I was never satisfied with). Maybe my problem there was > that I find the two intervals of 6/5 and 5/4 to be of such a different > nature that I simply don’t know how to „effectively“ treat a tuning > where they turn into one thing. Perhaps the main purpose is to use > neutral thirds? Another possibility would be to use a generator equal to > the cube root of either 15/8 (which would make 3/2 and 5/4 mistuned by > the same amount)or 9/5 (then this would go for 3/2 and 6/5). But this > would mean a clear preference for major thirds over minor thirds or vice > versa. Okay, the cbrt(15/8) choice sounds acceptable to me, maybe I’ll > try it one day too - and hear what it sounds like.
> > Petr

One way to deal with something like dicot temperament, to create a distinction similar to major vs. minor, is to change the spacing of chords. I.e. rather than CEG - ACE, do something like CGE - CAE. Also, dicot is one of those temperaments that sounds better with slightly stretched octaves.

Marcel de Velde wrote:
> It's one point that's getting lost in this whole JI vs. meantone
> thread. Not only is there a wealth of existing music which
> assumes the disappearance of the syntonic comma (and thus can't
> be adequately performed in strict JI), there is a like quantity
> of music that assumes the disappearance of other commas, which
> has yet to be written.
> > > I must disagree with this. All music can be performed in strict JI.
> Just await the Lasso example I'll post later today and I'll think even > you will be convinced :)

That's a pretty bold claim. If that's true, you should have no trouble with this simple comma pump:

http://www.io.com/~hmiller/music/ex/dim12.mid

This one's a little trickier: the goal is to keep it in the key of Eb. In 12-ET it rises one semitone each time around.

http://www.io.com/~hmiller/midi/porcupine-12.mid

For an idea of what it should sound like, check the 22-ET version.

http://www.io.com/~hmiller/midi/porcupine-22.mid

Well, you did say *all* music! The first example tempers out 648/625, which is a pretty modern thing to do compared with 81/80 or 128/125, but can be found in early 20th century music. The second example tempers out 250/243, which defines what we call "porcupine" temperament. I've used it, Paul Erlich has used it, and most recently Petr Pařízek. (I wouldn't be surprised to find someone has used it earlier.)

> That's a pretty bold claim. If that's true, you should have no trouble
> with this simple comma pump:
>

I consider it the other way around.
It's a pretty bold claim to me to say certain music can't be played in JI.
I consider JI to be the underlying structure of all music.
I don't see any proof to why it is not so.
It does not mean I can solve all comma pumps without any trouble.
I did just solve the Lasso piece.
This can be seen as strong evidence for pure JI.

http://www.io.com/~hmiller/music/ex/dim12.mid
>
> This one's a little trickier: the goal is to keep it in the key of Eb.
> In 12-ET it rises one semitone each time around.
>
> http://www.io.com/~hmiller/midi/porcupine-12.mid
>
> For an idea of what it should sound like, check the 22-ET version.
>
> http://www.io.com/~hmiller/midi/porcupine-22.mid
>
> Well, you did say *all* music! The first example tempers out 648/625,
> which is a pretty modern thing to do compared with 81/80 or 128/125, but
> can be found in early 20th century music. The second example tempers out
> 250/243, which defines what we call "porcupine" temperament. I've used
> it, Paul Erlich has used it, and most recently Petr Pařízek. (I wouldn't
> be surprised to find someone has used it earlier.)
>

I do like the examples though and will have a go at them later :)

Marcel

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> >
> > Have you heard the comma pump demo which I posted yesterday?
> > If you have, then I wonder what you're still trying to convince
> > us about.
>
> Yes I have.
> And nothing proves it can't be played in pure JI.
> Not saying I'm going to attempt the interpretation but you can't
> say it can't be played in pure JI either.
>
> Marcel

No JI performance would have remotely the same effect.
Meanwhile, Marcel, you're so full of it that everybody by
now wishes you'd either tone it down or take your ball and
go home. You clearly haven't done the homework required
to back up the statements you make, since every time someone
asks you to produce a demonstration you start out by saying
you've 1. never tried that before 2. aren't good at reading
music 3. made a mistake when you fixed the mistake you made
fixing the mistake you made the day before when you fixed the
mistake in your original mp3 or 4. the piece of music you're
being asked to deal with is an aberration.

Just cut it out, dude. Your position that strict JI can
be the only answer is dogmatic, and is not based on a
consideration of evidence or reasoning.

-Carl

Carl wrote:

> I keep about 1/3 of the files you post in a folder on my hard drive
> with your name on it, and these two are in it. Can you show your
> work or explain more about your method?
> [Actually, I have Run Down the Whistle ... I don't know which
> version.]

If you downloaded it in May 2008, then it's the one I was speaking about. The overall length should be about 3:57.

Well, I don't know if the text of message # 81696 makes sense to you, but generally, if I want to find any of the "model" comma pumps, I také the particular generator mapping as a starting point and then find the lowest and the highest difference between these (let's call them DL and DH for the sake of this explanation). For example, in the case of 81/80, the 2,3,5 prime generator mapping is "0, 1, 4", which means that DL is 1 (i.e. the difference between the mapping of 2 and the mapping of 3) and DH is 4 (between mappings of 2 and 5). To pick a different temperament, for 16875/16384, the generator mapping is "0, -4, 3" and therefore DL is 3 (between mappings of 2 and 5) and DH is 7 (between mappings for 3 and 5). So if we are staying with 5-limit temperaments, then I start with a major triad and keep adding or subtracting DL from the generator numbers (let's call them A for the lowest, B for the middle, and C for the highest number). If I'm adding, then I change C to a value lower than A in order the highest difference between the numbers was still DH. If I'm subtracting, I change A to a value higher than C (actually, only if this doesn't get me out of the minimum generator range). I do this a few times until I eventually end up with the same numbers with which I started. So for the case of meantone, for example, I'll start with "0_1_4" and let's say I'll decide to subtract DL (which is 1 here). But instead of using the actual result of "-1_0_3", I'll change the -1 to 4, which means that "0_3_4" will be the generator mapping for the second chord. Then I'll subtract 1 again, which makes "-1_2_3" (this time I don't make any other changes as I want the overall generator range used so far to be at its minimum, which is 5; if I changed the -1 to 6, then this would be 6). Then I subtract 1 again and change the resulting -2 to 5, so that I get "1_2_5". And when I subtract 1 once again, I get "0_1_4", which is what I started with.

In the message # 81696, I was trying to explain similar things with porcupine.

... Phew, ... I think I'm going to save this; you see, this is the first time ever that I've "publicly" described my method for mathematically "modelling" comma pumps.

Petr

Maybe I've lost the plot on this piece and its JI, but out of
curiosity I downloaded dim12 and then tried LucyTuning it using 3b2s,
which seemed to be the most appropriate tuning to my ears..

I put the result as I found it into this folder:

http://www.lucytune.com/dim/

On 1 Mar 2009, at 19:45, Marcel de Velde wrote:

>
>
> That's a pretty bold claim. If that's true, you should have no trouble
> with this simple comma pump:
>
> I consider it the other way around.
> It's a pretty bold claim to me to say certain music can't be played
> in JI.
> I consider JI to be the underlying structure of all music.
> I don't see any proof to why it is not so.
> It does not mean I can solve all comma pumps without any trouble.
> I did just solve the Lasso piece.
> This can be seen as strong evidence for pure JI.
>
>
> http://www.io.com/~hmiller/music/ex/dim12.mid
>
> This one's a little trickier: the goal is to keep it in the key of Eb.
> In 12-ET it rises one semitone each time around.
>
> http://www.io.com/~hmiller/midi/porcupine-12.mid
>
> For an idea of what it should sound like, check the 22-ET version.
>
> http://www.io.com/~hmiller/midi/porcupine-22.mid
>
> Well, you did say *all* music! The first example tempers out 648/625,
> which is a pretty modern thing to do compared with 81/80 or 128/125,
> but
> can be found in early 20th century music. The second example tempers
> out
> 250/243, which defines what we call "porcupine" temperament. I've used
> it, Paul Erlich has used it, and most recently Petr Pařízek. (I
> wouldn't
> be surprised to find someone has used it earlier.)
>
> I do like the examples though and will have a go at them later :)
>
> Marcel
>
>

Charles Lucy
lucy@lucytune.com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

>
> Your position that strict JI can
> be the only answer is dogmatic, and is not based on a
> consideration of evidence or reasoning.
>

Yes it is based on much of both.

Marcel

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> If you downloaded it in May 2008, then it's the one I was
> speaking about. The overall length should be about 3:57.

Yep, May 21st.

> Well, I don't know if the text of message # 81696 makes sense
> to you,

Not really, sorry.

> For example, in the case of 81/80, the 2,3,5 prime generator
> mapping is "0, 1, 4", which means that DL is 1 (i.e. the
> difference between the mapping of 2 and the mapping of 3) and
> DH is 4 (between mappings of 2 and 5). To pick a different
> temperament, for 16875/16384, the generator mapping is
> "0, -4, 3" and therefore DL is 3 (between mappings of 2 and 5)
> and DH is 7 (between mappings for 3 and 5). So if we are
> staying with 5-limit temperaments, then I start with a major
> triad and keep adding or subtracting DL from the generator
> numbers (let's call them A for the lowest, B for the middle,
> and C for the highest number). If I'm adding, then I change
> C to a value lower than A in order the highest difference
> between the numbers was still DH. If I'm subtracting, I
> change A to a value higher than C (actually, only if this
> doesn't get me out of the minimum generator range). I do
> this a few times until I eventually end up with the same
> numbers with which I started. So for the case of meantone,
> for example, I'll start with "0_1_4" and let's say I'll
> decide to subtract DL (which is 1 here). But instead of
> using the actual result of "-1_0_3", I'll change the -1 to 4,
> which means that "0_3_4" will be the generator mapping for
> the second chord. Then I'll subtract 1 again, which makes
> "-1_2_3" (this time I don't make any other changes as I want
> the overall generator range used so far to be at its minimum,
> which is 5; if I changed the -1 to 6, then this would be 6).
> Then I subtract 1 again and change the resulting -2 to 5,
> so that I get "1_2_5". And when I subtract 1 once again,
> I get "0_1_4", which is what I started with.

Whoa... I have no idea why you would do any of that!

> ... Phew, ... I think I'm going to save this; you see, this
> is the first time ever that I've "publicly" described my
> method for mathematically "modelling" comma pumps.

Your secret is safe with me, I'm afraid. :(

-Carl

Carl wrote:

> Whoa... I have no idea why you would do any of that!

Simply said, i'm trying to convert my musical knowledge into numbers in order I could do similar things in other temperaments than meantone. To show the meantone example at its extreme, I could happily start with major triads a fifth apart (like C, F, Bb, Eb, Ab). And I can immediately see that not only is G contained in C major but also in Eb major. And not only is C contained in C major but also in Ab major. But if I leave the progression of major triads as it is, then the highest tone in the chain of fifths will be E (i.e. 4 generators away from C) and the lowest tone in the chain will be Ab (-4 generators), which occupies a total range of 8 generators. If I change one of the generator numbers, I can stay in a smaller generator range without actually disrupting the effective descending by fifths in any way and eventually I'll get to the same chord as the first one. So if, instead of playing C major and F major, I play C major and A minor, then the fact that E-G go down a fifth to A-C is still there but the total generator range used is still 4, not 5 as in the case of C major and F major. When I then go down one more fifth, I get D minor. I could play D major instead of D minor but this would change the total generator range to 6 so I'll leave the D minor there to keep the total generator range at 5, which means that all three tones go down a fifth. When I go down another fifth then, I get G minor; but this time there's no problem in changing Bb to B so I'll use G major instead, which means that D-A goes down a fifth to G-D. And finally I play C major again, making all the tones go a fifth down once more. -- When I convert this chord progression to generator numbers, I get the thing I described in my previous message. And this eventually allows me to do similar things for other comma pumps, no matter what comma I choose to temper out and no matter what primes the comma contains.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > Whoa... I have no idea why you would do any of that!
>
> Simply said, i'm trying to convert my musical knowledge into
> numbers in order I could do similar things in other temperaments
> than meantone. [snip]

Sorry to be dense, but maybe you could start by saying what
your goal is. To find the minimum span of generators needed
to perform a pump for comma X in temperament Y? And why
is this desirable? etc.

-Carl

Carl wrote:

> To find the minimum span of generators needed
> to perform a pump for comma X in temperament Y? And why
> is this desirable? etc.

Well, my first question was how to find a comma pump for any particular temperament with as few chords as possible. And my first idea for answering that question was to start simply by repeatedly shifting major triads up or down by the same interval (like fifths in meantone) until I got to a chord which had one pitch equal to any one in the initial triad. But this didn't sound nice to me because after, let's say, going down a fifth for three times, I then had to go down by a minor third (or up a major sixth) to get to the initial chord (i.e. one step in the negative direction repeated three times and then three steps in the positive direction), which made the last two chords sound a bit "tonally distant". Because I didn't find this to be one of the best ways to do it, it soon occured to me that I also needed to minimize the generator range, which eventually gave me the possibility to basically keep going down by fifths without having to do things like going three fifths up at a time then. And because a major triad in meantone spans 4 generators, then I can do this task, if I'm subtracting, simply by changing the lowest generator number to a value which is 4 higher than the middle value (then becoming the lowest). And voila, when I keep doing this, I'll eventually get to the same chord as the initial one. And this also confirms that, in meantone, I need at least 7 tones in the scale to be able to realize the chord progression.

Petr