Re: "major" and "minor"

Hello, there, Monz and Paul and everyone, and please let me quickly
respond to the question concerning the terminology for "major" and
"minor" intervals.

First of all, I agree with Monz that the specific terms "major third"
and "minor third" (or Latin _tertia maior_ and _tertia minor_) are
characteristic of post-medieval European theory, although it would be
interesting to research the first uses in Latin or vernacular sources.

The usual terms I encounter in a more learned style of medieval
European theory are _ditonus_ and _semiditonus_ for the larger and
smaller regular thirds equal to 81:64 (~407.82 cents) and 32:27
(~294.13 cents) in Pythagorean tuning.

However, generic interval names like _tertia_ or _septima_ ("third"
and "seventh") may have come into use by around the 12th or 13th
century, and I might guess that the staff notation which became
standardized around the 11th-12th century era could have contributed
to this trend. Counting lines or spaces between notes might have
encouraged a focus on general interval categories, with either a
ditone (F-A) or a semiditone (E-G) at a visual distance of three
inclusive steps.

Now for the reference to the "perfect" and "imperfect" thirds of some
late medieval theory: the ditone or larger regular third is "perfect"
because it seeks expansion to the fifth, in 14th-century terms, while
the smaller third is "imperfect" because it tends to contract to a
unison. Thus according to Ugolino of Orvieto (c. 1425-1440), one
"perfects" a naturally minor third before a fifth by making it major
(to use the modern categories), or "colors" a naturally major third
before a unison by making it minor (e.g. E-G# before D-A; C-Eb before
a unison on D).

The categories of _tertia_ or "third" here are "perfect"
vs. "imperfect," synonymous with the later major and minor.

Around 1325, Jacobus of Liege speaks of _una tertia in semiditono_ or
"a third in the semiditone [form]," which tends to resolve to the
unison; and _una tertia in ditono_ or "a third in the ditone [form},"
seeking expansion to the fifth.

Interesting, Jacobus recognizes in theory three varieties of tone,
although regarding only the regular _tonus_ (9:8) as used in
practice. The _tonus maior_ or "major tone" consists of two major
semitones or apotomes at 2187:2048 (~113.69 cents), giving us an
interval at around 227.37 cents (smaller than 8:7 by a 3-7 schisma of
about 3.80 cents). The _tonus minor_ consists of two limmas or minor
semitones, the usual 256:243 semitones (~90.22 cents), giving us an
interval of about 180.45 cents (smaller than 10:9 by the 3-5 schisma
of about 1.95 cents).

Jacobus demonstrates, for example, that the _tonus maior_ is larger
than 9:8 but smaller than 8:7.

Anyway, the increasing use in medieval literature of informal Latin
terms like _tertia_ for both varieties of thirds might be linked both
to the popularization of basic theory (for example, in introductions
to discant or polyphony), where terms like _quarta_ and _quinta_ are
also used as alternatives to the Greek _diatessaron_ and _diapente_
(since the Greek terms mean "through four" and "through five," this is
closer to a translation than to a different naming system, as with
_tertia_ as a category including both _ditonus_ and _semiditonus_).

A brief remark on the 18:17:16 division of the 9:8 tone. Boethius uses
it not as an illustration of what prevails in practice (he takes the
Pythagorean division into 256:243 and 2187:2048 as the norm), but a
theoretical example to show the impossibility of dividing a
superparticular ratio such as 9:8 into two equal parts (i.e. a
geometric division) -- using integer ratios, that is, we might add.

The 18:17:16 division gets mentioned in Marchettus of Padua (1318) as
one possible division, and in 1373 is recommended as a way to add
semitones to an organ when one has arranged a basic Pythagorean
diatonic scale in the usual fashion: take the average of the lengths
of two pipes a 9:8 tone apart, and use this as the length for the pipe
adding an accidental between them.

I hope that this addresses some of the points in question.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello Margo!,

It's been a while since I got a chance to address you!

> From: M. Schulter <MSCHULTER@VALUE.NET>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 10:06 PM
> Subject: [tuning] Re: "major" and "minor"
>
>
> Hello, there, Monz and Paul and everyone, and please let me quickly
> respond to the question concerning the terminology for "major" and
> "minor" intervals.
>
> First of all, I agree with Monz that the specific terms "major third"
> and "minor third" (or Latin _tertia maior_ and _tertia minor_) are
> characteristic of post-medieval European theory, although it would be
> interesting to research the first uses in Latin or vernacular sources.

I'm going to take a look again at the Frankish sources of music-theory
that I have (c. 800-1000) to see if there's mention of _tertia maior_
and _tertia minor_. My guess is that what you say below is on the
right track, and that we won't find these until after the development
of staff notation c. 1100-1200.

> The usual terms I encounter in a more learned style of medieval
> European theory are _ditonus_ and _semiditonus_ for the larger and
> smaller regular thirds equal to 81:64 (~407.82 cents) and 32:27
> (~294.13 cents) in Pythagorean tuning.

Ah, *there's* the term I couldn't remember! "Semiditonus",
known in ancient times usually as "trihemitone".

> However, generic interval names like _tertia_ or _septima_ ("third"
> and "seventh") may have come into use by around the 12th or 13th
> century, and I might guess that the staff notation which became
> standardized around the 11th-12th century era could have contributed
> to this trend. Counting lines or spaces between notes might have
> encouraged a focus on general interval categories, with either a
> ditone (F-A) or a semiditone (E-G) at a visual distance of three
> inclusive steps.

This is brilliant, Margo! Yes, it makes a lot of sense.

<discussion of "perfect / imperfect" and Jacobus de Liege snipped>

... but it was all very interesting! Thanks!

> A brief remark on the 18:17:16 division of the 9:8 tone. Boethius uses
> it not as an illustration of what prevails in practice (he takes the
> Pythagorean division into 256:243 and 2187:2048 as the norm), but a
> theoretical example to show the impossibility of dividing a
> superparticular ratio such as 9:8 into two equal parts (i.e. a
> geometric division) -- using integer ratios, that is, we might add.

Yes, I meant to add all of this to my description of it. Glad you did.

> The 18:17:16 division gets mentioned in Marchettus of Padua (1318) as
> one possible division, and in 1373 is recommended as a way to add
> semitones to an organ when one has arranged a basic Pythagorean
> diatonic scale in the usual fashion: take the average of the lengths
> of two pipes a 9:8 tone apart, and use this as the length for the pipe
> adding an accidental between them.

18:17 was also recommended later by Galilei (c. 1590?) as the
"perfect tuning" for the lute, because it was the easiest way
to rationally approximate 12-EDO.

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Hello, there, Monz, and what a delight to be exchanging messages with
you about Boethius, semiditones, 18:17:16 divisions, and the like.

Your mention of Vincenzo Galileo's use of 18:17 to approach a 12-EDO
semitone, actually more accurate on a lute than the logarithmically
correct fret spacing according to Mark Lindley because of fretting
pressure, reminds me of my idea for a 21st-century fretting solution:
just use successive ratios of 25:24. However, I'll refrain from
calling this "JI"; formal RI plus a bit of octave stretching is more
like it.

While I much regret not mentioning Dave Keenan's definition in my
earlier post on "What is JI," I'm delighted that my omission prompted
others to remedy this fault, calling attention to a viewpoint that
deserves to be noticed whenever this topic comes up.

For now, while I consider a longer post on "What JI means to me,"
please let me add that you have frequently made it clear that you set
no arbitrary limits on JI, literally or figuratively, but like me have
been intrigued by writers such as Fabio Colonna who discuss higher
primes and integer ratios.

Recently I have felt a special wonder at reading Kathleen Schlesinger,
and seeing her mention many of my favorite ratios -- in 1939. Maybe
I'd compare her treatise on _The Greek Aulos_ to some late
16th-century Italian efforts to achieve a dramatic music along Greek
lines -- the origins of the opera, of course.

Anyway, this is mainly to say thanks, for your most recent reply and
your many contributions.

Most appreciatively,

Margo Schulter
mschulter@value.net

P.S. Aside to Jon Wild: the distinction you report in Aristoxenos between
the simple and composite use of an interval such as the ditone or major
third very much reminds me of the Vicentino-Lusitano controversy (does any
use of a melodic minor or major third represent an element of the
chromatic or enharmonic genus, so that almost all music is actually
"mixed" rather than diatonic?). Here Lusitano and later Zarlino took the
position you report, arguing that the chromatic semitone and enharmonic
diesis are characteristic of these genera, while major and minor thirds
are also routine as composite intervals in the diatonic.