Graham's search for a new paradigm - the seeds of another candidate.

I read Graham page on "new" paradigms with interest.
As an "interested" party, I felt that he underemphasised the significance of the meantone paradigm.

In my usual egocentric fashion, I shall attempt to explain how I conceptualise all microtonal tuning systems.

Although I originally wrote it some 20 years ago, see:

http://www.lucytune.com/new_to_lt/pitch_01.html

"octave (was) subdivided in three basic ways;

1) By a geometric progression, with any number of equal intervals.Eg. 12 (as on conventional guitars) [100 cents per semitone or interval]; 31 as advocated by Huyghens (1629-1695) [38.71 cents per interval], or 53 by Mercator and Bosanquet (1876 Treatise) [22.64 cents];

2) By low whole number ratios. Eg. (Just Intonation) 3:2 for the Vth; 5:4 for the major third etc.

3) By cumulative fifths. Eg. Pythagorean Tuning 3:2 for the Vth; but 81:64 for the major third.

There are also hybrids of the other three, Eg. Meantone Temperaments"

I can still imagine these types of tunings in a similar categorisation except that nowadays, rather than thinking of Meantone-type temperaments as "hybrids", I begin my conceptualisation of all tunings from a very specific meantone perspective.

From this biased point of view:

1) Equal temperaments are meantone-type systems which within a small number of steps become circles, instead of spirals; I see these as subsets of particular meantones;
There are many where the ratio of the cents of the Large and small intervals can be expressed as small integer ratios and follow the pattern of 2L+5s = one octave:

seee: http://www.lucytune.com/tuning/equal_temp.html

e.g. in 12 EDO 1 Large interval= 2 small. i.e 2/1 (or its reciprocal)
19 EDO 2 Large = 6 small. i.e. 3/2
31 EDO 3 Large = 5 small. i.e. 5/3

etc. etc.

Anyone reading this list can calculate the others for themselves, and tell us the size of the corresponding fifths or Large intervals in cents.

It could be argued that eventually any meantone where the size of the Large interval has been arrived at using rational numbers, including all the n/n comma meantones, will eventually become a circle and exactly repeat.
I am sure that Gene Wizard-Smith and others would be able to provided the mathematical proof of this.
So these too can be viewed as a form of equal temperaments, although the size of the intervals could become iotic. (is that a legitimate adjective?)

2) JI and other low integer frequency ratio systems.

I regard these intervals, not so as tuning systems; more as "landmarks" of some zero beating, which are intimately associated with the beating which is heard in all tuning systems.

3) Pythagorean tuning displays the characteristics of both JI and meantone systems

Now for the crux of my paradigm, for which I am extending my conceptualisation of meantone-type systems to include more patterns, not only of 5L + 2s; yet also (when necessary) to numbers other than 5 and 2; and to more than two different sizes of interval.
I am also prepared to also include octave rations of other than the traditional integer 2.

So I wish to open the limitaions of the traditional understanding of the (to my mind very unsatisfactory term) "meantone.

(Prehaps we need a new word for this concept;-). (extended meantone?) (Someone else must be able to do better than than;-)

I propose that it is possible to map any tuning system that anyone can ever devise, usng this concept of "extended meantone".

So I suggest that all tuning systems can be considered to be "extended" meantones".

My own personal way of analysing tuning systems (as you would expect) is from a LucyTuning perspective.

Remember, I am using a 5L + 2s meantone, a very particular meantone which is derived from the irrational, transcenental number pi, as the "yardstick" from which I am considering all other systems.

When I come accross a "new to me" tuning system, my comparisons to small integer ratios are not to evaluate "how good a fit" is this tuning to JI concepts and dogma.

My interest in JI intervals is in how the tuning beats, and how close that beating pattern is to the beat patterns heard in LucyTuning.

So my "new" paradigm (for the week?) is to analyse all tunings as types of "extended" meantones.

I expect your skepticism, yet before rejecting it out of hand, please treat it, at least, as a thought experiment, and visualise how it will also fit into the associated concept of ScaleCoding.

For ScaleCoding details see:

http://www.lucytune.com/new_to_lt/pitch_05.html

It works well for me.

(or do I have to emulate my "faithful, old" friends, and produce multi-million sellers before anyone is convinced? ;-)

Charles Lucy - lucy@lucytune.com ------------ Promoting global harmony through LucyTuning ------- for information on LucyTuning go to:
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for LucyTuned Lullabies, contest and Flash cartoon go to
http://www.lullabies.co.uk
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Charles Lucy wrote:
> I read Graham page on "new" paradigms with interest.
> As an "interested" party, I felt that he underemphasised the > significance of the meantone paradigm.

If anybody's wondering about this page, it's linked to from the usual

http://x31eq.com/tuning.htm

but it's still a bit rough and ready so I haven't announced it here yet.

Contrary to your title, it isn't a search for anything, but a description of a new paradigm we happened to arrive at.

> In my usual egocentric fashion, I shall attempt to explain how I > conceptualise all microtonal tuning systems.
> > Although I originally wrote it some 20 years ago, see:
> > http://www.lucytune.com/new_to_lt/pitch_01.html
> > > "octave (was) subdivided in three basic ways;
> > 1) By a geometric progression, with any number of equal intervals.Eg. > 12 (as on conventional guitars) [100 cents per semitone or interval]; > 31 as advocated by Huyghens (1629-1695) [38.71 cents per interval], > or 53 by Mercator and Bosanquet (1876 Treatise) [22.64 cents];

That's equal temperament, one of my paradigms.

> 2) By low whole number ratios. Eg. (Just Intonation) 3:2 for the Vth; > 5:4 for the major third etc.

And that's in my list as well.

> 3) By cumulative fifths. Eg. Pythagorean Tuning 3:2 for the Vth; but > 81:64 for the major third.
> > There are also hybrids of the other three, Eg. Meantone Temperaments"

If you only talk about fifths, you're restricted to my "Do Re Mi..." paradigm.

> I can still imagine these types of tunings in a similar > categorisation except that nowadays, rather than thinking of Meantone- > type temperaments as "hybrids", I begin my conceptualisation of all > tunings from a very specific meantone perspective.
> > From this biased point of view:
> > 1) Equal temperaments are meantone-type systems which within a small > number of steps become circles, instead of spirals; I see these as > subsets of particular meantones;
> There are many where the ratio of the cents of the Large and small > intervals can be expressed as small integer ratios and follow the > pattern of 2L+5s = one octave:

So what do you do about scales without a recognizable fifth? Or where the fifth closes a circle before you have enough notes (e.g. 24-equal)?

> It could be argued that eventually any meantone where the size of the > Large interval has been arrived at using rational numbers, including > all the n/n comma meantones, will eventually become a circle and > exactly repeat.
> I am sure that Gene Wizard-Smith and others would be able to provided > the mathematical proof of this.
> So these too can be viewed as a form of equal temperaments, although > the size of the intervals could become iotic. (is that a legitimate > adjective?)

It's obvious that any meantone with a tone as a rational fraction of the octave will be an equal temperament. The n/d comma meantones don't fit that pattern. Their tones are irrational and they never form a circle.

> 2) JI and other low integer frequency ratio systems.
> > I regard these intervals, not so as tuning systems; more as > "landmarks" of some zero beating, which are intimately associated > with the beating which is heard in all tuning systems.

Okay.

> 3) Pythagorean tuning displays the characteristics of both JI and > meantone systems

Right.

> Now for the crux of my paradigm, for which I am extending my > conceptualisation of meantone-type systems to include more patterns, > not only of 5L + 2s; yet also (when necessary) to numbers other than > 5 and 2; and to more than two different sizes of interval.
> I am also prepared to also include octave rations of other than the > traditional integer 2.

These are what I call regular tunings. I don't think it's a standard term. All I did was take the "temperament" out of "regular temperament".

> So I wish to open the limitaions of the traditional understanding of > the (to my mind very unsatisfactory term) "meantone.
> > (Prehaps we need a new word for this concept;-). (extended meantone?) > (Someone else must be able to do better than than;-)

Extended meantone usually means meantone with more than 12 notes.

> I propose that it is possible to map any tuning system that anyone > can ever devise, usng this concept of "extended meantone".

Maybe you can for existing systems (trivially by using a different size of interval (we call them generators) for every, uh, different size of interval) but it isn't always a helpful way of going about it. And one day somebody will devise a tuning system based on each note being tuned as a chaotic function of previous notes, just to prove you wrong.

> So I suggest that all tuning systems can be considered to be > "extended" meantones".

In that case you might have the regular mapping paradigm without the regular mappings.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> These are what I call regular tunings. I don't think it's a standard
> term. All I did was take the "temperament" out of "regular
temperament".

I like it, though. Maybe I'll rewrite my web pages to conform.