Hello, there, everyone, and it is with some trepidation that I find

myself writing at a time of more bombings and violence in the Middle

East. May these words, however trivial they may seem, serve to

encourage understanding among us as a musical community, possibly

providing a model for a larger world community of peace with music as

a force for communication and active reconciliation.

Mainly I would like to suggest that there are a range of "middle

paths" between classic multiprime just intonation (JI) systems and

equal temperaments or divisions of the octave (n-tET/n-EDO).

Further, tunings with octaves stretched or narrowed by a considerable

amount from a pure 2:1, or where an "octave" may not be deemed a

relevant concept (e.g. Bohlen-Pierce tuning in JI and tempered

versions based on a pure 3:1 "tritave"), provide other musical paths,

as do "spectrum tunings" based on the timbre of a specific type of

sound generator (e.g. acoustical or electronic).

Here I would like to present three types of "middle path" tunings, a

list possibly more suggestive than exhaustive.

-------------------------------

1. Pythagorean-based JI systems

-------------------------------

Pythagorean tunings featuring a chain of pure 3:2 fifths or 4:3

fourths have the property of being at once just and regular.

By combining two 12-note Pythagorean chains at some convenient

distance, for example a septimal comma of 64:63 (~27.26 cents), we can

achieve a JI system with the usual Pythagorean intervals within each

chain, plus intervals mixing notes from the two chains at pure ratios

of 2-3-7 prime (or 2-3-7-9 odd) such as 9:7, 7:6, 12:7, 7:4, and 8:7.

We thus get a 24-note system combining elements of regularity and

purity with free transposibility: in effect, we have a regular chain

of 23 fifths, with the one minor compromise that a single fifth is

stretched by about 3.80 cents, the 3-7 schisma, to obtain pure ratios

of 2-3-7-9.

Of course, like a regular 24-note Pythagorean tuning with 23 pure

fifths, this is an open rather than circulating system; for

circulation, we'd need a cycle of 53 notes in pure fifths. However, a

24-note chain provides lots of transpositions, much like an open

meantone or other eventone tuning (see next section) of this size.

For the kind of neo-medieval European style that the arrangement of

two 12-note Pythagorean keyboards a 64:63 apart might suggest, both

the traditional Pythagorean intervals and the new 7-based ones are

likely to be highly useful and largely interchangeable.

While other distances between the two Pythagorean chains can be very

attractive also, free transposibility seems an element of "middle

path" approaches, so it is the 64:63 system that I mention.

-------------------

2. Eventone tunings

-------------------

In addition to the meantone tunings discussed in this dialogue on

"middle" paths, I would mention other "eventone" tunings with regular

fifths all of the same size, and usual major thirds formed from four

of these fifths up less two octaves.

Meantones are eventones where these regular thirds more specifically

are at or near ratios of 5:4 and 6:5, with fifths narrower than the

pure 3:2 of Pythagorean tuning. Other eventones may seek to realize or

approximate ratios such as 9:7 and 7:6, or 14:11 and 13:11, with

fifths tempered in the _wide_ direction.

Meantones and other eventones can sometimes very closely approach

equal divisions of the octave, with the latter category possibly taken

as a subset of the former. However, identical or near-identical tuning

systems may be viewed from quite different musical perspectives.

For example, it is well-known that we can derive a circulating tuning

of 31 notes per octave from either 1/4-comma meantone with pure 5:4

thirds, or from 31-tET. These two tunings can both outstandingly serve

Renaissance and related styles of music, as well as some later styles.

As it happens, similarly, an eventone tuning with just 14:11 major

thirds is almost identical to 46-tET.

However, either of these mathematically synonymous tunings may be used

for some quite different musical purposes.

Someone who speaks of a "14:11 eventone" is likely focusing on the

regular and rather complex major third of this ratio (~417.51 cents)

formed from four fifths up, analogous to the 5:4 major third of

1/4-comma meantone, for example. Such a third nicely can serve a

neo-medieval European kind of style.

Many advocates of 46-tET, however, regard the "usual" major third as

the best approximation of 5:4, 15/46 octave or ~391.30 cents, an

interval formed from 21 fifths up.

For a 14:11 eventone enthusiast, this interval in 46-tET or its close

counterpart of ~391.92 cents in a just 14:11 tuning is a rather

"exotic" one, occurring at three positions in a 24-note tuning --

actually an augmented-second-plus-diesis.

As someone with a penchant for eventone tunings, I would say that the

elements of regularity and transposibility within a given range of

accidentals, and the diverse types of intervals generated as one

carries the system to more notes, are strong attractions.

----------------------------

3. Unequal well-temperaments

----------------------------

Both traditional 12-note well-temperaments of the kind often favored

in the era of European music around 1680-1850, and well-temperaments

based on tuning circles of other sizes such as 17 or 19 notes, offer

the advantages of a subtle shading of intervals within a system

featuring transposibility and circulation.

George Secor has published a superb 17-note well-temperament[1], while

Paul Erlich and others have devised new 12-note systems.

While eventone tunings may also have the property of circulation

(e.g. 1/4-comma meantone in a 31-note cycle, or a 14:11 tuning in a

46-note cycle), an unequal temperament often achieves such circulation

in fewer notes by varying the size of fifths in the tuning chain.

This technique, at the same time, causes variation in the sizes of

thirds and other intervals as one moves around the tuning circle. In

either a conventional 12-note scheme of the 17th-19th century type, or

a 17-note system such as Secor's, this gradation of intervals provides

an impressive musical resource.

----

Note

----

1. George Secor, "17-Tone Well Temperament," _Interval_ Volume 1,

Number 1 (Spring 1978), pp. 4-5, quoted in article by Brian McLaren,

"Secor well temperament," as posted by Gary Morrison, "Post from Brian

McLaren," 13 July 1994.

Most appreciatively,

Margo Schulter

mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> -------------------------------

> 1. Pythagorean-based JI systems

> -------------------------------

>

> Pythagorean tunings featuring a chain of pure 3:2 fifths or 4:3

> fourths have the property of being at once just and regular.

>

> By combining two 12-note Pythagorean chains at some convenient

> distance, for example a septimal comma of 64:63 (~27.26 cents), we

can

> achieve a JI system with the usual Pythagorean intervals within each

> chain, plus intervals mixing notes from the two chains at pure

ratios

> of 2-3-7 prime (or 2-3-7-9 odd) such as 9:7, 7:6, 12:7, 7:4, and

8:7.

>

> We thus get a 24-note system combining elements of regularity and

> purity with free transposibility: in effect, we have a regular chain

> of 23 fifths, with the one minor compromise that a single fifth is

> stretched by about 3.80 cents, the 3-7 schisma, to obtain pure

ratios

> of 2-3-7-9.

>

> Of course, like a regular 24-note Pythagorean tuning with 23 pure

> fifths,

It is rather like it. But for the specific definition of "middle

path" I had in mind, one would want an MOS -- since for me the

prototypical JI scales are periodicity blocks, and ETs are of course

tempered periodicity blocks. One could sharpen the fifths by a

fraction of a cent, in effect stretching the Pythagorean comma close

to the septimal comma value, and a 17- or 29- tone chain would then

exemplify the "middle path" quite well as well as embodying the

harmonic relationships you've outlined. However, from the point of

view of a performer/improviser working with two 12-tone keyboards, I

can understand the modus operandi for your specific arrangement quite

well.

>

> While other distances between the two Pythagorean chains can be very

> attractive also, free transposibility seems an element of "middle

> path" approaches, so it is the 64:63 system that I mention.

And even more so when the septimal comma becomes equal to the

Pythagorean comma through (micro)tempering.

>

> George Secor has published a superb 17-note well-temperament[1],

while

> Paul Erlich and others have devised new 12-note systems.

Margo -- I also devised a 22-tone well-temperament in my paper, which

allows for 5-limit-oriented Indian scales in some keys, and my 7-

limit-oriented decatonic scales in others. You might want to look at

my paper again -- I suspect you'll like this.

>

> While eventone tunings may also have the property of circulation

> (e.g. 1/4-comma meantone in a 31-note cycle, or a 14:11 tuning in a

> 46-note cycle), an unequal temperament often achieves such

circulation

> in fewer notes by varying the size of fifths in the tuning chain.

>

> This technique, at the same time, causes variation in the sizes of

> thirds and other intervals as one moves around the tuning circle. In

> either a conventional 12-note scheme of the 17th-19th century type,

or

> a 17-note system such as Secor's, this gradation of intervals

provides

> an impressive musical resource.

Indeed.

By the way, Margo, Joe Monzo lists you as an advocate of 17-tone

equal temperament . . . is that your "favorite" ET?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> It is rather like it. But for the specific definition of "middle

> path" I had in mind, one would want an MOS -- since for me the

> prototypical JI scales are periodicity blocks, and ETs are of

course

> tempered periodicity blocks.

Or we can view the JI scales as ETs lifted to JI, which is a useful

point of view.

Hello, there, Paul and everyone.

Thank you for your explanation of the "middle path" as meaning an MOS

system. You clarify some points, and also remind me of a very

important omission which I may now correct before addressing some of

your distinctions and questions, which maybe suggest a new category of

tuning (or a subcategory of some approach you've discussed?).

Of course I should have mentioned your 22-tone well-temperament for

decatonic music, indeed combining this tonal system (your main theme

in the article) with elements of the sruti system of India.

Why did I not recognize it as a "well-temperament," as well as a

"cross-cultural" kind of tuning system? What sticks with me especially

is the use of some fifths tempered just slightly less than in 17-tET,

a topic about which you ask and I'll reply a bit later in this

article.

Anyway, from now on, I'll certainly mention your 22-tET

well-temperament when I discuss this category, along with 17 and 19

and so forth.

Moving to the MOS concept, I might propose a concept like "dual-MOS

tuning" or maybe "multiple-MOS tuning" to include things like my

24-note Pythagorean schemes where each 12-note keyboard indeed defines

an MOS, but the overall tuning does not fit this category.

Your interestingly point to two attributes a "microtempered 2-3-7-9

JI" tuning would need to quality as an MOS: first, all fifths of equal

size (e.g. a temperament by 1/15 of a 3-7 schisma); and secondly, the

right number of notes, such as 17 or 29.

I'm noting that such a microtemperament -- or I'm tempted to say

nanotemperament, although that would be a different use of the term

than the more specific concept of adjusting for synthesizer tuning

units or the like -- in 24 notes would be strictly regular, but not an

MOS.

What I'm looking for is a term to communicate the idea of two MOS

tunings, with identical 12-note keyboard arrangements, for example,

which do not together form a larger MOS.

I'm tempted to speak of a "two-dimensional" tuning system, whether

based on Pythagorean (3-prime) JI or some temperament, since the

"rows" are MOS systems of a certain size, and the "columns" are pairs

of notes at a certain interval by which the rows are spaced.

Here maybe, as with consistency, we have the question: does an overall

MOS organization maybe become less important as the number of steps

and intervals increases?

In my view, a significant feature of a "two-dimensional" system with

each row a 12-note MOS, for example, is that lots of melodies can be

played and vertical progressions can take place within an MOS row.

For example, an 11-note Pythagorean tuning of the kind which may have

been common on keyboards around 1300 (Eb-C#) seems to invite expansion

to 12, especially once G# becomes a common accidental in progressions

from E-G# to D-A (major third to fifth in its most decisive form with

an ascending semitone).

However, does the distinction between 24 and 29, or between 36 and 41,

have a comparable significance? At this point we have what might be

described from a pragmatic viewpoint as "multiple versions of the same

note or interval category," whether the system is an MOS or otherwise.

The idea of a "multi-MOS" system may explain some of the appeal of

24-note systems with two 12-note keyboards (Pythagorean, meantone, or

neo-Gothic eventone, etc.)

Such system may represent a certain kind of compromise because it

permits one either to pursue regularity and symmetry within one of the

keyboards, or to seek new intervals, comma or diesis shifts, and so

forth, by mixing notes from the two MOS "rows." It is at once, one

might say, both regular and irregular, maybe with some of the appeal

of both alternatives.

You ask about my favorite equal temperament, and certainly I would say

that 17 expresses a lot of what I'm about for a very large portion of

my musicmaking. It was the first non-12 equal temperament that I

tuned, and I'm very happy to be associated with it.

Curiously, nowadays, I find myself also a passionate advocate for

13-tET as a neo-Gothic tuning: it seems very "modernistic" and

21st-century, with the implied process of Chowningization or the like

to obtain a 3:2-like effect for the 8/13-octave interval.

Here's a question: does being an "advocate" for an n-tET mean using

and favoring it specifically as a complete system, or simply using

some subset, and maybe being drawn to the division of the whole-tone

into equal fractional parts or similar features?

For example, I'm very enthusiastic about 29-tET, which I might guess

that I've used rather more than 17-tET -- but 12-out-of-29, or

24-out-of-29, rather than the full set.

Am I advocating "29-tET as a system," or simply "a regular temperament

around 703.45 cents"?

With 13, 17, 20, 22, and 24, I often do tune the full set, and do

encounter "system properties" such as the availability of any interval

from any note.

However, with 29, it seems more to me like one pleasant point on the

pleasant portion of the continuum roughly from 703.45 cents to

something like the e-based temperament at 704.61.

One very humorous side of this question is the kind of system I would

propose to carry 17 further, and get some new interval categories.

Lots of people advocate 34-tET, an excellent solution if one wants a

near-just 5-limit system, but own option is rather different.

Some months ago, it occurred to me that two 17-tET tunings at an

arbitrary distance of around 55.106 cents would yield some pure and

near-pure ratios of 2-3-7-9 (pure 7:6 and 12:7, with 9:7, 14:9, 7:4,

and 8:7 impure by the same amount as the wide fifth, ~3.93 cents).

Also, we supplement the fine neutral intervals of 17 with some

supraminor and submajor thirds and sixths, and add some interesting

near-Pythagorean intervals, including a minor third close to 19:16

(available for near-16:19:24 sonorities).

Maybe my problem is that it's hard to pick a single "favorite" n-tET

because each has its own attractions -- and this includes 22-tET,

where I find the 55-cent diatonic semitone quite satisfactory,

although not everyone agrees.

A curious thing is that with 17-tET or 29-tET, I can easily articulate

a defining quality of the "equalness" involved -- equal dieses or

fifthtones fifthtones in 29, and equal thirdtones in 17.

With 46, I'm more likely to say something like this: "This tuning is

almost identical to a 14:11 eventone, and has the same features:

regular thirds near 14:11 and 13:11, very nice supraminor/submajor

thirds, and some not-so-precise approximations of 2-3-7-9."

While I do clearly recognize 17 for its symmetry and "17-ness," this

might apply even more to 13, "the division of the octave into 13 equal

limmas," since 1/13 octave is very close to the Pythagorean limma at

256:243 (~90.22 cents).

With 17, there is for me a kind of meeting of styles, with those

beautiful 70.59-cent semitones and 423.53-cent major thirds for usual

14th-century cadential progressions -- and in the right timbres, those

thirds can be gentle and efficient at the same time -- and all those

neutral intervals to lend a "Romantic" or "Impressionist" air.

It's a bit like moving between a slightly alternative version of the

Gothic Era itself, say around the time of Guillaume de Machaut

(c. 1300-1377), and some curious conception of the "Gothic" in an

alternative 19th-century Romanticism.

Also, I should mention something that maybe seems a bit more familiar,

or at least a single keyboard within it -- 24-of-36, one of the best

optimizations I've seen for both 12:14:18:21 and 14:17:21.

Curiously, I haven't seen a lot of raving about 36-tET, or this

subset; maybe it's mainly a question of what one seeks to optimize.

Of course, while we're at this, I should mention a "near-ET" that I

use an immense amount: 1/4-comma meantone, which in 24 notes has many

diesis steps and progressions available. Here, while I know that this

isn't precisely 24-of-31-tET, I tend to count fifthtones as if it

were; carrying to a full 31, one could also circulate around the

system as if it were.

Anyway, for n-tET's which I tune as n-tET's (complete sets or

subsets), I would say that 17-tET and 12-or-24-of-29-tET are my

favorites, with 13-tET a favorite "unconventional" choice, and 20-tET

also in this category as slightly less unconventional.

Thank you for the clarifications, questions, and ideas, as usual.

Most appreciatively,

Margo Schulter

mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> Hello, there, Paul and everyone.

>

> Thank you for your explanation of the "middle path" as meaning an

MOS

> system.

It doesn't necessarily, Margo. If we're talking about, for example, a

planar temperament, the result won't be MOS.

> What I'm looking for is a term to communicate the idea of two MOS

> tunings, with identical 12-note keyboard arrangements, for example,

> which do not together form a larger MOS.

In many cases two MOSs interlace to form a larger MOS, but in this

case you don't. So whatever term you choose should reflect this.

>

> I'm tempted to speak of a "two-dimensional" tuning system, whether

> based on Pythagorean (3-prime) JI or some temperament, since the

> "rows" are MOS systems of a certain size, and the "columns" are

pairs

> of notes at a certain interval by which the rows are spaced.

Well, yes, it is a two-dimensional tuning system -- specifically, JI

with a 3-axis and a 7-axis . . . though the "chunk" of this plane

you're taking is not quite a periodicity block (well, it probably is,

for some odd choice of unison vectors).

> Here maybe, as with consistency, we have the question: does an

overall

> MOS organization maybe become less important as the number of steps

> and intervals increases?

Kraig Grady has argues, based largely on Erv Wilson's work, that such

features continue to be important. For me, they simply contribute to

a sense of "closure" -- a logical place to stop adding more notes.

> In my view, a significant feature of a "two-dimensional" system with

> each row a 12-note MOS, for example, is that lots of melodies can be

> played and vertical progressions can take place within an MOS row.

>

> For example, an 11-note Pythagorean tuning of the kind which may

have

> been common on keyboards around 1300 (Eb-C#) seems to invite

expansion

> to 12, especially once G# becomes a common accidental in

progressions

> from E-G# to D-A (major third to fifth in its most decisive form

with

> an ascending semitone).

>

> However, does the distinction between 24 and 29, or between 36 and

41,

> have a comparable significance?

If one is writing very sophisticated microtonal music along Wilsonian

lines, perhaps.

>

> Some months ago, it occurred to me that two 17-tET tunings at an

> arbitrary distance of around 55.106

Which is the same thing as two 17-tET tunings at a distance of 15.482

cents, right?

> Also, I should mention something that maybe seems a bit more

familiar,

> or at least a single keyboard within it -- 24-of-36, one of the best

> optimizations I've seen for both 12:14:18:21 and 14:17:21.

>

> Curiously, I haven't seen a lot of raving about 36-tET, or this

> subset; maybe it's mainly a question of what one seeks to optimize.

Yes -- I suspect most people are hung up on 36-tET being just as bad

as 12-tET when it comes to ratios of 5.

Thanks for your fascinating discussion, as always!

Hello, there, Paul, and thank you for another very engaging response.

Maybe we could distinguish between "the middle path," meaning as you

explain actually a number of approaches based on MOS or periodicity

block structures (if I now follow correctly), and "middle paths" more

generally based on some kind of eventone tuning, for example.

The latter type of "middle paths" category doesn't necessarily imply

either an MOS for the overall system, although it's very likely to

have MOS subsets, or a periodicity block structure, which you've

commented wouldn't apply to at least some of my 24-note tunings.

Your response to my mention of 36-tET raises a point that could lead

into a tuning that occurred to me early this year for a style of music

that 36-tET doesn't really optimize.

Suppose that we want to find an optimal tuning for a style mixing

neo-Gothic _and_ Xeno-Renaissance elements in the same piece. This

means that we want not only sonorities like 12:14:18:21 and 14:17:21,

for which 36-tET has a really outstanding sum of squares, but also

fairly close approximations of 4:5:6 and 10:12:15.

As you observed, of course, the regular 12-tET intervals of 36-tET

give a not-so-close approximation of 5-limit. For a neo-Gothic style

alone, this might be described as a virtue, since these same intervals

can serve as not too far from Pythagorean or 3-limit.

However, let's suppose we really want a Renaissance-style 5-limit. An

excellent solution, it turns out, is a Renaissance-style meantone --

in fact, the first regular meantone tuning known to us with a precise

mathematical definition.

Yes, it's Gioseffo Zarlino's 2/7-comma meantone of 1558, carried to 24

notes.

The regular major and minor thirds, of course, are only 1/7 syntonic

comma from a pure 5:4 and 6:5 -- about 3.07 cents.

The diminished fourths are very close to 9:7 (~435.08 cents), at about

433.52 cents, or ~1.57 cents narrow; and the augmented seconds quite

close to 7:6 (~266.87 cents), at around 262.29 cents, or ~4.58 cents

narrow.

Thus Zarlino's 2/7-comma meantone yields a very nice approximation for

6:7:9 or 7:9:12, although the near-7:4 is much less accurate than in

1/4-comma meantone. Here 2/7-comma gives us an augmented sixth at

around 958.10 cents, or ~10.72 cents narrow -- a variation on what

might be the more curious side, possibly with a resemblance to the

20-tET interval of 960 cents.

For a supraminor third at around 17:14 (~336.13 cents), we use an

interval of a major third (~383.24 cents) less the enharmonic diesis

of ~50.28 cents, at about 332.97 cents (~3.16 cents narrow),

e.g. C*3-E3, with the ASCII * showing a note raised by a diesis. The

submajor third near 21:17 (~365.83 cents) is equal to a minor third

(~312.57 cents) plus a diesis, or ~362.84 cents (~2.98 cents narrow).

Maybe the most dramatic difference between 2/7-comma and 1/4-comma as

24-note or larger tunings is that in 1/4-comma, the chromatic semitone

of ~76.05 cents gets divided into two near-equal dieses of the kind

called for in Vicentino's enharmonic styles -- a large fifthtone at

around 41.06 cents, and a small fifthtone at around 34.99 cents.

In 2/7-comma, in contrast, we have a division of the chromatic

semitone at precisely 25:24 (~70.67 cents) into a diesis at around

50.28 cents, plus a kind of comma (the "19-comma," the difference

between 19 fifths up and 11 pure octaves) at around 20.40 cents.

Thus 1/4-comma, or 31-tET, is the appropriate choice for Vicentino's

enharmonic progressions based on a conceptually equal division of the

chromatic semitone into two diesis steps.

However, 2/7-comma is a fascinating system to carry to 24 notes for a

mixed kind of "neo-Gothic/Xeno-Renaissance" style, and, of course,

each 12-note manual provides a beautiful Renaissance meantone tuning.

Finally, Paul, you very correctly point out that my tuning with two

17-tET chains (or 12-of-17-tET subsets thereof) at ~55.106 cents apart

is equivalent to a distance of ~15.482 cents apart -- the sum of these

intervals being the step of 1/17 octave, or ~70.588 cents.

My 55.11-cent arrangement is "metachromatic," with 7:6 minor thirds

played as major-second-plus-diesis, for example (C3-D*3), with lots of

resolutions involving melodic motion from a key on one manual to the

corresponding key on the other manual:

A*3 A3 F4 F*4

G3 A3 D*4 C*4

D*3 D3 C4 C*4

C3 D3 or G*3 F*3

In the first example, a near-12:14:18:21 (with the 7:6 pure and other

intervals impure by ~3.93 cents) contracts to a stable fifth. In the

second, a near-14:18:21:24 (with the 12:7 pure) expands to a

near-2:3:4 trine

In this arrangement, the supraminor third has a "conventional"

keyboard arrangement as minor-third-plus-diesis (e.g. E3-G*3), and the

submajor third also as major-third-less-diesis (e.g. E*3-G#3).

In a version of this tuning with two 12-of-17-tET subsets, an

advantage of the metachromatic arrangement is that we get the 2-3-7-9

types of sonorities in more locations.

For example, we get a near-9:7 as fourth-less-diesis, in 11 locations;

with a spacing of 15.48 cents, we'd get it in 8 locations as a regular

major third plus the "comma" between the keyboards.

With the 55.11-cent spacing, we get a pure 7:6 as tone-plus-diesis in

10 locations; with the other spacing, we'd get it in 9 locations as

minor-third-less-comma.

However, the "metachromatic" diesis arrangement gives a near-7:4 as

major-sixth-plus-diesis in 9 locations, while we get 10 with the

15.48-cent spacing, where it is played as minor-seventh-less-comma.

With a full tuning of two complete 17-tET sets, the difference would

seem purely one of taste, I guess.

Most appreciatively,

Margo Schulter

mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> Hello, there, Paul, and thank you for another very engaging response.

>

> Maybe we could distinguish between "the middle path," meaning as you

> explain actually a number of approaches based on MOS or periodicity

> block structures (if I now follow correctly), and "middle paths" more

> generally based on some kind of eventone tuning, for example.

Hmm . . . the category of eventone tunings seems to be in some respects more restrictive,

rather than more general . . . many linear temperaments are not eventone, and planar and other

"middle path" tunings are generally not eventone either. "Middle path" is a term I introduced to

simply suggest that there is much of interest to be explored in the conceptual area _between_

ETs and JIs (not inclusive).

On a different issue:

> With a full tuning of two complete 17-tET sets, the difference would

> seem purely one of taste, I guess.

And one would then have an MOS scale (of 34 notes), as well.

Hello, there, Paul, and please let me clarify my intent possibly to

expand the scope of "middle paths" a bit, but certainly not to

restrict it or to exclude any of your suggested approaches.

In suggesting the inclusion of eventones as an overall category, I

mean to suggest that the eventone nature of tuning might be taken as a

sufficient condition, but by no means a necessary condition.

Specifically, I wish to include eventone tunings which may not,

especially as overall systems, fit the MOS criterion.

Such systems might range from an "array" of two identical 12-note

eventone tunings at any arbitrary distance apart, to a 15-note

Pythagorean or meantone system with a "split key" arrangement or

equivalent (e.g. Gb-G#). Here we have a single eventone set, or two

such sets, not forming an overall MOS.

In other words, I propose to define "middle paths" to include, at

least:

(1) All the categories you've discussed, whether or

not they happen to be eventone tunings; and

(2) Eventone tunings, whether or not they happen

to be MOS tunings.

What all these categories seem to share are a "middle ground" between

classic multi-prime JI systems and n-tET systems -- although some

eventones may also be n-tET's, or vice versa.

Most appreciatively,

Margo Schulter

mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> Hello, there, Paul, and please let me clarify my intent possibly to

> expand the scope of "middle paths" a bit, but certainly not to

> restrict it or to exclude any of your suggested approaches.

>

> In suggesting the inclusion of eventones as an overall category, I

> mean to suggest that the eventone nature of tuning might be taken

as a

> sufficient condition, but by no means a necessary condition.

Perhaps I misunderstood you before, Margo . . . but I don't see it as

a sufficient condition, either. For example, Pythagorean is eventone,

but it is a JI tuning system, hence not a "middle path" between JI

and ET.

Hello, there, Paul, and I'd agree that Pythagorean intonation is a JI

system: as someone often advocating this point, I can hardly

contradict it now <grin>.

Thus if the "middle paths" concept means non-JI and non-n-tET, I would

revise my statement to say that I would consider the eventone nature

of a _temperament_ to be a sufficient condition, but not a necessary

condition.

Here I might distinguish between "eventone tunings" (including

Pythagorean), and "eventone temperaments" (all eventone tunings except

for Pythagorean), with the latter but not the former fitting the kind

of "middle paths" concept that excludes JI systems.

In other words, "middle paths" would include eventone temperaments and

the other categories you discuss, but not JI tunings, even if they

happen also to be eventones (specifically Pythagorean).

Additionally, I realize that under this kind of non-JI/non-n-tET concept,

we should also exclude eventones that happen to be n-tET's.

Thus I might say, "One subcategory of 'middle paths' includes those

eventone temperaments which happen not to be equal divisions of the

octave," or something like that.

Most appreciatively,

Margo Schulter

mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> Thus I might say, "One subcategory of 'middle paths' includes those

> eventone temperaments which happen not to be equal divisions of the

> octave," or something like that.

You betcha!