some historical 19-limit ratios

John Chalmers had asked me in a private email if Woolhouse
had proposed 16:19:24 for the 'minor' triad. I told him
that no, at least in the _Essay on Musical Intervals_,
Woolhouse used 5-limit JI as his harmonic/melodic basis,
but that his goal was to construct a temperament that had
only one size for the 'whole tone', to prevent commatic
drift, and two sizes of 'semitone', to simulate the differences
apparent in JI.

He mentions nothing about harmonic identities higher than
5 except in his chapter on 'harmonics', where he is content
to show how to produce them, but offers no advice or comment
on how musically useful they may be.

John's reply got me started on something that I thought
might interest others here. See my 'Tutorial on ancient
Greek Tetrachord-theory' if you don't understand the Greek
note-names:
http://www.ixpres.com/interval/monzo/aristoxenus/tutorial.htm

Since the theory of tetrachord division was normally
assumed to be replicated identically in all other tetrachords,
I didn't bother giving a description of the larger aspects
of Greek theory there, so the tetrachord names aren't in my
tutorial. They are as follows, with abbreviations and English
translations:

Hyperbolaion hyper. superfluous/furthest (lyre position)
Diezeugmenon diez. disjunct (= not connected to _mese_)
Synemmenon syn. conjunct (= connected to _mese_)
Meson mes. middle
Hypaton hypat. highest/nearest (lyre position)

Note that because the lowest-pitched lyre string was the
highest in position and nearest the player (as on the modern
guitar), the note names do not reflect the pitch-height.

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

> [John Chalmers, private communication]
>
> I was wrong, it was Ousely who proposed the 19/16 minor third
> and the 24/19 complement (16:19:24 triad). The reference is
> in Shirlaw's book.

Interesting, thanks. Don't know anything about Ousley
- tell me more. I'll have to include him in my book, in
the '19' chapter.

Here are some other '19'ers' I dug out of my book.
You're probably aware of most if not all, but I decided
to write this for you anyway, because I can print it out
and add it to my book - some of this stuff isn't there.

ERATOSTHENES
------------

I don't recall if Eratosthenes used the GPS/LPS [Greater
Perfect System / Lesser Perfect System] format in his
descriptions. But he lived in the century following
Aristoxenus, who was the first to use it, so I suppose it's
fair to assume it for Eratosthenes.

At any rate, he proposed ratios of 13 for his enharmonic
_tritai_ and _parhypate_, and ratios of 19 for his chromatic
_tritai_ and _parhypate_ and enharmonic _paranetai_ and
_lichanoi_.

If his ratios are assumed to compose a GPS/LPS, there is
a 16:19:24 'minor' triad available on:

30/19 chromatic trite hyperb. / enharmonic paranete hyperb.
5/4 chromatic paranete diezeugmenon
20/19 chromatic trite syn. / enharmonic paranete synemmenon

And most likely those combinations would not have been
used in ancient Greek music... or would they?
At any rate, a modern explorer in this tuning could
make use of the 16:19:24 'minor' chord.

But the most interesting thing I find in Eratosthenes is
the 1215:1216 [= ~1.42 cents] 3==19 xenharmonic bridge, between:
(all cents values measured up or down from _mese_)

256/243 [= ~90.2 cents] diatonic trite synemmenon
20/19 [= ~88.8 cents] chr. trite syn. / enh. paranete syn.

128/81 [= ~792.2 cents] diatonic trite hyperbolaion
30/19 [= ~790.8 cents] chr. trite hyperb. / enh. paranete hyperb.

64/81 [= ~-407.8 cents] diatonic parhypate meson
15/19 [= ~-409.2 cents] chr. parhypate meson / enh. lichaonos mes.

32/27 [= ~294.1 cents] dia. trite diez. / dia. paranete syn.
45/38 [= ~292.7 cents] chr. trite diez. / enh. paranete diez.

16/27 [= ~-905.9 cents] diatonic parhypate hypaton
45/76 [= ~-907.3 cents] chr. parhypate hypat. / enh. lich. hypat.

Of course, his measurements were based on superparticular
proportions, so that's why he insisted on using those 19-limit
ratios that are so close to the 3-limit ones, because they
formed superparticular relationships that the 3-limit ones
didn't.

BOETHIUS
--------

Boethius calculated his chromatic _paranetai_ and _lichanoi_,
and his enharmonic _tritai_ and _parhypatai_, as simple
arithmetic means between the notes found in his diatonic.
This resulted in 19-limit ratios in his chromatic genus.

The monochord string-lengths for his chromatic _paranetai_
were calculated according to the formula:

diatonic paranete + ((diatonic paranete - nete) / 2)

and similarly, the chromatic _lichanoi_ were:

diatonic lichanos + ((diatonic lichanos - mese OR hypate) / 2)

There are therefore 16:19:24 'minor' triads on:

32/19 chr. paranete hyperb.
4/3 nete synemmenon
64/57 chr. paranete syn.

24/19 chr. paranete diez.
1/1 mese
16/19 chr. lichanos meson

And Boethius's system contains the 512:513 [= ~3.38 cents]
3==19 xenharmonic bridge between:

9/8 [= ~203.9 cents] paramese
64/57 [= ~200.5 cents] chr. paranete synemmenon

GANASSI
-------

Ganassi 1543 proposed a 12-tone scale that used three ratios
of 17 and two of 19 for the five chromatic notes. It gave
a C#/Db-minor triad of

20/19 : 5/4 : 30/19
= 16 : 19 : 24

(interestingly, exactly the same one as in Eratosthenes...)

and a G#/Ab-minor of

30/19 : 15/8 : 40/17
= 816 : 969 : 1216

which is obviously nowhere near as euphonious.
It has the 16:19 'minor 3rd', but the 'fifth' is a 51:76
[= ~691 cents], which I suppose wouldn't be objectionable
to a later listener who's used to meantone.

The string-lengths for Ganassi's entire scale have
nice low-integer proportions which reduce to a simple
pattern:

/1 /4 /5 /6

60 15 12 10 'octave'
64 16
68 17
72 18 12
76 19
80 20 - 16 3/2
85 17
90 18 - 15 4/3
96 24 16
102 17
108 27 18
114 19
120 30 20 entire string

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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