Yahoo Tuning Groups Ultimate Backup tuning x-Intonation (discussion related to Margo's terminology ideas)

> "M. Schulter" <MSCHULTER@VALUE.NET>
>
> Hello, there, everyone, and reflecting a bit on this thread, I might
> propose three aspects of Just Intonation or JI:

> Harmonic Intonation (HI)
> The inclusion in a JI system of some small integer ratios
> with audibly pure or "coupling" partials, e.g. 2:3:4.
>
> Combinatorial Intonation (CI)
> The inclusion in a JI system of some "intermediate"
> integer ratios where the blending of combinatorial
> tones can give certain sonorities a "pure" rather
> than "tempered" effect, as discussed by George Secor.
>
> Rational Intonation (RI)
> More generally, the use of _any_ integer ratios,
> whether or not they also produce HI or CI effects.

Hello Margo !

As I mentioned a couple of days ago I was given much to think about by reading your posts (c. Dec 2000) re. RI, etc. I've been taking longer to collect my thoughts than I thought I would(I did say 24 hrs). But now your present post precipitates my posting those thoughts as is, as I think it could be useful for discussion. I will need a little time to think about what you just said and how it fits into my rudimentary ideas. The main issue is that I cannot agree with the limiting mention of JI in the above explanations. 'Combinatorial Intonation' certainly gives something to ponder on.

Try it:
Simply remove 'JI' from the HI explanation. Similarly, replace 'JI' with 'HI' in the explanation for CI. Why ? JI is but a tiny subset of HI, RI, and CI. Which is why the rest of what you say holds up none the less !

The way I've been able to think outside the JI box, while still being able to consider it's features is by thinking in terms of the harmonic series. Any reference to a harmonic-anything should at the very least refer to the entire, theoretically infinite, harmonic series.

I cannot see these as '...three aspects of Just Intonation or JI'. I see them as akin to three approaches to intonation systems, i.e. ways that allow me to view, study, and work with the wider universe of harmonics and tonality.

Sincerely,

Joel

🔗Joel Rodrigues <joelrodrigues@mac.com>

Hello Margo, all,

As I said, this post is not quite complete, but I felt I had to post it before revising and extending it based on your (Margo) current post. Thanks for your time !

Sincerely,
Joel

***

I am working here mainly from Margo's posts from Dec. 2000 and the related discussions around that time, as well as other posts.

Just Intonation : Based on the harmonic series, but limited to intervals with 'small number' ratios, ideally with a 'beatless locking of partials'. An 'aesthetic of beatlessness.', as Margo puts it.

'Extended Just Intonation' ? Borders on being an oxymoron.

Rational Intonation : (quoting Margo) '...based on integer ratios, both simple & complex, but not based on a systematic use of, or preference for the harmonic or subharmonic series.'

'My longer term "rational continuum intonation" (RCI) emphasizes that not only is the system based exclusively on integer ratios (as are many "harmonic-tuning-oriented" JI systems), but that it favors a continuum of intonational nuances and flavors as defined by such ratios of any desired complexity on a dense number line, rather than a tuning set derived mainly from harmonic/subharmonic relations.'

Margo, this is exquisite !

The following are my initial thoughts and comments.

Monzo (Dec 2000) stated he uses 'rational tuning' for all rational tunings, including 'those above the "extended JI" limits (37, 43...?) specifically.' This is inconsistent and a misconception of Margo's clear explanation.

RI is not simply, "tuning by integer ratios". It is not an alternative to JI. JI is a form of RI. JI is a subset of RCI. JI is also a subset of Harmonic Continuum Intonation (see below). I don't see how Margo suggested it to be the opposite, 'in discourse settings where people tend to associate "JI" with integer ratios, I would describe Gothic/neo-Gothic RI and Renaissance 5-limit music as subsets of JI.'

I find this unnecessarily restrictive:

Harmonic Intonation (HI): JI where harmonicity and the
purest or most beatless tuning of sonorities possible
is the ideal -- or, from another view, the Partch/Doty
ideal that ratios should be kept as small as is fitting
for the musical context

The above use is restricted to a narrow sense of consonance of the Helmholtzian, beatless kind. I know that this is the common use of the term, which is deceptively limited in it's scope.

I had just been continuing my thoughts to the concept of harmonic intonation, logically extending from the harmonic series. So, I would have harmonic/inharmonic scales/chords/timbres. Harmonic intonation should refer to the entire continuum of the harmonic series, 2:1 ('octave') reduced to humanly audible range. It yields a companion to your (Margo's) RCI, HCI (Harmonic Continuum Intonation). There is much existing use of the term 'Harmonic Intonation' which is in accord with this conception.

Rational Intonation (RI): JI where integer-based ratios,
simple or complex, are the distinguishing feature, many
of these ratios possibly having little if any obvious
connection with relations between partials, and with
beatlessness not necessarily a pervasive ideal,
especially for complex or unstable sonorities (e.g.
Gothic/neo-Gothic), or in primarily melodic rather than
polyphonic textures (e.g. ancient Greece a la Bill Alves).

This makes sense to me only if "JI" is removed from the beginning of this definition. Your original explanations cited above are complete.

So, *I* get the following:

Rational Intonation / Rational Continuum Intonation (RCI)

Harmonic Intonation / Harmonic Continuum Intonation (HCI)

RCI encompasses HCI.

Just Intonation (JI)

JI is limited to small-number ratio intervals, ideally with beatless locking of partials.

Small number ratio, beatless (ideally) JI is a subset of HCI, and thus of RCI.

'Extended-JI' is an exploration further up the harmonic series, making it easier for interested parties to think simply in terms of HCI.

Scales are subsets of all these intonations.

Modes to do with a scale exhibiting different tonalities depending on which scale step is used as a tonal centre.
(This is just a cursory statement, scribbled out in a hurry)

Tuning is the process of putting an instrument into tune with a scale.

Harmonic scales/intervals/timbres

Inharmonic scales/intervals/timbres

Rational scales/intervals/timbres

***

Much of this was typed out in a hurry, trying to get my thoughts down before I lost them. There are bound to be incomplete statements & possibly ;-) some errors. I hoped to have this properly edited for public consumption before inflicting it on you all ! - Joel

🔗genewardsmith <genewardsmith@juno.com>

🔗Joel Rodrigues <joelrodrigues@mac.com>

🔗genewardsmith <genewardsmith@juno.com>

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Gene,

I just saw this remark of yours and it happens
to relate to something I've been mulling over:

> This makes sense if you are looking at the matter theoretically, for instance
> in terms of group theory. If you are looking at dense subsets of the reals as a
> pitch continuum, however, it seems to me any dense subset is in effect the same
> as any other dense subset. I see no need for a 5-limit continuum intonation,
> 7-limit continuum intonation, and so forth, *except* as theoretical constructs.
> It's not as if you could possibly hear a difference, after all.

I may be taking it out of context, but the thought I had is
that you could characterise the golden ratio say as 3 or 5 limit
etc. depending on whether it is more rapidly approximated by a sequence of
3 limit or 5 limit ratios.

The thing is that there will be many formulae say of the form
2^a*3^b*5^c*7^d that approximate the golden ratio, but
if you want to keep the product of denominator and denumerator
as small as possible while achieving the closest possible values
to the golden ratio, you will find very few.

I wrote a program a while back to look for ratio approimations
to another ratio, and just updated it to accept arbitrary decimals.
so that it can look for approximations to golden ratio etc. too.

Here is its output for ratios approximating to the
golden ratio:

.........................................
Successive approx. to 1.61803398875/1 in form:
2^a * 3^b * 5^c * 7^d

Max quotient to look for: 1e+050

1/1 2/1 7/2^2 7^2/2^5 2^12/7^4 7^7/2^19 2^26/7^9 7^12/2^33 2^40/7^14 3*
7^15/2^43 3^2*7^44/2^126 3^6*7^25/2^79 3^10*7^6/2^32 2^15*3^14/7^13 2^62*3^
18/7^32 2^109*3^22/7^51 2^156*3^26/7^70 2^141*3^101/7^107 3^18*7^39/(2^135*5
) 2^76*5^4*7^6/3^64
(9952371 values tested)

Values in cents
0.0 0 1200 968.82591 737.65181 924.69637 781.78135 880.56684 825.91088 836.43731
834.3436 832.24989 832.37767 832.50545 832.63323 832.76101 832.88879 833.01657
833.0831 833.08665 833.09024

Cents diffs from target 833.0903
-833.0903 366.9097 135.73561 -95.438483 91.606078 -51.308951 47.476545 -7.179418
7 3.3470131 1.2533015 -0.84040998 -0.71262944 -0.58484889 -0.45706834 -0.3292877
9 -0.20150724 -0.073726695 -0.007201149 -0.0036423487 -5.7467458e-005

1.61803398875/1 = 833.090296357 cents
.........................................

An interesting observation is that it looks as if 7 frequently
occurs to the highest power, and 3 is the next highest, and
5 is very rare, with only one instance so far.

So one could say that the golden ratio is a 7 limit ratio perhaps.

To try another one, let's see what pi is like:

.........................................
Successive approx. to 3.14159265359/1 in form:
2^a * 3^b * 5^c * 7^d

Max quotient to look for: 1e+050

7/1 2/1 7/2 7^2/2^4 7^7/2^18 2^33/(3^13*5*7^3) 2^143*7^7/(3^84*5^12) 5^20
*7^22/(2^59*3^30) 2^42*5^36/(3^41*7^21) 2^143*5^52/(3^52*7^64) 5^57*7^4/(2^50
*3^58)
(9952371 values tested)

Values in cents
0.0 3368.8259 1200 2168.8259 1937.6518 1981.7813 1981.7936 1981.7967 1981.7942 1
981.7946 1981.7951 1981.7953

Cents diffs from target 1981.7954
1387.0306 -781.79536 187.03055 -44.143542 -0.014010083 -0.001799889 0.0013508465
-0.001161711 -0.00072756525 -0.00029341954 -8.9387212e-005

3.14159265359/1 = 1981.79535537 cents
...............................................................

There, pi is somewhat more 5 limit.

Obviously these huge numbers aren't of immediate musical relevance,
but kind of interesting. It rather looks as if there is enough
of a trend there so that with some work one could define
a mathematically precise notion of the relative proprotions
of the various priimes needed to approximate an irrational,
which mightn't necessarily converge, so next thing would
be to see if one could prove it did converge, and if
every irrational has a definite flavour in the n-limit
or if only some do and so forth.

Robert

🔗M. Schulter <MSCHULTER@VALUE.NET>

[Please use "Expand Messages" option for diagrams if reading on Yahoo]

Hello, Joel, Gene, Robert, and everyone, and thank you all for giving
me an opportunity to reconsider and possibly "clarify" some of my
views regarding "What is JI?" and related questions.

My special thanks to you, Joel, for so thoughtfully reviewing some of
my earlier postings, thus illustrating more generally how a person's
views on this kind of question can shift a bit as the dialogue takes
different turns.

I wrote a rather long article, but maybe could better facilitate
discussion by posting some excerpts, while offering to make the rest
available either here or via e-mail for anyone curious.

For an integer-based approach to "just intonation" (JI) or "rational
intonation" (RI), I might suggest this definition along with some
subcategories:

(I) Just intonation (JI) or rational intonation (RI):
any tuning system where all intervals are derived
from integer ratios, large or small, of which some
typically will also have an "audibly pure" effect
based on coupling of harmonic partials (HI) or
combinatorial tones (CI).

(a) Harmonic intonation (HI) involves the audible
"locking in" or "coupling" of partials, and
typically involves rather small ratios;

(b) Combinatorial intonation (CI) involves the
perception of combinatorial tones, and as
George Secor has suggested, can give a
feeling of "pure" tuning to sonorities
where the ratios might be too complex to
support an HI effect.

(c) More complex JI/RI intervals or sonorities
typically arise in a system from simpler
ratios of HI or CI (e.g. 3:2, 4:3, and 9:8
in a 3-prime or Pythagorean tuning).

However, one could also argue as Dave Keenan, for example, eloquently
does, that audible HI (or CI) effects are the essence of a "JI sound,"
whether produced by an explicit tuning system using small integer
ratios, or by performers who might be quite unconcerned with the
mathematics but achieve the "purity" associated with these ratios.

As I also wrote:

If asked to make a Venn diagram of my JI/RI scheme applying to systems
based on integer ratios, it might look like this:

|----------------------------------------------|
| U(niverse)=All tunings |
| |
| |-------------------------| |
| | JI/RI |---------| | |
| | | HI/CI | | |
| | |---------| | |
| |-------------------------| |
| |
|----------------------------------------------|

Here the overall universe or "U" is that of all tunings, with JI/RI
including all tunings based on integer ratios for all intervals; the
HI/CI subset includes JI/RI intervals or sonorities with an audible
effect of "purity" from coupling of partials or perception of
combinatorial tones.

However, from another point of view, our map might also look like this
if HI/CI are taken as _audible_ phenomena which might be produced by
irrational as well as rational ratios:

|----------------------------------------------|
| U(niverse)=All tunings |
| |
| |---------| |
| | | |
| |--------|---------|------| |
| | JI/RI | | | |
| | | HI/CI | | |
| | |---------| | |
| |-------------------------| |
| |
|----------------------------------------------|

Here a "perceptual JI" definition might include the set of HI/CI
intervals and sonorities, whether or not within the integer-based
JI/RI subset.

From a real world perspective, of course, integer ratios are
mathematical models rather than precise definitions of what performers
actually produce using human voices, other acoustical instruments, or
even electronic ones (apart from certain instruments which do use
harmonic or subharmonic series, as described for example in some
recent posts on the Scalatron in certain situations.)

For typical JI systems including HI/CI sonorities, what attuned
performers do is likely to approach a model based on integer ratios,
but with inevitable and often expressive variations. An approach like
Keenan's nicely communicates this reality, and my own view is that the
mathematical and perceptual outlooks can complement one another, with
either available depending on the context of discussion.

Also, the following comment from my longer statement might tie in with
some of the latest dialogue regarding the implications of approaching
JI/RI as "tuning by integer ratios":

To take this approach is to accept certain ambiguities or even
paradoxes. Since the set of rational ratios on the number line is
dense, this means that _any_ interval is from one viewpoint either
just or infinitesimally close to a just integer ratio. I find this
conclusion an attraction of this definition.

Here, Gene, I much agree that "tuning by integer ratios" is more of an
outlook than a definition of a musically distinct set of intervals,
although it _might_ invite focusing on such themes as audibly "pure"
intervals, combination tones, superparticular steps, and so on.

Also, Robert, I much enjoy the idea of using different prime or odd
factors to approximate an irrational ratio such as Phi or pi. Among
other things, your examples suggested to me the convenient rational
approximation of pi as 22:7 (~1982.49 cents),

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗jpehrson2 <jpehrson@rcn.com>

🔗Joel Rodrigues <joelrodrigues@mac.com>

Hello Margo, all,

Thank you, Margo for your unfailingly thought provoking posts. You're right about shifting views. I constantly find myself revising my understanding of all manner of subjects. Unsurprisingly, given the nature of things, it seems that in this case we've shifted in different directions, looking at the same picture from quite different angles. So needless to say, what follows is not an attempt at 'I'm right, you're wrong'. I know you wouldn't mistake it for that, but in this forum, it's a matter of precaution :-)

> I wrote a rather long article, but maybe could better facilitate
> discussion by posting some excerpts, while offering to make the rest
> available either here or via e-mail for anyone curious.

I'd certainly love to see the entire article ! I'm OK with it being posted here.

> For an integer-based approach to "just intonation" (JI) or "rational
> intonation" (RI), I might suggest this definition along with some
> subcategories:
>
> (I) Just intonation (JI) or rational intonation (RI):
> any tuning system where all intervals are derived
> from integer ratios, large or small, of which some
> typically will also have an "audibly pure" effect
> based on coupling of harmonic partials (HI) or
> combinatorial tones (CI).
>
> (a) Harmonic intonation (HI) involves the audible
> "locking in" or "coupling" of partials, and
> typically involves rather small ratios;
>
> (b) Combinatorial intonation (CI) involves the
> perception of combinatorial tones, and as
> George Secor has suggested, can give a
> feeling of "pure" tuning to sonorities
> where the ratios might be too complex to
> support an HI effect.
>
> (c) More complex JI/RI intervals or sonorities
> typically arise in a system from simpler
> ratios of HI or CI (e.g. 3:2, 4:3, and 9:8
> in a 3-prime or Pythagorean tuning).
>

My own logic organises the above very differently. I've tried to pay attention to etymology, prior and current use, and logical extrapolation.

Harmonic Intonation : A tonal system in which intervals are derived from the harmonic series, inherently using only integer ratios. The full term for this tonal space is Harmonic Continuum Intonation.

To me this conception is obvious from the use of 'harmonic' as a reference to the harmonic series, *not* 'harmony', which might be what would give rise to your interpretation. I feel this works better, going full circle to, or at least more in line with, the Greek sense of a study of 'harmonics', which I do realise was a bit broader, concerned with tonality.

'Harmonic' being used as an adjective for both harmony and harmonics is potentially very confusing. I've come upon many articles where both senses of 'harmonic' are used within a single article, for example with reference to 'harmonic tonality' and 'harmonic rhythm' (used here to refer to harmony), and 'harmonic intonation' (here used in reference to instrument performance/tuning/build quality).

So, there already exists (quite common, it would appear) use of 'harmonic intonation' to refer to the harmonic series, and not to harmony. 'Harmonic tonality' is commonly used to refer to harmony, often in discussions relating to so-called 'tonal' music. No doubt, this use of 'harmonic tonality' can be seen as confusing from a microtonalist's vantage, since it *can* be interpreted as implying the use of harmonic intonation, rather than of harmony. For what it's worth, there exists another less used adjective form of harmony, 'harmonial'.

Just Intonation : A tonal system in which intervals are derived from the harmonic series with a theoretic preference for, but not insistence on, small number integer ratios and an auditory preference for beatlessness.

This is a more complex concept than Harmonic Intonation, though it is commonly approached as a theoretic subset of HI. But, JI brings into play issues of timbre, and words like 'pure'. I personally dislike both this use of 'Just' and 'Pure, but that's besides the point.

Combinatorial Intonation :

Your distinction between HI & CI above seems (to me) too fine and unnecessary. I feel that Combinatorial Intonation by itself supports both your definitions.

I came across something at <http://members.aol.com/MetPhys/71phasewave.html#71.2> (don't ask, at the top of my to-do list is 'get a life' - Monzo's got the right idea), which supports the proposition that 'Combinatorial Intonation' can be an umbrella categorisation for both phenomena described by Margo above under HI & CI. The cited page is part of an off-beat site, home page at <http://members.aol.com/MetPhys/metaphysicshome.html>.

Just in case it's overlooked, there is an unrelated (thankfully !) use of the adjective 'combinatorial', in serial music terminology. 'A descriptive term for tone rows in which the second half is a transposed version of the first half.', <http://www.omnidisc.com/MUSIC/Glossary.html>.

> However, one could also argue as Dave Keenan, for example, eloquently
> does, that audible HI (or CI) effects are the essence of a "JI sound,"
> whether produced by an explicit tuning system using small integer
> ratios, or by performers who might be quite unconcerned with the
> mathematics but achieve the "purity" associated with these ratios.

Indeed ! This is what I realised yesterday, except that as I said above, CI covers this by itself. I almost see CI as a working definition of JI.

Combinatorial Intonation can also be seen as a broader perception of the concept presented by William Sethares in 'Relating Tuning and Timbre', <http://eceserv0.ece.wisc.edu/~sethares/consemi.html>.

Rational Intonation : A tonal system in which the target intonation for a given note is any _integer_ ratio. The 'rational' in 'Rational Intonation' refers to rational numbers, which involve only integers (whole numbers), and is not a reference to 'ratio' per se.

How/why would anyone put the limitation of 'small-number' on HI and RI ?

> As I also wrote:
>
> If asked to make a Venn diagram of my JI/RI scheme applying to systems
> based on integer ratios, it might look like this:
>
>
> |----------------------------------------------|
> | U(niverse)=All tunings |
> | |
> | |-------------------------| |
> | | JI/RI |---------| | |
> | | | HI/CI | | |
> | | |---------| | |
> | |-------------------------| |
> | |
> |----------------------------------------------|
>
> Here the overall universe or "U" is that of all tunings, with JI/RI
> including all tunings based on integer ratios for all intervals; the
> HI/CI subset includes JI/RI intervals or sonorities with an audible
> effect of "purity" from coupling of partials or perception of
> combinatorial tones.
>

From my perspective this becomes (with the above explanation still holding with 'HI/RI' and 'JI/CI' :

|----------------------------------------------|
| U(niverse)=All tunings |
| |
| |-------------------------| |
| | HI/RI |---------| | |
| | | JI/CI | | |
| | |---------| | |
| |-------------------------| |
| |
|----------------------------------------------|

A similar thing happens to the other diagram.

As I said before, everything else you say still holds up - in my mind even better.

Needless to say, this is just my own view. To a degree terminology can be flexible. Personally, when I get around to writing an essay of any kind I'll be sure to explicitly define every potentially problematic term. Better safe, no ?

Thank you for your patience & pardon any errors !

Sincerely,
Joel

🔗Joel Rodrigues <joelrodrigues@mac.com>

> "genewardsmith" <genewardsmith@juno.com>
>
> --- In tuning@y..., Joel Rodrigues <joelrodrigues@m...> wrote:
>>> "genewardsmith" <genewardsmith@j...>
>>> Why is this different in practice from continuum intonation,
>>> sans the "rational"?
>>
>> Hello Gene,
>>
>> 'continuum intonation' sans qualification would refer to the
>> entire virtual pitch continuum.
>
> My question was not how it was different in theory, but in practice.

I'll just quote Margo here, because I think it's an easily understood (taken for granted, even) matter:

'From a real world perspective, of course, integer ratios are
mathematical models rather than precise definitions of what performers
actually produce using human voices, other acoustical instruments, or
even electronic ones (apart from certain instruments which do use
harmonic or subharmonic series, as described for example in some
recent posts on the Scalatron in certain situations.)'

>> RCI refers to a tonal system in which the target intonation for
>> a given note is any _integer_ ratio. The 'rational' here refers
>> to rational numbers, which involve only integers (whole numbers).
>
> This makes sense if you are looking at the matter > theoretically, for instance in terms of group theory. If you > are looking at dense subsets of the reals as a pitch continuum, > however, it seems to me any dense subset is in effect the same > as any other dense subset.

Eh ? I don't understand, or know about 'group theory', 'dense subset', or 'reals'. Yes, that's the depth of my knowledge of mathematics. But seriously, I think you misunderstand. From a cursory investigation, I don't think either Margo or I are talking about 'group theory' or 'dense subsets of the reals'. It's simply 'any _integer_ ratio', because the 'rational' in RI/RCI does not refer to ratios, but to rational numbers.

Waitamminit ! I just checked because I suspected as much; you're the chap who thought Joe P was talking math when he said 'scalar'. Gene ! You may be overdosing on math !;-) Isn't there an old Mickey Mouse cartoon where he's in some kind of nightmarish scenario with numbers & mathematical symbols swirling all around him ? If not, it's probably something out of my childhood. Brrrr...

> I see no need for a 5-limit continuum intonation, 7-limit > continuum intonation, and so forth, *except* as theoretical > constructs. It's not as if you could possibly hear a > difference, after all.

'5-limit continuum intonation' is a statement not possible by any definition/explanation offered by Margo, or myself. '5-limit harmonic intonation' or '5-limit rational intonation'? Maybe.

>> 2:1 reduction is a given.
>
> Eh? Why? How did that enter the conversation?

It's not a big deal, I only included this to avoid earlier comments by some that interval X is beyond humanly audibility. The pitch continuum is theoretically infinite, and one of the ways we use it to build scales/tonal systems, is using 2:1 reduction.

Phew !

Cheers,
Joel

🔗Joel Rodrigues <joelrodrigues@mac.com>

> "jpehrson2" <jpehrson@rcn.com>
>
> --- In tuning@y..., Joel Rodrigues <joelrodrigues@m...> wrote:
>
> /tuning/topicId_38453.html#38454
>
>> I find this unnecessarily restrictive:
>>
>> Harmonic Intonation (HI): JI where harmonicity and the
>> purest or most beatless tuning of sonorities possible
>> is the ideal -- or, from another view, the Partch/Doty
>> ideal that ratios should be kept as small as is fitting
>> for the musical context
>>
>
> ***Thank you so much, Joel, for your interesting post. The
> word "harmonic" has so much "excess baggage" in music, though, that I
> fear you might want to think of another descriptor...
>
> J. Pehrson

Hey there Joe ! Thank *you* for finding it interesting. You
raise an interesting point yourself. Oh yeah, sorry for the
delay in responding. Much has been lying in the

It is true that 'harmonic' has accumulated much '"excess
baggage"', but so far I think it's a easily sorted out. There
seem to be four senses (there may be more) that the word is used
in reference to:

1) The harmonic series, with a term like 'harmonic intonation'
in common use.

2) Harmony. Terms in use include:

'harmonic tonality' - in discussions of so-called 'tonal' music.
<http://cc2.hku.nl/wim/reservoir/ict-
b/j1theorie/1harmton/harton_eng.html>
<http://www2.rhbnc.ac.uk/Music/Archive/Disserts/brover-l.html>

'harmonic tension'

'harmonic language'

'harmonic progression'

'harmonic rhythm' - The rate at which the basic harmonies of a
piece of music change.

'harmonic interval' - If the two notes of an interval are
sounded simultaneously (a 2-note chord).
[Taylor, Eric. The Associated Board Guide to Music Theory - Part
I. (London, ABRSM Publishing, 1989)]

Some of these may be perceived to be decipherable through
context, but it doesn't usually work out that way.

3) The study of harmonics in the Greek tradition. See the
introduction to Scientific Method in Ptolemy’s Harmonics by
Andrew Barker, available in PDF format at
<http://books.cambridge.org/0521553725.htm>.

4) This is the most bizarre, since I can see no discernible
logic behind terms like:
'harmonic chromatic scale'
'harmonic minor scale'
[Taylor, Eric. The Associated Board Guide to Music Theory - Part
I. (London, ABRSM Publishing, 1989)]

Can anyone shed light on why these are called so ?

For me no. 4 is easily discarded. No. 1 gets top priority. No. 2
is a nuisance and I'd like to find a nice alternate term.
'Harmonial' *is* an adjective for 'harmonic', but doesn't roll
well off the tongue. No. 3 is related to No. 1.

I'd appreciate any thoughts you have on all this !

You know, I was watching a programme on the BBC about Douglas
Adams and they mentioned a book called "The Meaning of Liff"
containing terms for commonplace things in life that have no
name. This may be the type of document needed to deal with the
mess microtonality makes of common use mainstream music
terminology and knowledge.

Sincerely,
Joel