"major" and "minor" (was: small interval (s) flat second [bII])

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, January 28, 2002 4:47 PM
> Subject: [tuning] Re: small interval (s) flat second [bII]
>
>
> The term "minor" just means "smaller". These names came into use long
> before (someone correct me if I'm wrong) the concepts of "Major
> scale" and "Minor scale" and tonality as we know it.

You're correct there, Paul.

The earliest reference I know of is in Boethius (c. 505 AD).
He used the terms "semitonium maius" and "semitonium minus"
in his _De institutione musica_, Book 1, end of Chapter XVI
[p 203 of Friedlein's edition, p 26 in Bower's English translation]:
http://www.music.indiana.edu/tml/6th-8th/BOEMUS1_TEXT.html

This is at the end of Boethius's discussion of how the
"whole-tone" 9:8 cannot be split exactly in half numerically
[without the more modern method of taking the square-root].
He doubles each of the terms to make it 18:16, and by
interpolating the mediant 18:17:16 he gets the two semitones:

semitonium minus = 18:17 = ~98.95459223 cents
semitonium maius = 17:16 = ~104.9554095 cents

>> Sed utraque semitonia nuncupantur, non quod omnino
>> semitonia ex aequo sint media, sed quod semum dici solet,
>> quod ad integritatem usque non pervenit. Sed inter haec
>> unum maius semitonium nuncupatur, aliud minus.
>>
>> Both of these are called "semitones", not because these
>> intermediate semitones are equal at all, but because
>> something that does not come to a whole is usually called
>> "semi". In the case of these, one semitone is called
>> "major", and the other "minor".

This is about 1000 years before the concepts of "major"
and "minor" tonality were developed. Indeed, AFAIK Boethius
does not apply these terms to any other intervals, since
his basic tuning was Pythagorean and thus had only one
size of the other "imperfect" intervals.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> This is about 1000 years before the concepts of "major"
> and "minor" tonality were developed. Indeed, AFAIK Boethius
> does not apply these terms to any other intervals, since
> his basic tuning was Pythagorean and thus had only one
> size of the other "imperfect" intervals.

Huh? Pythagorean has minor thirds and major thirds, not only one size
of third!

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 12:44 PM
> Subject: [tuning] Re: "major" and "minor" (was: small interval (s) flat
second [bII])
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > This is about 1000 years before the concepts of "major"
> > and "minor" tonality were developed. Indeed, AFAIK Boethius
> > does not apply these terms to any other intervals, since
> > his basic tuning was Pythagorean and thus had only one
> > size of the other "imperfect" intervals.
>
> Huh? Pythagorean has minor thirds and major thirds, not only one size
> of third!

Oops! ... my bad! I sure worded that one wrong!

Here's a proper explanation of what I wrote there:

"Pythagorean has minor and major thirds" *today*,
but not in Boethius's time!

It's a matter of terminology: back then, the "minor third" was
a "trihemitone" [= "3 semitones"] and a "major third" was a
"ditone" [= "2 tones"].

In other words, no reference was made to them being larger
and smaller versions of one generic interval, but rather instead
to the number of tones or semitones which make them up.

It's an important distinction which gets at the root of the
difference between medieval 3-limit-based theory and
Renaissance 5-limit based theory.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> "Pythagorean has minor and major thirds" *today*,
> but not in Boethius's time!
>
> It's a matter of terminology: back then, the "minor third" was
> a "trihemitone" [= "3 semitones"] and a "major third" was a
> "ditone" [= "2 tones"].
>
> In other words, no reference was made to them being larger
> and smaller versions of one generic interval, but rather instead
> to the number of tones or semitones which make them up.
>
> It's an important distinction which gets at the root of the
> difference between medieval 3-limit-based theory and
> Renaissance 5-limit based theory.

Really? Why wouldn't some Pythaogorean theorist classified diatonic
intervals as we do today, into "major" and "minor" categories, well
before the "5-limit" era? Are you sure that none did?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 1:44 PM
> Subject: [tuning] Re: "major" and "minor" (was: small interval (s) flat
second [bII])
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > "Pythagorean has minor and major thirds" *today*,
> > but not in Boethius's time!
> >
> > It's a matter of terminology: back then, the "minor third" was
> > a "trihemitone" [= "3 semitones"] and a "major third" was a
> > "ditone" [= "2 tones"].
> >
> > In other words, no reference was made to them being larger
> > and smaller versions of one generic interval, but rather instead
> > to the number of tones or semitones which make them up.
> >
> > It's an important distinction which gets at the root of the
> > difference between medieval 3-limit-based theory and
> > Renaissance 5-limit based theory.
>
> Really? Why wouldn't some Pythaogorean theorist classified diatonic
> intervals as we do today, into "major" and "minor" categories, well
> before the "5-limit" era? Are you sure that none did?

NO! Sorry, Paul, didn't mean to imply that *no* Pythagorean
theorist ever did this! Of course they did, later. The thought
that prompted my original (badly worded) statement was that there
*were NOT two different versions of the WHOLE-TONE*, as there
are in 5-limit JI.

(Of course, there can be many other versions of 5-limit "whole-tones"
too, but theorists have primarily been concerned with the "major tone"
9:8 and the "minor tone" 10:9; the latter was not part of Boethius's
theory.)

But it's telling that when the "trihemitone" and "ditone" were
first considered to be two different flavors of one generic "3rd"
(around 1200-1300 or so -- Margo will know the details), they
had to be tagged with the name "imperfect", not only because
contemporary usage dictated this nomenclature, but *ALSO* because
*contemporary terminology gave no other way to describe them*!
It took a few centuries for the "major/minor 3rd" terminology
to evolve.

But in Boethius's time, the qualifiers "major" and "minor"
were only used for semitones. At least, I haven't been able
to find them used in his book for any other intervals.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> But it's telling that when the "trihemitone" and "ditone" were
> first considered to be two different flavors of one generic "3rd"
> (around 1200-1300 or so -- Margo will know the details), they
> had to be tagged with the name "imperfect", not only because
> contemporary usage dictated this nomenclature, but *ALSO* because
> *contemporary terminology gave no other way to describe them*!

Hmm . . . not seeing the "imperfect" label as relevant or necessary
in this particular context.

> It took a few centuries for the "major/minor 3rd" terminology
> to evolve.

Are you sure about that? And what about ancient Greece? Weren't there
concepts corresponding to "generic third", "minor third", "major
third", and so on for other intervals?

> But in Boethius's time, the qualifiers "major" and "minor"
> were only used for semitones. At least, I haven't been able
> to find them used in his book for any other intervals.

That's only one theorist, Monz.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 2:18 PM
> Subject: [tuning] Re: "major" and "minor" (was: small interval (s) flat
second [bII])
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > But it's telling that when the "trihemitone" and "ditone" were
> > first considered to be two different flavors of one generic "3rd"
> > (around 1200-1300 or so -- Margo will know the details), they
> > had to be tagged with the name "imperfect", not only because
> > contemporary usage dictated this nomenclature, but *ALSO* because
> > *contemporary terminology gave no other way to describe them*!
>
> Hmm . . . not seeing the "imperfect" label as relevant or necessary
> in this particular context.

My reasoning is that, since there was no way to describe
them both as anything other than the Latin equivalent of
"3 semitones" and "2 tones", they could not be classified
*together* as "3rds" until some innovative theorist came
along who realized that they could indeed function as
two different flavors of one generic interval. So at some
point around 1200 someone did that, and we got "major" and
"minor 3rds".

> > It took a few centuries for the "major/minor 3rd" terminology
> > to evolve.
>
> Are you sure about that? And what about ancient Greece? Weren't there
> concepts corresponding to "generic third", "minor third", "major
> third", and so on for other intervals?

Hmmm... that's a good question. I think the Greeks had
the *concept* of these varying and generic "3rds", but not
sure about how careful their taxonomy was.

The Latin terminology is derived directly from the Greek.
Generally, you'll find that different flavors of "3rds"
functioned as the "characteristic interval" of the various
genera, so that the "diatonic genus" has a CI of a "tone",
the "chromatic" CI is a "trihemitone", and the "enharmonic"
CI is a "ditone". All Greek scales were categorized according
to the genus to which they belonged.

So, the most informed answer I can give (perhaps someone else
out there knows better) is no, the Greeks did not have a
*terminology* corresponding to "generic / major / minor 3rd".

> > But in Boethius's time, the qualifiers "major" and "minor"
> > were only used for semitones. At least, I haven't been able
> > to find them used in his book for any other intervals.
>
> That's only one theorist, Monz.

Oh, but Paul, *please* don't underestimate the importance of
this *particular* "only one theorist"!! Boethius's treatise
became the *standard* reference on music-theory in Europe for
1000 years! That's A THOUSAND YEARS.

(Wouldn't any of us hope that our work would be so enduring? ...)

-monz

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Well, Monz, I'm quite surprised to learn that "major" and "minor"
were first used to distinguish different sizes of _semitone_ rather
than different sizes of mod-7 diatonic interval. I'd be grateful if
John Chalmers or someone could confirm what you're saying here.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 4:29 PM
> Subject: [tuning] Re: "major" and "minor" (was: small interval (s) flat
second [bII])
>
>
> Well, Monz, I'm quite surprised to learn that "major" and "minor"
> were first used to distinguish different sizes of _semitone_ rather
> than different sizes of mod-7 diatonic interval. I'd be grateful if
> John Chalmers or someone could confirm what you're saying here.

Well, as I said, the earliest reference I know to the actual use
of the terms "minor" and "major" (in Latin equivalents) is in
Boethius, which is c. 505 AD.

I'll have to take a look into the Greek treatises I know to see
what they say to differentiate different sizes of generic intervals.

But it's not at all surprising to me that this terminology would
arise in connection with semitones. To Classical (Greek and Roman)
theorists, keeping the sizes of different semitones straight was
very important, as very often they thought in terms of tetrachord
species, that is, the pattern of "tones" and "semitones" with a
given tetrachord, which ordinarily would be replicated in other
tetrachords at other parts of the scale.

And since they *didn't* call what we know as the "minor 3rd"
any kind of third, but rather a "trihemitone" (= "3 semitones"),
measurement of that interval's specific size in their tetrachords
was also done by means of semitones.

So the proper manipulation of the 9:8 "whole-tone" and the
different sizes of semitones (and the smaller intervals too,
for the enharmonic) was/is fundamental in understanding the
different genera of Greek and Roman theory.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Tuesday, January 29, 2002 4:29 PM
> > Subject: [tuning] Re: "major" and "minor" (was: small interval
(s) flat
> second [bII])
> >
> >
> > Well, Monz, I'm quite surprised to learn that "major" and "minor"
> > were first used to distinguish different sizes of _semitone_
rather
> > than different sizes of mod-7 diatonic interval. I'd be grateful
if
> > John Chalmers or someone could confirm what you're saying here.
>
>
>
> Well, as I said, the earliest reference I know to the actual use
> of the terms "minor" and "major" (in Latin equivalents) is in
> Boethius, which is c. 505 AD.
>
> I'll have to take a look into the Greek treatises I know to see
> what they say to differentiate different sizes of generic intervals.
>
>
> But it's not at all surprising to me that this terminology would
> arise in connection with semitones. To Classical (Greek and Roman)
> theorists, keeping the sizes of different semitones straight was
> very important, as very often they thought in terms of tetrachord
> species, that is, the pattern of "tones" and "semitones" with a
> given tetrachord, which ordinarily would be replicated in other
> tetrachords at other parts of the scale.

If thinking of a "semitone" as literally one of two parts of a tone,
then this makes sense. However, I'd argue that from a strictly
_diatonic_ perspective (i.e., defined either by the usual pair of
comma and chroma unison vectors, or solely by the Pythagorean apotome
as chroma) if ever such a thing existed (I like to think it did),
a "semitone" is really either a "minor second" or an "augmented
unison" -- it is not really a single category unto itself, despite
the coincidence of pitch in some tuning systems.

This sort of thing is very real in diatonic music perception. My 12th
grade music teacher, Phil Rosenberg, would play some harmonic minor
snippets on the piano, and then hit an interval, and ask "consonant"
or "dissonant". If that interval was a minor third in the context of
the preceding scale, we'd all say "consonant". If it was an augmented
second, "dissonant". This despite the two intervals having exactly
the same ratio, the fourth root of two, on the piano. That experiment
pointed out to me the importance of _diatonic categorical perception_
(as opposed to 12-tET categorical perception, which trained musicians
seem to have overwhelming amounts of). Thus observations like Carol
Krumhansl's "if chord construction is determined in some principled
way by scale structure, this further serves to maintain the tonal
framework for encoding pitch information" take on a very strong voice
during my speculations into new tonal systems.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 7:36 PM
> Subject: [tuning] Re: "major" and "minor" (was: small interval (s) flat
second [bII])
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > But it's not at all surprising to me that this terminology would
> > arise in connection with semitones. To Classical (Greek and Roman)
> > theorists, keeping the sizes of different semitones straight was
> > very important, as very often they thought in terms of tetrachord
> > species, that is, the pattern of "tones" and "semitones" with a
> > given tetrachord, which ordinarily would be replicated in other
> > tetrachords at other parts of the scale.
>
> If thinking of a "semitone" as literally one of two parts of a tone,
> then this makes sense. However, I'd argue that from a strictly
> _diatonic_ perspective (i.e., defined either by the usual pair of
> comma and chroma unison vectors, or solely by the Pythagorean apotome
> as chroma) if ever such a thing existed (I like to think it did),
> a "semitone" is really either a "minor second" or an "augmented
> unison" -- it is not really a single category unto itself, despite
> the coincidence of pitch in some tuning systems.

Well... OK, I can certainly buy your argument here. *But* ...
I'm not sure if it has any application to music or music-theory
of ancient Greek and Roman times, and my inclination is to say no.

Greek theorists certainly argued that *functional* notation
was important -- this refers to the names "mese", "lichanos meson",
etc. -- but I don't think they reckoned so much in terms of
"scale steps", as we do.

Their entire theory was based on, to quote the title of a great
book by our dear friend John Chalmers, "Divisions of the Tetrachord".
The tetrachord was seen as a microcosm of the whole pitch-continuum,
and was divided up in various ways which always consisted of
three intervals (or in some really old cases, 3 notes and
2 intervals).

These three particular intervals, which form a set unique to a
particular genus (and possibly also "shade" of genus, i.e.,
Aristoxenos's "relaxed diatonic", etc.), along with the
separate 9:8 "whole tone" which always appeared at some point
in a Greek system as a "tone of disjunction" between tetrachords,
were the building blocks of all other intervals in a Greek
tuning.

So while a case could be made that they had a concept analagous
to our "minor second" (I'll have to look into that one), there
*definitely* was no such concept as "augmented unison".

> This sort of thing is very real in diatonic music perception. My 12th
> grade music teacher, Phil Rosenberg, would play some harmonic minor
> snippets on the piano, and then hit an interval, and ask "consonant"
> or "dissonant". If that interval was a minor third in the context of
> the preceding scale, we'd all say "consonant". If it was an augmented
> second, "dissonant". This despite the two intervals having exactly
> the same ratio, the fourth root of two, on the piano. That experiment
> pointed out to me the importance of _diatonic categorical perception_
> (as opposed to 12-tET categorical perception, which trained musicians
> seem to have overwhelming amounts of). Thus observations like Carol
> Krumhansl's "if chord construction is determined in some principled
> way by scale structure, this further serves to maintain the tonal
> framework for encoding pitch information" take on a very strong voice
> during my speculations into new tonal systems.

Wow, Paul, have you ever mentioned this before? This is a terrific
story! The quote from Krumhansl reminds me of something I wrote
once about Haba... can't remember now if Haba agreed or disagreed
with Krumhansl; it was either that he felt exactly the same way,
or exactly the opposite: that is, "if *scale structure* is determined
by *chord construction* ..." etc.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> So while a case could be made that they had a concept analagous
> to our "minor second" (I'll have to look into that one), there
> *definitely* was no such concept as "augmented unison".

Wouldn't the comparison of, say, the second scale degree in one mode
vs. the second scale degree in another mode, be a comparison that
involved the concept "augmented unison"?

> Wow, Paul, have you ever mentioned this before? This is a terrific
> story! The quote from Krumhansl

You can find that in my paper.