Yahoo Tuning Groups Ultimate Backup tuning Decatonic scales: temperaments with relatively just thirds and fifth?

Dear all,

I am looking for a temperament for Paul Erlich's decatonic scales. In his paper "Tuning, Tonality, and Twenty-Two-Tone Temperament", Paul recommends 22 ET for this purpose. However, I would prefer a tuning where the intervals are somewhat closer to their JI equivalents. In particular, I would like the (diatonically speaking) thirds and fifths to be "somewhat more just". So, I searched around for a while and came across the TOP tuning of Pajara (AKA Paultone, or Twintone) which Erlich reports in his text "A Middle Path". I find this tuning rather pleasing for my purposes.

However, this whole subject is very new to me and so I wanted to ask for alternative recommendations. Important for my purposes is that all intervals within the decatonic scales can be used, so 50/49 and 64/63 (and 224/225) must be tempered out (which Pajara does, as does 22 ET). Additionally, as mentioned above, I would like relatively just tunings (thirds and fifths in particular).

In general, I would be grateful for any alternative temperament recommendations with which these scales can be played, just to get a better overview of the options.

Perhaps I should mention that the TOP tuning of Pajara is seemingly not contained in the Scala list of scales, so I wondered whether there are alternative or even better recommendations.

Thank you!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

🔗Herman Miller <hmiller@IO.COM>

Torsten Anders wrote:
> Dear all,
> > I am looking for a temperament for Paul Erlich's decatonic scales. In > his paper "Tuning, Tonality, and Twenty-Two-Tone Temperament", Paul > recommends 22 ET for this purpose. However, I would prefer a tuning > where the intervals are somewhat closer to their JI equivalents. In > particular, I would like the (diatonically speaking) thirds and > fifths to be "somewhat more just". So, I searched around for a while > and came across the TOP tuning of Pajara (AKA Paultone, or Twintone) > which Erlich reports in his text "A Middle Path". I find this tuning > rather pleasing for my purposes.
> > However, this whole subject is very new to me and so I wanted to ask > for alternative recommendations. Important for my purposes is that > all intervals within the decatonic scales can be used, so 50/49 and > 64/63 (and 224/225) must be tempered out (which Pajara does, as does > 22 ET). Additionally, as mentioned above, I would like relatively > just tunings (thirds and fifths in particular).
> > In general, I would be grateful for any alternative temperament > recommendations with which these scales can be played, just to get a > better overview of the options.
> > Perhaps I should mention that the TOP tuning of Pajara is seemingly > not contained in the Scala list of scales, so I wondered whether > there are alternative or even better recommendations.
> > Thank you!
> > Best
> Torsten

For any two independent 7-limit intervals (like 50/49 and 64/63), there's only one rank-2 temperament that tempers out both of them; in this case, pajara. You can tune pajara in a number of different ways; another optimal tuning by a different standard is TOP-RMS, with a period of 598.859347 cents and a generator of 106.844183 cents.

There are other temperaments that temper out one or the other of 50/49 and 64/63 but not both. An example is porcupine, which reduces 50/49 to 16.61 cents in the TOP-RMS tuning (P = 1197.839112, G = 162.586787) and tempers out 64/63 completely. Doublewide (TOP-RMS P = 600.046727, G = 274.302244) tempers out 50/49 completely and reduces 64/63 to 17.09 cents.

🔗Petr Parízek <p.parizek@chello.cz>

🔗Graham Breed <gbreed@gmail.com>

Yes, that should work fine. I don't think you can get appreciably more just, anyway -- unless by "thirds and fifths" you mean the 5-limit intervals. In that case try 34 ET.

> However, this whole subject is very new to me and so I wanted to ask > for alternative recommendations. Important for my purposes is that > all intervals within the decatonic scales can be used, so 50/49 and > 64/63 (and 224/225) must be tempered out (which Pajara does, as does > 22 ET). Additionally, as mentioned above, I would like relatively > just tunings (thirds and fifths in particular).
> > In general, I would be grateful for any alternative temperament > recommendations with which these scales can be played, just to get a > better overview of the options.

In a purist sense, it needs to be pajara because that's the only temperament class where intervals have a consistent size, and the right commas are tempered out. If you want to be more pragmatic there are alternatives:

Well temperament -- make the tuning more lumpy. You can make some chords better in tune at the expense of others. Naturally you'll probably want the most important chords to be well tuned. As a bonus you can get the temperament to circulate, so that you can modulate over a fixed number of notes. 22 and 34 are the most obvious choices.

Adaptive temperament -- have the tuning adjust on the fly. This is probably not something you're ready to deal with because you'll have to program it yourself.

Use more than 10 notes for the decatonic scale. I don't think there's a cute name for this. It means you can get the improved precision of adaptive temperament but you have to do the work yourself, instead of a machine making the adjustments for you. As an extreme measure you can write decatonic music and tune it to just intonation.

> Perhaps I should mention that the TOP tuning of Pajara is seemingly > not contained in the Scala list of scales, so I wondered whether > there are alternative or even better recommendations.

You can easily set up a pajara tuning using the "pythagorean" command in Scala. So there isn't a need to supply every interesting tuning of every temperament class.

The reason 22 ET was originally considered optimal is that the tuning of 7:5 is independent of the generator size. As no intervals in 22-equal are worse than this you can't improve on 22-equal -- if you believe in the minimax (minimum value of the maximum error).

A minimax tuning for the pajara generator is 109.4 cents.

For comparison, the generator's 109.1 cents in 22-equal.

However there are some intervals which can be improved relative to 22-equal. If you care about the average mistuning then the RMS (root mean squared) optimum is more relevant.

The 7-limit RMS tuning is 108.8 cents.

Now we come to the "TOP" tunings. The idea is that simple intervals are more important than complex one. Herman's given you the TOP-RMS tuning, which optimizes the RMS of weighted intervals. This is most valid if you care about the average mistuning of intervals beyond the 7-odd limit. The octave is one of the intervals that gets optimized. If you don't want that you can unstretch the scale so that octaves are pure again.

The unstretched TOP-RMS tuning is 107.0 cents.

The plain TOP you found optimizes the worst weighted error. However, this error is always that of 7:5, so to get a single answer you have to consider only intervals that depend on the generator. I don't think that has any special meaning, but it'll work fine because it'll be close to the TOP-RMS.

The unstretched TOP tuning is 106.8 cents.

Which of these tunings is best is entirely a matter of taste. The current fashion is for something like TOP. You should try different ones (there are only two numbers to choose). Some tuning that isn't mathematically optimal may happen to give intervals a character you like.

Graham

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Torsten Anders wrote:
> > Dear all,
> >
> > I am looking for a temperament for Paul Erlich's decatonic
scales. In
> > his paper "Tuning, Tonality, and Twenty-Two-Tone Temperament",
Paul
> > recommends 22 ET for this purpose. However, I would prefer a
tuning
> > where the intervals are somewhat closer to their JI equivalents.

Carl, Herman, Graham, et al, please take note: I think it's clear
from the context that Torsten is using the word "temperament" to
mean "tempered tuning" (as I claimed the rest of the musical world
was using it), not "temperament class" (as it's been used most
recently around here). Fortunately, both Graham and Herman took care
in their replies not to confuse the issue.

> > In
> > particular, I would like the (diatonically speaking) thirds and
> > fifths to be "somewhat more just". So, I searched around for a
while
> > and came across the TOP tuning of Pajara (AKA Paultone, or
Twintone)
> > which Erlich reports in his text "A Middle Path". I find this
tuning
> > rather pleasing for my purposes.
>
> Yes, that should work fine. I don't think you can get
> appreciably more just, anyway -- unless by "thirds and
> fifths" you mean the 5-limit intervals. In that case try 34 ET.

The problem with 34-ET is that, while the 5-limit error is greatly
reduced, the maximum 7-limit error exceeds that of 22-ET. 56-ET
(taking prime 7 as 46deg, as would be required for pajara) comes very
close to minimizing 5-limit error without exceeding the 22-ET 7-limit
max. error. (See the first link below.)

> > However, this whole subject is very new to me and so I wanted to
ask
> > for alternative recommendations. Important for my purposes is
that
> > all intervals within the decatonic scales can be used, so 50/49
and
> > 64/63 (and 224/225) must be tempered out (which Pajara does, as
does
> > 22 ET). Additionally, as mentioned above, I would like
relatively
> > just tunings (thirds and fifths in particular).
> >
> > In general, I would be grateful for any alternative temperament
> > recommendations with which these scales can be played, just to
get a
> > better overview of the options.
>
> In a purist sense, it needs to be pajara because that's the
> only temperament class where intervals have a consistent
> size, and the right commas are tempered out. If you want to
> be more pragmatic there are alternatives:
>
> Well temperament -- make the tuning more lumpy. You can
> make some chords better in tune at the expense of others.
> Naturally you'll probably want the most important chords to
> be well tuned. As a bonus you can get the temperament to
> circulate, so that you can modulate over a fixed number of
> notes. 22 and 34 are the most obvious choices.

I discussed some of the tuning issues with the pajara temperament
(with pure octaves), in particular 56-ET and a 34-tone well-
temperament (with .scl listing) here:
/tuning/topicId_67957.html#68032

Shortly thereafter I created some .mp3 files for a comparative
listening test; see:
/tuning/topicId_67957.html#68285
and
/tuning/topicId_67957.html#68371
The progressions aren't really illustrative of the pajara scale, but
rather are intended to demonstrate what a few selected chords (played
slowly in several pajara-capable tunings) sound like. I included
ones with primes 11 and 13, since there are pajara tunings that
include them (thus extending the prime limit to 17).

The sound files are still out there, so you can listen for yourself
and draw your own conclusions before you read the comments of the
test participants, which are summarized here:
/tuning/topicId_67957.html#68521

> Adaptive temperament -- have the tuning adjust on the fly.
> This is probably not something you're ready to deal with
> because you'll have to program it yourself.
>
> Use more than 10 notes for the decatonic scale. I don't
> think there's a cute name for this. It means you can get
> the improved precision of adaptive temperament but you have
> to do the work yourself, instead of a machine making the
> adjustments for you. As an extreme measure you can write
> decatonic music and tune it to just intonation.
>
> > Perhaps I should mention that the TOP tuning of Pajara is
seemingly
> > not contained in the Scala list of scales, so I wondered whether
> > there are alternative or even better recommendations.
>
> You can easily set up a pajara tuning using the
> "pythagorean" command in Scala. So there isn't a need to
> supply every interesting tuning of every temperament class.
>
> The reason 22 ET was originally considered optimal is that
> the tuning of 7:5 is independent of the generator size. As
> no intervals in 22-equal are worse than this you can't
> improve on 22-equal -- if you believe in the minimax
> (minimum value of the maximum error).
>
> A minimax tuning for the pajara generator is 109.4 cents.
>
> For comparison, the generator's 109.1 cents in 22-equal.
>
> However there are some intervals which can be improved
> relative to 22-equal. If you care about the average
> mistuning then the RMS (root mean squared) optimum is more
> relevant.
>
> The 7-limit RMS tuning is 108.8 cents.

Since Torsten seems to be interested mainly in improving the 5-limit
consonances, then I would conclude that he would want the pajara
generator that minimizes the 5-limit errors while limiting the
absolute error of each 7-limit consonance to no more than the 5:7
error of 22-ET. With untempered octaves, this is accomplished with a
generator of ~106.843c (a combined 5,7-limit minimax generator of
sorts), which makes 4:5 just.

> Now we come to the "TOP" tunings. The idea is that simple
> intervals are more important than complex one. Herman's
> given you the TOP-RMS tuning, which optimizes the RMS of
> weighted intervals. This is most valid if you care about
> the average mistuning of intervals beyond the 7-odd limit.
> The octave is one of the intervals that gets optimized. If
> you don't want that you can unstretch the scale so that
> octaves are pure again.
>
> The unstretched TOP-RMS tuning is 107.0 cents.
>
> The plain TOP you found optimizes the worst weighted error.
> However, this error is always that of 7:5, so to get a
> single answer you have to consider only intervals that
> depend on the generator. I don't think that has any special
> meaning, but it'll work fine because it'll be close to the
> TOP-RMS.
>
> The unstretched TOP tuning is 106.8 cents.

That's in agreement with the 5,7-limit minimax generator I gave above.

> Which of these tunings is best is entirely a matter of
> taste. The current fashion is for something like TOP. You
> should try different ones (there are only two numbers to
> choose). Some tuning that isn't mathematically optimal may
> happen to give intervals a character you like.

Now back to lurking. ;-)

--George

🔗Carl Lumma <carl@lumma.org>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > Torsten Anders wrote:
> > > Dear all,
> > >
> > > I am looking for a temperament for Paul Erlich's decatonic
> > > scales. In his paper "Tuning, Tonality, and Twenty-Two-Tone
> > > Temperament", Paul recommends 22 ET for this purpose.
> > > However, I would prefer a tuning where the intervals are
> > > somewhat closer to their JI equivalents.
>
> Carl, Herman, Graham, et al, please take note: I think it's clear
> from the context that Torsten is using the word "temperament" to
> mean "tempered tuning" (as I claimed the rest of the musical world
> was using it), not "temperament class" (as it's been used most
> recently around here).

George, George, George. I thought we were going to let this
rest? Now we're going to argue over what Torsten meant right
in front of his face?

Perhaps Torsten can clarify, but I understood him to ask
about a temperament other than pajara. Both Herman and Graham
(and you!) seemed to read him this way too, by answering that
no other temperament sends both of the targeted commas to unison.
Further, he uses "tuning" in precisely the tuning-math way.
His terminology continues to exactly follow the tuning-math
way here:

> > > So, I searched around for a while and came across
> > > the TOP tuning of Pajara (AKA Paultone, or Twintone)
> > > which Erlich reports in his text "A Middle Path". I find
> > > this tuning rather pleasing for my purposes.
//
> > > Perhaps I should mention that the TOP tuning of Pajara is
> > > seemingly not contained in the Scala list of scales,

Scala scales, TOP tuning of Pajara... he's gotten everything
right. You (George) seem to be reading through your own lens.

-Carl

🔗Torsten Anders <torstenanders@gmx.de>

Thank you all very much for your replies!! I had no time yet to test
all your recommendations, but I definitely will :)

On Mar 27, 2008, at 6:28 AM, Petr Parízek wrote:
> If you want thirds and fifths to be closer to just, then you’ll get
> only a very poor approximation to 7/4 because the same interval is
> also used here to approximate 16/9.

Does there also exist a variant where the intonation of the septimal
intervals is optimised (possibly to the detriment of thirds and
fifths) like the (major) thirds are optimised in meantone (to the
detriment of the fifths)?

> If you are interested, I’ve left it here for you to download:
> http://download.yousendit.com/7C803D583387FE19

Interesting, thanks for sharing. I must say, I most like the bits
with piano solo, like the beginning :)

On Mar 27, 2008, at 7:37 AM, Graham Breed wrote:
> Well temperament -- make the tuning more lumpy. You can
> make some chords better in tune at the expense of others.
> Naturally you'll probably want the most important chords to
> be well tuned. As a bonus you can get the temperament to
> circulate, so that you can modulate over a fixed number of
> notes. 22 and 34 are the most obvious choices.
I have never created a well temperament yet -- is there some "gentle
introduction" to this subject available I could read?

> Adaptive temperament -- have the tuning adjust on the fly.
> This is probably not something you're ready to deal with
> because you'll have to program it yourself.
I am programming anyway, so this might be an option. Also, I couldpossibly use software specially designed for adaptive tunings, like
Mutabor by Bernhard Ganter and others. I was quite impressed by the
capabilities of this program, when I used it years ago (see link
below, unfortunately the documentation is in German only).

http://www.math.tu-dresden.de/~mutabor/index.html.de

So, I would be grateful if there already exist any decatonic-specific
details on adaptive tunings.

Best
Torsten

🔗Torsten Anders <torstenanders@gmx.de>

Dear Carl and George,

On Mar 27, 2008, at 9:13 PM, Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > >
> > > Torsten Anders wrote:
> > > > Dear all,
> > > >
> > > > I am looking for a temperament for Paul Erlich's decatonic
> > > > scales. In his paper "Tuning, Tonality, and Twenty-Two-Tone
> > > > Temperament", Paul recommends 22 ET for this purpose.
> > > > However, I would prefer a tuning where the intervals are
> > > > somewhat closer to their JI equivalents.
> >
> > Carl, Herman, Graham, et al, please take note: I think it's clear
> > from the context that Torsten is using the word "temperament" to
> > mean "tempered tuning" (as I claimed the rest of the musical world
> > was using it), not "temperament class" (as it's been used most
> > recently around here).
>
> George, George, George. I thought we were going to let this
> rest? Now we're going to argue over what Torsten meant right
> in front of his face?
thank you for your considerateness, but I don't know what you are talking about :) What is a "temperament class"?

> Perhaps Torsten can clarify, but I understood him to ask
> about a temperament other than pajara. Both Herman and Graham
> (and you!) seemed to read him this way too, by answering that
> no other temperament sends both of the targeted commas to unison.
> Further, he uses "tuning" in precisely the tuning-math way.
> His terminology continues to exactly follow the tuning-math
> way here:
:) I should possibly clarify that I know about music and wrote microtonal music before, but I know next to nothing about "tuning math".

Perhaps if helps if I briefly sketch how I came up with my question (hopefully this will not embarrass me :). I would like to write some 7-limit music (possibly higher limit) where I don't primarily think chord progressions (as I did before), but have scales which complement the harmony (so I could write melodies, harmonic counterpoint...). I searched around, and by some chance I came across the writing of Paul Erlich on decatonic scales. Theoretically, I appreciate how he generalises the notion of diatonic scales, and practically I just like listening to these scales. I feel I can almost sing them, although they are 7-limit (perhaps after a while I really can :).

However, playing _chords_ in 22 ET made me feel uneasy, it seemed to me that it didn't sound in tune. I am not even sure that the problem actually are the thirds and fifths, but I feel I don't like the strong beating of many chords in 22 ET, and I prefer the sound of these chords in the TOP tuning of Pajara where this beating is much reduced. I wanted to get a better idea of alternatives..

As I am rather new to this field I thought I better ask more knowledgable folk :) However, in order to make my question somewhat precise, I tried to narrow my question down and also tried to use appropriate terminology which I picked up, e.g., from the Erlich papers.

Still, I must confess that I don't fully understand some details of your replies, because my tuning math knowledge is rather limited. For example, you specified some period and some generator for various tunings. I guess that these describe a linear temperament, correct? Where can I read more about this matter? Is it correct that I can render these tunings in Scale by creating a new linear scale and simply entering the scale size (22), the period and the generator -- leaving all the other values at their default?

Thank you!

Best
Torsten

🔗Graham Breed <gbreed@gmail.com>

Torsten Anders wrote:
> Thank you all very much for your replies!! I had no time yet to test > all your recommendations, but I definitely will :)
> > On Mar 27, 2008, at 6:28 AM, Petr Parï¿½zek wrote:
>> If you want thirds and fifths to be closer to just, then youï¿½ll get >> only a very poor approximation to 7/4 because the same interval is >> also used here to approximate 16/9.
> > Does there also exist a variant where the intonation of the septimal > intervals is optimised (possibly to the detriment of thirds and > fifths) like the (major) thirds are optimised in meantone (to the > detriment of the fifths)?

The 7-limit minimax would be the equivalent of 1/4-comma meantone. The TOP tunings balance different intervals in different ways, so there's a slightly higher emphasis on the 5-limit.

> On Mar 27, 2008, at 7:37 AM, Graham Breed wrote:
>> Well temperament -- make the tuning more lumpy. You can
>> make some chords better in tune at the expense of others.
>> Naturally you'll probably want the most important chords to
>> be well tuned. As a bonus you can get the temperament to
>> circulate, so that you can modulate over a fixed number of
>> notes. 22 and 34 are the most obvious choices.
> I have never created a well temperament yet -- is there some "gentle > introduction" to this subject available I could read?

I don't think there's a general theory. Most of the attention's on 12 note scales and George Secor also has a paper on a 17 note well temperament. I thought a 34 note well temperament came out of that as well, but he didn't mention it so maybe it doesn't apply to pajara.

>> Adaptive temperament -- have the tuning adjust on the fly.
>> This is probably not something you're ready to deal with
>> because you'll have to program it yourself.
> I am programming anyway, so this might be an option. Also, I could > possibly use software specially designed for adaptive tunings, like > Mutabor by Bernhard Ganter and others. I was quite impressed by the > capabilities of this program, when I used it years ago (see link > below, unfortunately the documentation is in German only).
> > http://www.math.tu-dresden.de/~mutabor/index.html.de
> > So, I would be grateful if there already exist any decatonic-specific > details on adaptive tunings.

Oh, I didn't know about that. Of course, being documented in German I still don't know much about it. I went to the "Tonsystem" page of the reference handbook and it says "Ein Tonsystem besteht aus einer Verankerungstaste, einer Menge von Tï¿½nen und einem Periodenintervall." Well, I don't know what a Verankerungstaste or Menge is, but it'd be nice if "Periodenintervall" maps to what we call "period". Possibly it only starts with an equal temperament though.

If you're working with 22 notes to the octave you'll need to tell it so, along with the number of steps to a 3:2, 5:4, and 7:4. Hopefully it'll know what to do then. And see under "Intervall". It certainly allows you to specify arbitrary ratios.

Graham

🔗Carl Lumma <carl@lumma.org>

Hi Torsten,

Your name is familiar to me, but I can't remember from
where at the moment. :)

> > George, George, George. I thought we were going to let this
> > rest? Now we're going to argue over what Torsten meant right
> > in front of his face?
>
> thank you for your considerateness, but I don't know what you are
> talking about :) What is a "temperament class"?

There was recently a big thread/argument here about how to
which terminology/definitions to use. "Temperament class"
is what George likes to call the abstract thing that
pajara is. I just call it a "temperament". Then things
like TOP pajara, 22-ET, 34-ET, are all *tunings* of the
pajara temperament, as you've said. I think George wants
to call these "tempered tunings" or something like that.
It's all minutia and nitpicking that will probably make
you wonder if either of us are sane.

> Perhaps if helps if I briefly sketch how I came up with my question
> (hopefully this will not embarrass me :). I would like to write
> some 7-limit music (possibly higher limit) where I don't primarily
> think chord progressions (as I did before), but have scales which
> complement the harmony (so I could write melodies, harmonic
> counterpoint...). I searched around, and by some chance I came
> across the writing of Paul Erlich on decatonic scales.
> Theoretically, I appreciate how he generalises the notion of
> diatonic scales, and practically I just like listening to these
> scales. I feel I can almost sing them, although they are 7-limit
> (perhaps after a while I really can :).

They're nice scales!

> However, playing _chords_ in 22 ET made me feel uneasy, it seemed to
> me that it didn't sound in tune. I am not even sure that the problem
> actually are the thirds and fifths, but I feel I don't like the
> strong beating of many chords in 22 ET, and I prefer the sound of
> these chords in the TOP tuning of Pajara where this beating is much
> reduced. I wanted to get a better idea of alternatives..

I also agree that the TOP tuning is better than the 22-ET one.

> Still, I must confess that I don't fully understand some details of
> your replies, because my tuning math knowledge is rather limited.
> For example, you specified some period and some generator for
> various tunings. I guess that these describe a linear temperament,
> correct?

They all describe pajara, except I think Petr may have given
one describing the diaschismic temperament, which is like
pajara in the 5-limit. Or one could say pajara is like
diaschismic with the addition of a mapping from the chain of
generators to 7:4... this changes the optimal tuning since
the generator chain now has a new target to hit.

> Where can I read more about this matter? Is it correct that I can
> render these tunings in Scale by creating a new linear scale and
> simply entering the scale size (22), the period and the generator --
> leaving all the other values at their default?

Yes. In fact the size does not have to be 22, but for a
couple of reasons it is a good choice. In Scala you can
type "pythag" in the text box along the bottom of the window
and then...
enter 22
enter the period size (Scala calls this the "formal octave")
hit return to accept 0
enter the generator size ("formal fifth")
hit return to accept 0
then type "show" to see your scale.

-Carl

🔗Torsten Anders <torstenanders@gmx.de>

Dear Graham,

thanks for your reply.

On Mar 28, 2008, at 3:37 AM, Graham Breed wrote:
> Torsten Anders wrote:
>
>>> Adaptive temperament -- have the tuning adjust on the fly.
>>> This is probably not something you're ready to deal with
>>> because you'll have to program it yourself.
>> I am programming anyway, so this might be an option. Also, I could
>> possibly use software specially designed for adaptive tunings, like
>> Mutabor by Bernhard Ganter and others. I was quite impressed by the
>> capabilities of this program, when I used it years ago (see link
>> below, unfortunately the documentation is in German only).
>>
>> http://www.math.tu-dresden.de/~mutabor/index.html.de
>>
>> So, I would be grateful if there already exist any decatonic-specific
>> details on adaptive tunings.
>
> Oh, I didn't know about that. Of course, being documented
> in German I still don't know much about it. I went to the
> "Tonsystem" page of the reference handbook and it says "Ein
> Tonsystem besteht aus einer Verankerungstaste, einer Menge
> von Tönen und einem Periodenintervall." Well, I don't know
> what a Verankerungstaste or Menge is, but it'd be nice if
> "Periodenintervall" maps to what we call "period". Possibly
> it only starts with an equal temperament though.
>
> If you're working with 22 notes to the octave you'll need to
> tell it so, along with the number of steps to a 3:2, 5:4,
> and 7:4. Hopefully it'll know what to do then. And see
> under "Intervall". It certainly allows you to specify
> arbitrary ratios.

Mutabor is basically a decalrative domain-specific programming
language for mutating/adaptive tunings, in particular just tunings.
It transforms a stream of MIDI data in realtime (or a MIDI file),
inserting pitchbend information (and distributing it over multiple
MIDI channels). In the following I am translating some basic
terminology from the documentation.

Intervall: specifies an interval, e.g., as ratio or an expression
("Quint" is fifth, "Terz" is third, "wurzel" is root).

Quint = 3 : 2
SynKomma = 4 Quint - 2 Oktave - Terz
Semitone = 12 WURZEL 2

Ton: declares a note name, either absolutely or relatively to other
pitches

a = 440
e = a + Fifth

Tonsystem: declares a tone system consisting in a reference MIDI key
(Verankerungstaste), a sequence or set (Menge) of tones (implicitly
defining the size of the system), and an interval at which these
tones are repeated (Periodenintervall).

C_DUR = 60 [c, des, d, es, e, f, ges, g, as, a, b, h] Oktave
ET12 = 60 [c] HalbTon

Umstimmung: defines rules how the present tuning shall be changed,
either absolute or relative to the present state. There are various
possibilities. For example, the reference key (Verankerungstaste),
the sequence of notes (including their size), or the
Periodenintervall might be changed. The keyword "@" denotes the
present state. The following example transposes the reference key by
a variable x.

Transpo (x) = @ + x [ ]

Harmonie: declares a key pattern of tone system positions. These can
be used to trigger retunings. Positions marked with a star (*) are
ignored. The following example matches the dominant seventh chord,
either with or without the fifth present.

Dom7 = {*2,5,7,11}

Form: an extension of the notion of "Harmonie": key patterns are also
recognised if they are transposed. For example, the following matches
not only G7, but also B7.

FORM Dom7

Logik: a logic defines how a Mutabor tuning reacts to certain events.
For example, an event occurs if the pressed key pattern matches a
defined "Harmonie". It consists in a name, some computer key (Taste)
with an associated Tonsystem (used for starting), and a set of
clauses Event -> Action.

Below is a complete example implementing a mutating Tonnetz (lattice)
proposed by Martin Vogel. The program recognises major, minor and
dominant seventh chords by the set of keys pressed and possibly
retunes the tuning when these occur (so you wander commas up and down!).

[Example starts below this line]

INTERVALL
Oktave = 2:1
Quint = 3:2
Terz = 5:4

TON
a = 440
c = a + Quint - Terz - Oktave
d = c + Quint + Quint - Oktave
e = c + Terz
f = a - Terz
g = c + Quint
h = g + Terz
b = f - Quint + Oktave
es = g - Terz
as = c - Terz + Oktave
des = as - Quint
fis = d + Terz

TONSYSTEM
C_Dur = 60 [c,des,d,es,e,f,fis,g,as,a,b,h] Oktave

UMSTIMMUNG
Transpo (x) = @ + x [ ]

HARMONIE
Dur = {0,4,*7}
Moll = {0,4,9}
Dom7 = {*2,5,7,11}

LOGIK
TonNetz Taste N = C_Dur
[
FORM Dom7 -> Transpo ( ABSTAND )
FORM Dur -> Transpo ( ABSTAND )
FORM Moll -> Transpo ( ABSTAND )
]

[Example ends here]

Perhaps I should mention that I am in no way affiliated with the
Mutabor project. Only you mentioned that you don't know it, and I
recognise that there is this language barrier...

Best
Torsten

🔗Torsten Anders <torstenanders@gmx.de>

Dear Carl,

thanks for your reply.

On Mar 28, 2008, at 6:47 AM, Carl Lumma wrote:
> Your name is familiar to me, but I can't remember from
> where at the moment. :)
:) We may have met on some other mailing list (Csound, SuperCollider, Common Music). Also, you possibly came across my constraint-based composition system Strasheela, which also supports defining microtonal music theories. For efficiency reasons, the system always assumes an equal temperament (constrained pitch or pitch class variables are integers). But you could map these to arbitrary tunings and with something like, e.g., 1200 ET or even 120000 ET you get relatively close to JI.

http://strasheela.sourceforge.net

> > > George, George, George. I thought we were going to let this
> > > rest? Now we're going to argue over what Torsten meant right
> > > in front of his face?
> >
> > thank you for your considerateness, but I don't know what you are
> > talking about :) What is a "temperament class"?
>
> There was recently a big thread/argument here about how to
> which terminology/definitions to use. "Temperament class"
> is what George likes to call the abstract thing that
> pajara is. I just call it a "temperament". Then things
> like TOP pajara, 22-ET, 34-ET, are all *tunings* of the
> pajara temperament, as you've said. I think George wants
> to call these "tempered tunings" or something like that.
> It's all minutia and nitpicking that will probably make
> you wonder if either of us are sane.

I certainly don't mind precise terminology, but I was unaware of these distinctions. As you were discussing my naive understanding of these terms, here it is :) Naively, I would have assumed that "tuning" indicates any pitch set (including JI), and "temperament" is some tuning which only approximates JI for some reasons (e.g., to remove some commas). So some random set of pitches would be a tuning but no temperament.

I appears you have a more narrow definition of these terms, as I just found out in the Tonalsoft Encyclopedia of Microtonal Music Theory.

"A tuning is a particular tuning of [the] generators and periods [of a certain tone-space] -- RMS optimal, TOP, etc." Carl Lumma, Yahoo tuning message 73329 (Wed Sep 19, 2007 1:51 am PDT), http://tonalsoft.com/enc/t/tuning.aspx

Does that mean that JI and some random pitch set are no tunings?

BTW: Tonalsoft Encyclopedia of Microtonal Music Theory also has an entry for temperament, but not (yet?) for "temperament class".
http://tonalsoft.com/enc/t/temperament.aspx

Best
Torsten

🔗Carl Lumma <carl@lumma.org>

Hi Torsten,

> On Mar 28, 2008, at 6:47 AM, Carl Lumma wrote:
> > Your name is familiar to me, but I can't remember from
> > where at the moment. :)
> :) We may have met on some other mailing list (Csound,
> SuperCollider, Common Music). Also, you possibly came
> across my constraint-based composition system Strasheela,
//
> http://strasheela.sourceforge.net

No, none of those. Perhaps you were mentioned here by
Werner Mohrlok?

> > There was recently a big thread/argument here about how to
> > which terminology/definitions to use. "Temperament class"
> > is what George likes to call the abstract thing that
> > pajara is. I just call it a "temperament". Then things
> > like TOP pajara, 22-ET, 34-ET, are all *tunings* of the
> > pajara temperament, as you've said. I think George wants
> > to call these "tempered tunings" or something like that.
> > It's all minutia and nitpicking that will probably make
> > you wonder if either of us are sane.
>
> I certainly don't mind precise terminology, but I was unaware
> of these distinctions. As you were discussing my naive
> understanding of these terms, here it is :) Naively, I would
> have assumed that "tuning" indicates any pitch
> set (including JI), and "temperament" is some tuning which
> only approximates JI for some reasons (e.g., to remove some
> commas). So some random set of pitches would be a tuning
> but no temperament.

That's a very common usage. But if you think back to your
first message, I would argue you had already violated it
on some level! You realized that there is something called
pajara, which admits to different tunings (TOP, 22-ET, etc.).
There's no widespread understanding of this, so it's unclear
what it should be called. George has spoken in favor of
"temperament class", and I in favor of simply "temperament".

> Does that mean that JI and some random pitch set are no tunings?

To my way of thinking, a bag of pitches is a scale. Given
it alone, it can be impossible to know whether it implies
a temperament or not, etc. etc. There are temperaments
arbitrarily close in accuracy to JI, and in practice any
JI cannot be exact. Further, it is possible to express a
temperament using rational numbers. I don't call a bag of
pitches a tuning unless I know what the target is.

temperament * tuning = scale

In the case of a linear temperament, the tuning can be
expressed with only two numbers (the size of the period
and the generator), but the tuning could be more complex.
For example, in the case of well temperaments (which
I would like to call "circulating tunings").

George wants to preserve the common usage of these terms.
However, since the common usage isn't precise (uses these
terms interchangeably), I don't think that's a good idea.

> BTW: Tonalsoft Encyclopedia of Microtonal Music Theory also has an
> entry for temperament, but not (yet?) for "temperament class".
> http://tonalsoft.com/enc/t/temperament.aspx

Yes. The entry for temperament also shows several different
definitions. Gene's is the one I am advocating.

-Carl

🔗Torsten Anders <torstenanders@gmx.de>

🔗Herman Miller <hmiller@IO.COM>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <carl@lumma.org>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Carl Lumma wrote:
>
> > To my way of thinking, a bag of pitches is a scale. Given
> > it alone, it can be impossible to know whether it implies
> > a temperament or not, etc. etc. There are temperaments
> > arbitrarily close in accuracy to JI, and in practice any
> > JI cannot be exact. Further, it is possible to express a
> > temperament using rational numbers. I don't call a bag of
> > pitches a tuning unless I know what the target is.
> >
> > temperament * tuning = scale
> >
> > In the case of a linear temperament, the tuning can be
> > expressed with only two numbers (the size of the period
> > and the generator), but the tuning could be more complex.
> > For example, in the case of well temperaments (which
> > I would like to call "circulating tunings").
> >
> > George wants to preserve the common usage of these terms.
> > However, since the common usage isn't precise (uses these
> > terms interchangeably), I don't think that's a good idea.
>
> I wonder if some of the confusion over the "temperament"
> definition could be resolved by introducing the idea of
> a "rank zero" temperament? Like a point, a zero-dimensional
> geometric figure, it has specific unvarying coordinates.
> Quarter-comma meantone is a rank zero temperament that
> always has 2/1 octaves and approx. 696.578-cent fifths. That
> doesn't cover irregular temperaments, but it's a start. It
> also has the odd consequence that JI itself would be a sort
> of temperament.

Sounds interesting, but I don't understand how any kind
of meantone would be rank 0. The notion of JI being the
null temperament had occurred to me.

-C.

🔗Cameron Bobro <misterbobro@yahoo.com>

🔗Graham Breed <gbreed@gmail.com>

🔗Cameron Bobro <misterbobro@yahoo.com>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Cameron Bobro wrote:
>
> > It seems that 56-equal might work well for what Torsten is after,
but
> > I have misgivings about the "pajara" temperament scheme and 7/4.
In
> > my opinion, having used 34-equal quite a bit, making 7/4 and 16/9
> > enharmonic ("tempering out 64/63" in Genese) effectively removes
the
> > sound of 7/4. It works functionally, and it's a suave stunt, but
it
> > is as if the shape of the 7/4 is fine but the color wrong, or
vice
> > versa, as far as the true sensuality of 7/4, or something along
those
> > lines.
>
> Yes, that's what pajara does. What are your misgivings?
> Working functionally is exactly the stunt that allows
> decatonic scales to work.

I have nothing against blurring out the audible effect of the seventh
partial, and do it all the time with 34, where it is "detuned" in a
very smooth way, riding in ambiguous zones that form a foggy path to
the higher partials- a third right smack in the middle of where 9/7
and 14/11 would be for example. The 11th partial is also blurred, 34
strikes the middle of 11/8 and 7/5 for example, and so on.

This approach is just dandy, very musical in sound in my opinion, but
it is not the way to incorporate the actual physical sound of the
seventh partial. In order to do this, the ambiguities must be higher
up, 7/4 very pure, and unlinked from a chain of fourths. In tertian
harmonies, 7 can also be linked to 11 and 13 in an exceptionally
euphonious way by unhitching "re" from the fifths, and so on.

I'm not knocking the "pajara" approach to temperament. That would be
nuts, as I am in effect continually using the same kind of thing and
have been steadfastly cheering for those ambiguous zones that make it
work, as beautiful intervals in their own right.

I am saying that really having 7/4 in a tuning means really having
7/4, or something extremely close. And as far as going higher, simply
playing various Just "tall chords" will quickly reveal that the sweet
harmonies dictate scales that simply don't match the basically
Pythagorean shape, and are full of "neutral" intervals.

-Cameron Bobro

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > >
> > > Torsten Anders wrote:
> > > > Dear all,
> > > >
> > > > I am looking for a temperament for Paul Erlich's decatonic
> > > > scales. In his paper "Tuning, Tonality, and Twenty-Two-Tone
> > > > Temperament", Paul recommends 22 ET for this purpose.
> > > > However, I would prefer a tuning where the intervals are
> > > > somewhat closer to their JI equivalents.
> >
> > Carl, Herman, Graham, et al, please take note: I think it's clear
> > from the context that Torsten is using the word "temperament" to
> > mean "tempered tuning" (as I claimed the rest of the musical
world
> > was using it), not "temperament class" (as it's been used most
> > recently around here).
>
> George, George, George. I thought we were going to let this
> rest? Now we're going to argue over what Torsten meant right
> in front of his face?

Hi, Carl. I've been away for a few days and didn't see your reply
till this (Monday) morning (and have had to catch up with the rest of
this thread).

I had no choice but to bring it up, because no sooner did we finish
discussing the matter than an unsolicited, ready-made example of the
communication problem manifest itself.

It was quite obvious to me that Herman didn't seem to understand what
Torsten was looking for (in asking for a temperament) in his original
message (#75735), most of which I'll quote here:

<< I am looking for a temperament for Paul Erlich's decatonic scales.
In his paper "Tuning, Tonality, and Twenty-Two-Tone Temperament",
Paul recommends 22 ET for this purpose. However, I would prefer a
tuning where the intervals are somewhat closer to their JI
equivalents. In particular, I would like the (diatonically speaking)
thirds and fifths to be "somewhat more just". So, I searched around
for a while and came across the TOP tuning of Pajara (AKA Paultone,
or Twintone) which Erlich reports in his text "A Middle Path". I find
this tuning rather pleasing for my purposes.

However, this whole subject is very new to me and so I wanted to ask
for alternative recommendations. Important for my purposes is that
all intervals within the decatonic scales can be used, so 50/49 and
64/63 (and 224/225) must be tempered out (which Pajara does, as does
22 ET). Additionally, as mentioned above, I would like relatively
just tunings (thirds and fifths in particular).

In general, I would be grateful for any alternative temperament
recommendations with which these scales can be played, just to get a
better overview of the options. >>

(End of quote)

I thought that, even if there were any doubt whether Torsten was
seeking a temperament (class) or temperament (tuning), the inclusion
of the phrases "would prefer a tuning" and "find this tuning rather
pleasing" should have made it clear that it was the latter. All 3
occurrences of the term "temperament" in his message make perfect
sense when you interpret it to mean a tuning, particularly when he
says that he wants "the thirds and fifths in particular" (which I
think we agree refers to the 5-limit consonances) "relatively just"
to specify what he's seeking in a tuning for the decatonic scale.

> Perhaps Torsten can clarify, but I understood him to ask
> about a temperament other than pajara.

Just to clarify that I understand what you mean by "temperament",
that's temperament (class) you're speaking of, not temperament
(tuning).

After reading through this thread, I discovered that he did indeed
clarify what he was seeking:
/tuning/topicId_75735.html#75755
from which I quote:

<< Naively, I would have assumed that "tuning" indicates any pitch
set (including JI), and "temperament" is some tuning which only
approximates JI for some reasons (e.g., to remove some commas). So
some random set of pitches would be a tuning but no temperament. >>

So there you have it: as Torsten (and the rest of the musical world)
understands it, a tuning is a set of pitches (either rational or
tempered), and a temperament is a tuning that's tempered (i.e., a
tempered tuning).

> Both Herman and Graham
> (and you!) seemed to read him this way too, by answering that
> no other temperament sends both of the targeted commas to unison.

Yes, Herman read him that way, but I see Graham's reply (which came
after Herman's) as an attempt to indicate that paraja is not a
temperament (in the sense that he and I believed Torsten used it) by
calling it a temperament class. Unfortunately, that didn't seem to
clarify very much, since that term was new to Torsten. I would
really prefer "temperament family", which I used extensively in both
my 17-tone and Miracle articles; after all, don't temperaments
(tempered tunings) inherit characteristics from their families. If
we then need another term for what Gene calls a "family", then why
not "clan"?

I had to reply in the wake of both Herman & Graham, so I might have
made some adjustments in my reply to reflect that. But if you think
I read Torsten that way too, you're mistaken.

> Further, he uses "tuning" in precisely the tuning-math way.

There's no difference of opinion about what "tuning" means (unless
one strictly adheres to Barbour's definition).

> His terminology continues to exactly follow the tuning-math
> way here:
> ...
> Scala scales, TOP tuning of Pajara... he's gotten everything
> right. You (George) seem to be reading through your own lens.

But only because he was talking almost exclusively about *tunings*
(about which there is no difference of opinion as to meaning).
Nowhere did he use the word "temperament" to refer to a temperament
class -- unless you're reading through a tuning-math lens.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Torsten,
> ...
> > > There was recently a big thread/argument here about how to
> > > which terminology/definitions to use. "Temperament class"
> > > is what George likes to call the abstract thing that
> > > pajara is.

Hi, Carl. Here we go again!

I'd really like "temperament family" now that:
1) Torsten asked Graham to explain what a "temperament class" is;
2) "Family" could be used by itself more readily than "class" in
referring to the more abstract meaning of temperament.

One could therefore use "pajara temperament" and "pajara family"
interchangeably, since pajara would be understood to be a family of
temperaments (tunings), but "meantone family" would be required to
indicate the more abstract meaning, since "meantone temperament" has
a historical meaning that's still widely used to specify a particular
(tempered) tuning.

> > > I just call it a "temperament". Then things
> > > like TOP pajara, 22-ET, 34-ET, are all *tunings* of the
> > > pajara temperament, as you've said. I think George wants
> > > to call these "tempered tunings" or something like that.

I'd be perfectly happy to call them temperaments or tunings, if
everyone else is agreeable to stop referring to temperament
classes/families as temperaments.

> > > It's all minutia and nitpicking that will probably make
> > > you wonder if either of us are sane.
> >
> > I certainly don't mind precise terminology, but I was unaware
> > of these distinctions. As you were discussing my naive
> > understanding of these terms, here it is :) Naively, I would
> > have assumed that "tuning" indicates any pitch
> > set (including JI), and "temperament" is some tuning which
> > only approximates JI for some reasons (e.g., to remove some
> > commas). So some random set of pitches would be a tuning
> > but no temperament.
>
> That's a very common usage. But if you think back to your
> first message, I would argue you had already violated it
> on some level! You realized that there is something called
> pajara, which admits to different tunings (TOP, 22-ET, etc.).
> There's no widespread understanding of this, so it's unclear
> what it should be called. George has spoken in favor of
> "temperament class", and I in favor of simply "temperament".

The bottom line is that you favor "temperament" because it's been
used that way for several years on these lists and therefore don't
want to change that. But you'd rather ignore the fact that it's been
used for a very long time elsewhere to mean a tempered tuning. Yes,
I'd also prefer not to use the term "tempered tuning", but what else
do I use when others have hijacked the term "temperament" to mean
other than what it's meant for centuries?

> > Does that mean that JI and some random pitch set are no tunings?
>
> To my way of thinking, a bag of pitches is a scale.

To my way of thinking, a bag of (specific) pitches is a tuning.

If you want to define the term "scale", first list some important
examples: major, chromatic, decatonic (Erlich), pelog. Then figure
out what's essential to all of these examples and what isn't. None
of these require specific pitches (or intervals between pitches), but
any one of these:
1) is a set of tones suitable for composing a simple melody (as
opposed to a few consecutive pitches separated by, e.g., 3 or 3000
cents);
2) is generally assumed to repeat at a particular period (usually an
octave);
3) generally indicates some particular number of tones per period:
major=7, chromatic=12); decatonic=10, pelog=7;
4) often specifies some general interval pattern (or progression of
interval classes) when the tones are put in ascending order:
major=LLsLLLs, etc.;
5) is capable of being played in more than a single tuning (such that
the pitches and/or intervals are not really specific).

So #5 reveals that a bag of pitches is not necessarily a scale, while
two different bags of pitches may indeed be the same scale.

> Given
> it alone, it can be impossible to know whether it implies
> a temperament or not, etc. etc.

Yes, it can be tricky to discern the origin and/or intent of some
irregular collection of pitches, but OTOH, you can often reverse-
engineer a tuning by playing around with it and/or executing a "show
data" command in Scala. It's pretty obvious that 152-ET closely
approximates 11-limit JI, but to understand why Paul Erlich likes it
you'd also have to be interested in multi-purpose tunings with both
narrow (meantone-like) and wide (pajara) fifths. I'd have to say
that 224-ET is much better across the board; but I digress.

> There are temperaments
> arbitrarily close in accuracy to JI, and in practice any
> JI cannot be exact.

Yep, such is the folly of being a diehard JI-ist.

> Further, it is possible to express a
> temperament using rational numbers.

If you're using the term to mean a (tempered) tuning (which I'm not
sure you are), then yes, that was a lot of fun for me a couple of
years ago.

> I don't call a bag of
> pitches a tuning unless I know what the target is.

What do you mean by that? A set of pitches, listed as frequencies
(or cents) and expressible as a Scala file, *are* the target (to
which one tunes). (Technically, it's the intervals between the
pitches that are specified, since changing the starting pitch doesn't
really change the essence of the tuning, but merely transposes it.)

> temperament * tuning = scale

What does that mean?

As I see it, in order to make music using a temperament (class), you
need to choose a tuning by specifying the exact generator(s) and
period. Then, if that tuning is not a closed system (one-
dimensional), you need to specify a scale that limits the tuning to a
finite number of tones (within the audible range of pitches).

We're probably saying much the same things, but our terms may not
mean the same. Rigorous definitions can require a lot of careful
thought.

> In the case of a linear temperament, the tuning can be
> expressed with only two numbers (the size of the period
> and the generator), but the tuning could be more complex.
> For example, in the case of well temperaments (which
> I would like to call "circulating tunings").
>
> George wants to preserve the common usage of these terms.
> However, since the common usage isn't precise (uses these
> terms interchangeably), I don't think that's a good idea.

I'm only saying that it's not a good idea to assign a term (such
as "temperament", or "meantone" as used in conjunction
with "temperament") a meaning that is seldom (if ever) used outside
these lists. That makes terms even less precise than they already
are.

> > BTW: Tonalsoft Encyclopedia of Microtonal Music Theory also has
an
> > entry for temperament, but not (yet?) for "temperament class".
> > http://tonalsoft.com/enc/t/temperament.aspx
>
> Yes. The entry for temperament also shows several different
> definitions. Gene's is the one I am advocating.

Tee hee! You practically need a math degree to figure out what that
means.

Besides, Gene's definition is for a *regular* temperament, and
therefore not a temperament in general.

-- George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Torsten Anders wrote:
> > Thank you all very much for your replies!! I had no time yet to
test
> > all your recommendations, but I definitely will :)
> >
> > On Mar 27, 2008, at 6:28 AM, Petr Parízek wrote:
> >> If you want thirds and fifths to be closer to just, then you'll
get
> >> only a very poor approximation to 7/4 because the same interval
is
> >> also used here to approximate 16/9.
> >
> > Does there also exist a variant where the intonation of the
septimal
> > intervals is optimised (possibly to the detriment of thirds and
> > fifths) like the (major) thirds are optimised in meantone (to
the
> > detriment of the fifths)?
>
> The 7-limit minimax would be the equivalent of 1/4-comma
> meantone. The TOP tunings balance different intervals in
> different ways, so there's a slightly higher emphasis on the
> 5-limit.
>
> > On Mar 27, 2008, at 7:37 AM, Graham Breed wrote:
> >> Well temperament -- make the tuning more lumpy. You can
> >> make some chords better in tune at the expense of others.
> >> Naturally you'll probably want the most important chords to
> >> be well tuned. As a bonus you can get the temperament to
> >> circulate, so that you can modulate over a fixed number of
> >> notes. 22 and 34 are the most obvious choices.
> > I have never created a well temperament yet -- is there
some "gentle
> > introduction" to this subject available I could read?
>
> I don't think there's a general theory. Most of the
> attention's on 12 note scales and George Secor also has a
> paper on a 17 note well temperament.

You'll find it here:
http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf
Although it doesn't go much into theory, it's an easy read.

> I thought a 34 note
> well temperament came out of that as well, but he didn't
> mention it so maybe it doesn't apply to pajara.

Huh? You must have missed my reply:
/tuning/topicId_75735.html#75741
You'd have to follow the links to get the full story. The bottom
line is that you'll find an .scl file of the 34-WT.

--George

🔗Charles Lucy <lucy@harmonics.com>

🔗Herman Miller <hmiller@IO.COM>

🔗Graham Breed <gbreed@gmail.com>

🔗monz <joemonz@yahoo.com>

🔗Graham Breed <gbreed@gmail.com>

George D. Secor wrote:

> I'd really like "temperament family" now that:

I'd really prefer we didn't reopen that discussion. "Temperament family" is on of the few terms we agree on.

> 1) Torsten asked Graham to explain what a "temperament class" is;

It's a concept that didn't have a name before. People are going to want it clarified whatever name it has. It helps this discussion if you can say "the decatonic scale depends on certain commas being tempered out, and they imply the pajara temperament class". I'm not sure if Torsten was initially clear that what he wanted was a tuning for pajara. If he'd been able to say "I want a tuning for pajara" we'd have understood that he understood this. (And no need for the term "temperament class" until somebody asks "what's pajara?")

Where I don't like "temperament class" it's because it implies temperament. I think it's useful to think of things that you wouldn't normally call temperaments, like JI periodicity blocks, as belonging to a temperament class. But none of the proposed terms really cover "temperament class without the temperament".

The best I can think of, FWIW, is "T-class" which is an obvious generalization of "temperament class". I'm reminded of Freud's term "ucs" (or "Ubw" or something in the original) which he introduced as an abbreviation for "unconscious" but with a precise meaning that was distinct from mere unconsciousness. And then, of course, he went on to use it as a plain synonym for "unconscious" with all the ambiguity that he was trying to avoid. So I try to be careful with new terms.

> 2) "Family" could be used by itself more readily than "class" in > referring to the more abstract meaning of temperament.
> > One could therefore use "pajara temperament" and "pajara family" > interchangeably, since pajara would be understood to be a family of > temperaments (tunings), but "meantone family" would be required to > indicate the more abstract meaning, since "meantone temperament" has > a historical meaning that's still widely used to specify a particular > (tempered) tuning.

In many contexts you can still use them interchangeably because pajara can be a family of equal temperaments.

<snip>
> As I see it, in order to make music using a temperament (class), you > need to choose a tuning by specifying the exact generator(s) and > period. Then, if that tuning is not a closed system (one-
> dimensional), you need to specify a scale that limits the tuning to a > finite number of tones (within the audible range of pitches).

You don't have to be specific. You can write a piece in meantone (or whatever you want to call it) and let the performers work out how to balance its melodic and harmonic implications. This is what composers have done throughout the centuries. And it's why temperament classes are more important than temperaments.

Graham

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...>
wrote:
>
> Thank you all very much for your replies!! I had no time yet to
test
> all your recommendations, but I definitely will :)

Hi Torsten,

I thought it was about time I replied directly to one of your
messages about a couple of things that weren't completely answered.

> On Mar 27, 2008, at 6:28 AM, Petr Parízek wrote:
> > If you want thirds and fifths to be closer to just, then you'll
get
> > only a very poor approximation to 7/4 because the same interval
is
> > also used here to approximate 16/9.
>
> Does there also exist a variant where the intonation of the
septimal
> intervals is optimised (possibly to the detriment of thirds and
> fifths) like the (major) thirds are optimised in meantone (to the
> detriment of the fifths)?

It looks as if that would have to be something very close to 22-
equal. Because ~7/5 is fixed at 600 cents in pajara (with pure
octaves; but, in any case, something very close to that), further
decreasing the error of 7/4 (by lowering it) causes the error of 5/4
(which also must go lower) to increase further in error. A major 3rd
that's narrower than in 22-equal is not very good melodically, which
I suspect is one of the things that caused you to look for
alternatives to 22-equal (in addition to the heavily tempered fifths).

> ...
> On Mar 27, 2008, at 7:37 AM, Graham Breed wrote:
> > Well temperament -- make the tuning more lumpy. You can
> > make some chords better in tune at the expense of others.
> > Naturally you'll probably want the most important chords to
> > be well tuned. As a bonus you can get the temperament to
> > circulate, so that you can modulate over a fixed number of
> > notes. 22 and 34 are the most obvious choices.
> I have never created a well temperament yet -- is there
some "gentle
> introduction" to this subject available I could read?

Please do check out the 34-tone well-temperament that I discussed
here, because (in its dozen or so best keys) it doesn't damage 7/4
nearly as much as does 34-equal:
/tuning/topicId_67957.html#68032

I've tried (on several occasions) to construct some sort of 22-tone
well temperament, but I wasn't able to come up with anything
satisfactory.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> George D. Secor wrote:
>
> > I'd really like "temperament family" now that:
>
> I'd really prefer we didn't reopen that discussion.
> "Temperament family" is on of the few terms we agree on.
>
> > 1) Torsten asked Graham to explain what a "temperament class" is;
>
> It's a concept that didn't have a name before. People are
> going to want it clarified whatever name it has. It helps
> this discussion if you can say "the decatonic scale depends
> on certain commas being tempered out, and they imply the
> pajara temperament class". I'm not sure if Torsten was
> initially clear that what he wanted was a tuning for pajara.
> If he'd been able to say "I want a tuning for pajara" we'd
> have understood that he understood this. (And no need for
> the term "temperament class" until somebody asks "what's
> pajara?")

Okay, I'll leave "temperament family" alone and use "class" in those
instances required to avoid ambiguity..

> Where I don't like "temperament class" it's because it
> implies temperament. I think it's useful to think of things
> that you wouldn't normally call temperaments, like JI
> periodicity blocks, as belonging to a temperament class.
> But none of the proposed terms really cover "temperament
> class without the temperament".
>
> The best I can think of, FWIW, is "T-class" which is an
> obvious generalization of "temperament class". I'm reminded
> of Freud's term "ucs" (or "Ubw" or something in the
> original) which he introduced as an abbreviation for
> "unconscious" but with a precise meaning that was distinct
> from mere unconsciousness. And then, of course, he went on
> to use it as a plain synonym for "unconscious" with all the
> ambiguity that he was trying to avoid. So I try to be
> careful with new terms.

Sure.

> > 2) "Family" could be used by itself more readily than "class" in
> > referring to the more abstract meaning of temperament.
> >
> > One could therefore use "pajara temperament" and "pajara family"
> > interchangeably, ...
>
> In many contexts you can still use them interchangeably
> because pajara can be a family of equal temperaments.
>
> <snip>
> > As I see it, in order to make music using a temperament (class),
you
> > need to choose a tuning by specifying the exact generator(s) and
> > period. Then, if that tuning is not a closed system (one-
> > dimensional), you need to specify a scale that limits the tuning
to a
> > finite number of tones (within the audible range of pitches).
>
> You don't have to be specific. You can write a piece in
> meantone (or whatever you want to call it) and let the
> performers work out how to balance its melodic and harmonic
> implications. This is what composers have done throughout
> the centuries. And it's why temperament classes are more
> important than temperaments.

Because I said "make music", not "write music", a tuning needs to be
chosen. I was thinking of MMM, where the composer is usually also
the (more often than not, electronic) performer.

Still, you have a valid point: if someone else writes something in
pajara, if I'm performing it, then I may choose to tune it as I like.

(Back to lurking again for a while -- I hope. :-)

--George

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

🔗tuneit99 <tuneit99@yahoo.com>

I am new here and in trying to get educated on how to interpret some
of this esoteric material, I wonder if you would indulge me in giving
the series of notes that correspond to your tuning. I have done some
analysis and have discovered some of the irregular fifths. With the
Well tempering I am just not sure in some cases which is A and which
is Bbb, for example. In the following (your listing) the zero value
is apparently implied to amount to 34 values. I believe C=0,
and "2/1" can be ignored for the sake of deriving fifth interval
values. My worksheet is actually three octaves in order to contain
all the fifth intervals.

Your fifth interval value is mostly 707.22045 cents except
where "tempered". I haven't gone as far as analyzing whether
subsequent fifths are relative to the tempered value or the
calculated value.

I might actually try the core tuning, the "in tune" part, but have to
tune strings, not program electronics.

Thanks in advance for the help.

! secor_34wt.scl
!
George Secor's 34-tone well temperament (with 10 exact 11/7)
34
!
40.47925
66.74120
107.22045
144.85624
171.11819
214.44090
249.23324
278.33864
321.66136
353.61023
385.55910
428.88181
457.98722
492.77955
536.10226
562.36421
600.00000
640.47925
666.74120
707.22045
744.85624
771.11819
814.44090
849.23324
878.33864
921.66136
953.61023
985.55910
1028.88181
1057.98722
1092.77955
1136.10226
1162.36421
2/1

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> Please do check out the 34-tone well-temperament that I discussed
> here, because (in its dozen or so best keys) it doesn't damage 7/4
> nearly as much as does 34-equal:
> /tuning/topicId_67957.html#68032
>
> I've tried (on several occasions) to construct some sort of 22-tone
> well temperament, but I wasn't able to come up with anything
> satisfactory.
>
> --George
>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "tuneit99" <tuneit99@...> wrote:
>
> I am new here and in trying to get educated on how to interpret
some
> of this esoteric material, I wonder if you would indulge me in
giving
> the series of notes that correspond to your tuning. I have done
some
> analysis and have discovered some of the irregular fifths. With the
> Well tempering I am just not sure in some cases which is A and
which
> is Bbb, for example. In the following (your listing) the zero value
> is apparently implied to amount to 34 values. I believe C=0,

Yes, C = 0 cents.

> and "2/1" can be ignored for the sake of deriving fifth interval
> values.

Yes.

> My worksheet is actually three octaves in order to contain
> all the fifth intervals.

Two octaves should suffice.

You can think of this tuning as two separate circles of fifths of 17
tones each. One circle consists of all of the even-numbered tones
and the other of all the odd-numbered tones. The second circle is
exactly like the first, the only difference being that it's
transposed upward by 600 cents (exactly 1/2 octave). Thus the well-
temperament repeats at the half-octave (i.e., has a period of 600
cents).

> Your fifth interval value is mostly 707.22045 cents except
> where "tempered". I haven't gone as far as analyzing whether
> subsequent fifths are relative to the tempered value or the
> calculated value.

All of the fifths are tempered. (Only fifths that are exactly 2:3,
or ~701.955 cents, are considered untempered, i.e., pure or just.)
In this 34-WT there are two different sizes:
707.22045c (~5.265c wide) in a chain from Ab to B
704.37699c (~2.422c wide) in a chain from Cv to G^
where Ab and B are respelled G^ and Cv, respectively, with the
additional pair of accidentals representing:
v = semiflat (~32:33, or 11-diesis, down)
and
^ = semisharp (~32:33 up)

The tones in the first circle of 17 are then spelled (in ascending
order, showing alternate spellings):
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
C C^ C# Eb Ev E E^ Gb Gv G G^ G# Bb Bv B B^
B^ Db Dv D D^ D# Fv F F^ F# Ab Av A A^ A# Cv C

(To view the above with the proper spacing, click on "Show Message
Option" in the right margin of this message; then click on "Use Fixed
Width Font".)

The above can be found (with older half-sharp/flat symbols) in the
next-to-last page of my 17-tone paper:
http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf

To notate the 17 tones in the other circle we'll need another pair of
accidentals:
\ for ~80:81 (5-comma) down
/ for ~80:81 (5-comma) up

So a portion of the WT (from D to G) would be notated:

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Eb\ Eb Eb/ Ev E\ E E/ E^ Gb\ Gb Gb/ Gv G\ G
D D/ D^ D#\ D# Fv F\ F F/ F^ F#\ F# F#/

Observe that the tone approximating 5/4 of C is notated E\ (E lowered
by a 5-comma) and the tone approximating 11/8 of C is notated F^ (F
raised by an 11-diesis).

The tones approximating 7/4 and 7/6 of C are simply Bb and Eb,
respectively. Since the 7-comma, 63:64, vanishes in the temperament
mapping, no microtonal accidentals are required for those tones.

The above microtonal accidentals are ASCII "shorthand" characters
that represent microtonal accidentals in the Sagittal notation system:
http://dkeenan.com/sagittal/Sagittal.pdf
(A few of the shorthand characters are shown on page 5, in Figure 3.)

> I might actually try the core tuning, the "in tune" part, but have
to
> tune strings, not program electronics.

Have you tried Scala (freeware -- no programming required)?
http://www.xs4all.nl/~huygensf/scala/
If so, you can save the secor_34wt.scl file listing to a text file
(with the .scl extension), load it in Scala, and play it on Scala's
chromatic clavier. To see the Sagittal notation for this tuning,
enter "set nota sa34" on the command line.

If you want to view a listing of the tones with the shorthand
notation, then enter "set sagi mixed short" on the command line and
then click on the Edit button at the top.

Also, the command "show data" will give you a ton of information
about the tuning. There's a lot of other neat stuff, too, including
a brief tutorial to help new users get started:
http://www.xs4all.nl/~huygensf/scala/dummies.txt

> Thanks in advance for the help.

My pleasure!

--George

🔗tuneit99 <tuneit99@yahoo.com>

In the "17 tone paper" I found issue with the chronology from
Pythagorean to 1/4 comma meantone temperament.

From page 2 of the linked pdf, but indicating 56 in the pdf image,

"Once tones related by simple ratios involving the prime number 5
were discovered to be harmonically consonant (beginning in the 13th
century), it was found that if the fifths were altered (or tempered)
slightly narrow, the resulting scale would be better for triadic
harmony. The meantone temperament, devised in 1523 by Pietro Aron,
has a fifth tempered by one-quarter of Didymus' comma (the amount by
which four just fifths less two octaves exceeds a just major third),
of approximately 696.6 cents."

Meantone, by all descriptions I have read, was derived from Just
Intonation, not Pythagorean. Although JI has mostly pure fifths, they
are not all pure, and thus the distinctive difference from
Pythagorean in practical terms. Since JI was already "better for
triadic harmony" than Pythagorean taken beyond monotone, pentatonic
scales, the motivation for meantone is muddied or misrepresented. As
a player of a diatonic chording instrument, I know that I have no ii
chord in Just Major and no usable V in Just minor. The instrument's
repertoire is very confined unless I consistently play in a, what
shall we call it, temperament, 2/9 comma being what I settled on
years ago, 1/4 comma found a little dreary for my ear.

Since you are probably quite aware of all this, perhaps that section
simply begs a bit of editing for clarity.

I'll be back with more, I'm sure. This is great. You might want to
set some of this up as a way for other newcomers to get their
bearings here.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
>
> The above can be found (with older half-sharp/flat symbols) in the
> next-to-last page of my 17-tone paper:
> http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf
>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "tuneit99" <tuneit99@...> wrote:
>
> In the "17 tone paper" I found issue with the chronology from
> Pythagorean to 1/4 comma meantone temperament.
>
> From page 2 of the linked pdf, but indicating 56 in the pdf image,
>
> "Once tones related by simple ratios involving the prime number 5
> were discovered to be harmonically consonant (beginning in the 13th
> century), it was found that if the fifths were altered (or
tempered)
> slightly narrow, the resulting scale would be better for triadic
> harmony. The meantone temperament, devised in 1523 by Pietro Aron,
> has a fifth tempered by one-quarter of Didymus' comma (the amount
by
> which four just fifths less two octaves exceeds a just major
third),
> of approximately 696.6 cents."
>
> Meantone, by all descriptions I have read, was derived from Just
> Intonation, not Pythagorean.

I would prefer to say that it was a synthesis of the two tunings in
that the former contains just major 3rds, while the latter has all
major (and all minor) triads tuned alike. I described meantone
temperament in terms of the Pythagorean tuning simply because
Pythagorean (not JI) was the standard tuning up to that time.

> Although JI has mostly pure fifths, they
> are not all pure, and thus the distinctive difference from
> Pythagorean in practical terms. Since JI was already "better for
> triadic harmony" than Pythagorean taken beyond monotone, pentatonic
> scales, the motivation for meantone is muddied or misrepresented.
As
> a player of a diatonic chording instrument, I know that I have no
ii
> chord in Just Major and no usable V in Just minor. The instrument's
> repertoire is very confined unless I consistently play in a, what
> shall we call it, temperament, 2/9 comma being what I settled on
> years ago, 1/4 comma found a little dreary for my ear.

My favorite alternative to 1/4-comma temperament is a synchronous-
beating temperament that's very close to 5/23-comma, where the 5th
and major 3rd in a major triad beat at the same rate (and the minor
3rd four times as fast).

> Since you are probably quite aware of all this, perhaps that
section
> simply begs a bit of editing for clarity.
>
> I'll be back with more, I'm sure. This is great. You might want to
> set some of this up as a way for other newcomers to get their
> bearings here.

Yes, that would be nice, but where to put it? Would you think to
look in the Files section if I created a folder? (I just did -- see
if you can find it.)

My 17-tone paper is, for the most part, rather narrow in scope. Of
much more general interest would be the brief history and rationale
for alternative tunings that I recently wrote:
http://dkeenan.com/sagittal/gift/GiftOfTheGods.htm
This serves as the introduction to the "mythical" account of how the
Sagittal notation was developed (read the rest if you like edu-
tainment).

Another good resource is the alternative tunings encyclopedia that
Joe Monzo has been compiling (incorporating the thoughts of many on
this forum) over the years:
http://tonalsoft.com/enc/encyclopedia.aspx

Anyway, if you think that would be a good place to look, I'll put a
file there with some comments & helpful links for new tuning members.

--George

🔗kraiggrady@anaphoria.com

George`
i quite like those synchronous beating tempermentsof yours. i played though a couple of them when you where first postingthem to the list. It deserves far more attention that it got at thetime.

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: George D. Secor [mailto:gdsecor@yahoo.com]
Sent: Monday, April 7, 2008 12:06 PM
To: tuning@yahoogroups.com
Subject: [tuning] New Member Orientation (was: Decatonic scales: temperaments ...)

Yes, that would be nice, but where to put it? Would you think to
look in the Files section if I created a folder? (I just did -- see
if you can find it.)

Anyway, if you think that would be a good place to look, I'll put a
file there with some comments & helpful links for new tuning members.

--George

🔗Carl Lumma <carl@lumma.org>

🔗kraiggrady@anaphoria.com

ï¿½I must of been off list for that one.ï¿½
all the mt. Meru scales BTW have this property
,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:ï¿½
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

ï¿½
-----Original Message-----
From: Carl Lumma [mailto:carl@lumma.org]
Sent: Monday, April 7, 2008 03:10 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: New Member Orientation (was: Decatonic scales: temperaments ...)

Hi Kraig,

Synch. beating well temperaments got a huge response here
(and on tuning-math) after Bob Wendell introduced the idea
in 2004 (IIRC). Several people designed WTs of this kind
and posted them, including George, Aaron Johnson, Gene,
and myself. Dave K. also had some very interesting things
to say, and I posted three batches of audio samples.
Gene gave equations that make it possible to find synch-
beating versions of any well temperament, and he put
some synch-tuned MIDI files on his site. Manuel added
beat ratios to Scala. I can't think how the response
could have been bigger.

-Carl

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> George`
> i quite like those synchronous beating tempermentsof yours. i
> played though a couple of them when you where first postingthem
> to the list. It deserves far more attention that it got at thetime.
>
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria

🔗tuneit99 <tuneit99@yahoo.com>

🔗Carl Lumma <carl@lumma.org>

🔗tuneit99 <tuneit99@yahoo.com>

🔗Carl Lumma <carl@lumma.org>

--- In tuning@yahoogroups.com, "tuneit99" <tuneit99@...> wrote:
>
> I don't know that Just Intonation requires any qualification
> as "5-limit". I feel unbound in that regard. It's simply JI
> as in any reference one will find outside of Yahoo! tuning.

Not true. The "limit" terminology is used throughout the
literature of the Just Intonation Network (and their
journal 1/1), and goes back to Partch. In the world at
large, just intonation usually implies the 5-limit, and
people say "extended just intonation" to refer to stuff
beyond that. Here when we say "JI" we don't mean specifically
the 5-limit, so it would seem you're in good company here.

> Diatonic, fixed pitch instruments did and still do use
> Just intonation. The psaltery and its descendant zither
> are examples.

Let me rephrase that: Among instruments on which meantone,
well temperament and/or equal temperament were standardized,
which of them are you claiming had previously standardized
on JI?

> JI ws no less useful on fixed pitch instruments than
> Pythagorean. I guess you will spring the snare and explain
> your point in asking.

Yes, sorry. There was a long period during which Pythagorean
("3-limit JI") was the standard for keyboard instruments.
This regime gave way directly to meantone. It is a common
misconception that there was a period of 5-limit JI use
first. If you are claiming that, now would be the time to
present evidence, citations, etc.

> I guess I am a bit edgy because of these peculiar,
> microscopic, seemingly deliberately intimidating distinctions
> you fellas want to make in terminology, unheard of outside
> of this group. I really don't mean to sound unfriendly or
> ungrateful.

I agree that the arcane terminology probably has gone too
far. In defense of it all, it is quite a challenge to
communicate via e-mail with people from all parts of the
world and many different backgrounds about a complicated
subject in which there is little agreement as to the
ultimate goal of it all. You don't sound unfriendly, but
you do sound like you've believe you already know all
the answers.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

Dear "Tuneit99" (sorry, but that's the best I can do, since I don't
think you have yet told us your actual name),

I apologize if I did not make myself clear. My intention was not to
correct you, but simply to explain my understanding that meantone
followed *both* Pythagorean (the *tuning actually used* for many
centuries up to that time) and JI (a rediscovered *tuning ideal* used
by the ancient Greeks). As such, I see it as a reconciliation (or
synthesis) between an actual and an ideal, so that to ignore one or
the other is to miss half the picture. If you choose not to see it
that way, then we can agree to disagree. But at least let me explain
how I came to this conclusion.

If all of the descriptions you have read say that meantone was
derived from JI, not Pythagorean, then I would have to think that
either they were mistaken, or else you're making an incorrect
inference from what was said or (more likely) not said. When I first
investigated alternative tunings (in the mid-1960's), there were not
very many sources of information available to me. Nevertheless,
there were differing viewpoints and biases, and it didn't take me
long to discover that some of these sources contained all sorts of
misleading information: either incomplete, or inaccurate, or
incorrect, or mere opinion, or some combination of these. Sadly,
much of this misinformation appeared in articles in music
dictionaries and encyclopedias, written by people who were supposedly
authorities on the subject. Since some of these seemed to be passing
on misinformation from others, I concluded that a majority opinion
cannot be assumed to be correct.

One blatant example I encountered was the widespread belief that
melody involving intervals smaller than a semitone (such as found in
musical scales in the East, and in experimental microtonal scales) is
incompatible with harmony, from which one was expected to draw the
following conclusions: 1) a 12-tone octave, specifically 12-equal, is
best for harmony; and 2) microtonality is not an appropriate and/or
desirable path for the music of the West.

Anyway, to get back to the point in question, my most reliable
historical source was J. Murray Barbour's _Tuning and Temperament_,
from which I'll quote from the beginning of chapter 3
(titled "Meantone Temperament", pp. 25-26):

<< It is not definitely known when temperament was first used.
Vicentino stated that the fretted instruments had always been in
equal temperament. As for the keyboard instruments, Zarlino declared
that temperament was as old as the complete chromatic keyboard. It
may well be that some organs in the fifteenth century had had
temperament of a sort, although the Pythagorean tuning continued to
have too many advocates not to have been dominant in the earlier
period. However that may be, Riemann discovered the first mention of
temperament in a passage from Gafurius' _Practica musica_ (1496).
[footnote: Hugo Riemann, _Geschichte der Musiktheorie_ (Berlin,
1898), p. 327] There, among the eight rules of counterpoint,
Gafurius said that organists assert that fifths undergo a small,
indefinite amount of diminution called temperament (_participata_).
Since he was reporting a contemporary fact, rather than advocating an
innovation, the practice may have begun decades earlier than his time.

... Grammateus' division of Pythagorean tones into equal semitones
came only twenty-two years after Gafurius' observation, and ranks
very high among irregular systems that approach equal temperament. ...

Dechales had no authority for stating that Guideo of Arezzo was the
father of temperament. The association of Ramis with temperament is
one of the most common misconceptions in the history of tuning. And,
although Schlick's system undoubtedly can properly be described as a
temperament, it is just as surely of an irregular variety. It is
well to mention these names, and discard each of them, before saying
that full credit for describing the meantone temperament must go to
Pietro Aron. >>

Barbour goes on to give Aron's directions for tuning meantone,
starting with a just C-E, then making the fifths C-G and G-D "a
little flat" and then tuning A "so that the fifths D-A and A-E are
equal." From these tuning directions I believe we can safely
conclude that this temperament was intended to be used on *keyboard
instruments*.

So you see that the first temperaments (in the several decades
preceding Aron's description of meantone, in 1523) were deliberate
alterations to the Pythagorean tuning, on keyboard instruments. I
would find it very difficult to believe that these early practical
experiments had no influence or bearing whatsoever on the development
of the meantone temperament. While it's possible to explain that
meantone temperament can be derived (abstractly) *solely* with
reference to just intonation, the historical context seems to
indicate that this was not the way it actually happened.

Do you have any evidence that just intonation was actually used on
keyboard instruments prior to meantone, or that Aron (or one of his
contemporaries) explicitly explained meantone temperament as having
been derived solely from JI, or that meantone temperament was
designed for (and consequently used on) an instrument previously
tuned to JI (such as a psaltery, as you suggest)? If such is the
case, then I'll gladly stand corrected.

Really, I'm not trying to intimidate you. Discussions in which there
exist differences of opinion can be contests, or they can be
opportunities to learn and to build a consensus. You were the one
who initiated this, when you said that you thought I needed to
clarify that meantone was derived (exclusively) from JI in one place
in my 17-tone paper, where I wrote:

> "Once tones related by simple ratios involving the prime number 5
> were discovered to be harmonically consonant (beginning in the 13th
> century), it was found that if the fifths were altered (or
tempered)
> slightly narrow, the resulting scale would be better for triadic
> harmony. The meantone temperament, devised in 1523 by Pietro Aron,
> has a fifth tempered by one-quarter of Didymus' comma (the amount
by
> which four just fifths less two octaves exceeds a just major
third),
> of approximately 696.6 cents."

This was a portion of my very brief (four-paragraph) "whirlwind"
history of tuning, against which I subsequently outlined a fictional
alternate history of tuning that could have been, and if it were so,
then where we might be today.

Is there anything in the above quote that's incorrect or misleading?
(Too late to change it anyway, since it's already published in
Xenharmonikon 18.) I referred to both 5-limit JI (as "simple ratios
involving the prime number 5") and tempered fifths, with the
(intended) implication that they were essential ingredients that led
to the development of the meantone temperament. One omission in the
above quote, which has come up in the course of this discussion, is
that temperaments have been developed mainly on (and for) keyboard
instruments, but I can't see that as serious.

If the tuning of the psaltery had any part to play (was anyone using
it in JI in the early 16th century?) in the development of meantone,
then that would be good to know.

--George

--- In tuning@yahoogroups.com, "tuneit99" <tuneit99@...> wrote:
>
> Attribute abstract complexity to it if you wish, but I choose not
to be
> corrected on that point. Meantone clearly followed JI and addressed
its
> limitations.
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "tuneit99" <tuneit99@> wrote:
> > >
> > > Meantone, by all descriptions I have read, was derived from
Just
> > > Intonation, not Pythagorean.
> >
> > I would prefer to say that it was a synthesis of the two tunings
in
> > that the former contains just major 3rds, while the latter has
all
> > major (and all minor) triads tuned alike. I described meantone
> > temperament in terms of the Pythagorean tuning simply because
> > Pythagorean (not JI) was the standard tuning up to that time.
> >
>

Decatonic scales: temperaments with relatively just thirds and fifth?

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