On Fabio Colonna's Sambuca tuning

[Joe Monzo]

(This originated in an email from me to John Chalmers, 1999.4.9.)

I've found some interesting things in the Lynn Wood Martin article, in Xenharmonikon 7/8, on Colonna's 'Sambuca'.

Fabio Colonna lived in Naples, and published a treatise in 1618 called _La Sambuca Lincea_, which included a description of the instrument by that name which he built on commission from Scipione Stella, who had had the opportunity in 1594 to examine Vincentino's 'Archicembalo', a 31-tone-per-octave (not equal-tempered) keyboard instrument.

Colonna's treatise (which I have not read) describes first the ancient genera of the Greeks and the monochord divisions by which they achieved their scales. Then he describes the instrument.

Martin's 'example 2' on page 4 of the article is a (edited?) reproduction of a table of string-lengths from Colonna's treatise, but the numbers don't all jive.

Colonna used a nominal string length of 2000 units, but he uses such small fractional measurements that the actual number of equal divisions is frequently far larger than this.

Getting to some of the minor problems first:

At the beginning of page 5, Martin is discussing the ratio which Colonna measured as (1949 & 47/197) / 2000, and how it compares to the remainder of the string (the part not sounding).

This can be written as a simple formula: (sounding length)/((stopped length)-(sounding length))

In referring to the Latin ratio names associated with this note, Martin calls 'trigeculpa octupla' the ratio 38:1 and 'superbipartiens quintas' the ratio 17:15, without really explaining how or why they go together:

[Martin:]

With the remainder (two columns to the right [in Colonna's table - 'example 2'], 50 & 150/197) a minor third sounds. The proportion between the part of the string which sounds and the remainder is _Trigecupla octupla_, or 38:1, _Superbipartiens quintas_, or 17:15. The proportion with the grave [the whole string] is _Sesquitrigesima ottava_, or 39:38.

My Latin is *very* rusty, but this is not right.

I can see by measuring the ratios that 'trigeculpa octupla' and 'superbipartiens quintas' go together, and actually mean '38 & 2/5 : 1' as follows:

(1949 & 47/197) / 2000 = 197/192 [ratio of the note]

192 / (197 - 192)

= 192 / 5

= 38.4 [= 38 & 2/5, the ratio of the two string lengths]

_Sesquitrigesima ottava_ appears on the left side of Colonna's table, where he is labelling particular intervals, apparently to illustrate the small divisions in the ancient tetrachords. Right next to it, apparently pointing to the same note, and totally ignored in the text by Martin, is 'Sesquiquadragesima', meaning 41:40.

The Latin term comparing the measured ratio with the remainder of the string, appearing on the right side of Colonna's table, is 'trigeculpa nontupla superbipartiens quintas', which indicates an actual note ratio of 39 & 2/5 to the 'trigeculpa octupla superbipartiens quintas', 38 & 2/5.

39 & 2/5 : 38 & 2/5 = 197:192.

Colonna was apparently using this interval to represent both 41:40 and 39:38. I'm not sure where Colonna found 41:40 in the ancient texts - I haven't.

But 40:39 and 39:38 both appear in the Enharmonic of Eratosthenes. Colonna's 197:192 falls (not quite exactly) right between these two:

~0.46 cents below 39:38

~0.68 cents above 40:39

~1.76 cents above 41:40

After mentioning 25/24, the other small ratios are not discussed.

The only other error I found in the text was on the next page where Martin is explaining the Latin ratio terminology, and says that the proportion of the ['whole'] 'tone' with the 'grave' [whole string] is 'Sesquisettima'.

This is not correct. It means a ratio of 8:7, which has certainly been used by some musicians as a 'whole tone', but the 'standard' ratio for the 'whole tone' has always been 9:8, which would be 'Sesquiottava' in Latin.

It should also be noted in connection with this that altho Colonna specifies 7 as a factor in the numerator of two of his ratios (7:4 and 21:20), he does not use it as a denominator at all. The Latin terms always specify the denominator, and describe the numerator in relation to that.

But there are more problems in 'example 2'.

I'll just tabulate the three instances where the Latin ratio description is just slightly different from the actual numbers I calculated. They're probably just typos.

They all concern the above operation. I have indicated the corrections by capitalizing.

> 1811 & 17/53 noncupla superbipartiens quintas

should be: noncupla superTRIpartiens quintas = 9 & 3/5 : 1

> 1846 & 2/13 duocupla

should be: duoDEcupla = 12 : 1

> 1900 & 100/101 noncupla sesquiquinta

should be: nonDEcupla sesquiquinta = 19 & 1/5 : 1

Those are the obvious ones.

Then there are a few cases where the Latin ratio given by Colonna is close but not quite exactly the real ratio. I wondered about this, since he went into such detailed calculations to get his ratios in the first place.

>1951 & 12/41 quadragecupla = 40 : 1

The exact ratio is 40.061592, only about 0.154% error. OK, this one's not that far off, but still... Next instance:

> 1063 & 14/17 sesquioctava = 8 : 1

The exact ratio is 1.1363494, about 1% error either way.

Next:

> 1658 & 18/29 quadrupla superquadripartiens quintas = 4 & 4/5 : 1

The exact ratio is 4.858585..., a 1.2% or 1.22% error, depending on which ratio you assume to be correct.

Now this is still not that much of a difference, but as I said, if he's going thru the trouble of dividing the string into (2000 * 29) = 58,000 parts(!) to get his ratio, why is he fudging the terminology like this?

Finally, I'm absolutely stupefied by this one:

> 1937 & 59/83 vigecupla, bipartians tertias = 20 & 2/3 : 1

The exact ratio is 31.108317, 33.6% or 50.5% error, depending on which ratio you assume is correct. So for either case, this one's not even in the ballpark.

I thought perhaps the Latin terminology is wrong, but the only thing there that's even close is that 'vigeculpa' (20) could easily be a typo for 'trigeculpa' (30). But 'bipartians tertias' doesn't describe the remainder of 1.1 at all (and it doesn't say 'superbipartians', as it should).

I also wonder why some of the notes in this table are 'out of order'. It goes generally from highest at the top to lowest at the bottom, but 5 of the notes appear higher up in the chart than where their string-lengths would lead one to expect them:

```
1538 &  6/13   [=    10/13    ]
1739 &  3/23   [=    20/23    ]
1860 & 20/43   [=    40/43    ]
1937 & 59/83   [= 16083/16600 ]
1963 & 31/163  [=   160/163   ]

```

I can find no reason for this.

I'm going to use my little 'micro.CAL' program to put in a sequence of the Stella(?) piece at the end of this article, which uses all three genera and modulates thru all 31 keys.

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

Colonna specified two 'shades' each of the chromatic and enharmonic genera, thus 5 different genera in all. The difference was in the lowest moveable note of the tetrachord, which could be either 'intense' (higher) or 'soft' (lower), in the same manner as Ptolemy's tetrachords, except that Ptolemy had 3 shades of the Diatonic and only 1 of the Enharmonic.

```
DIATONIC        CHROMATIC            ENHARMONIC
INTENSE      SOFT     INTENSE    SOFT

4:3           4:3        4:3        4:3       4:3

6:5

9:8        9:8

17:16        17:16                 17:16     17:16

25:24      25:24
39:38 or 41:40

1:1           1:1        1:1        1:1       1:1

```

-monzo

Below is John Chalmers's response to me.

Joe-

...

I spent some time with Colonna and the best I can do is guess that what is meant is 32/31. 31/32 x 2000 = 1937.5 and

``` 2000-1937.5
------------ = 31
1937.5

```

However, in Colonna's theory, 32/31 would have to be a primary number. On the other hand, it was used in Didymos's enharmonic 32/31 x 32/30 x 5/4, with which Colonna would have been familiar.

Another guess is that it has become confused with 1927 & 59/83. Colonna's ratios are derived from primary ratios of the form b/a such as 1/1, 3/2, 5/4, 5/3 etc. by the formula ((2^k)*b +a)/(2^k)*b according to J. M. Barbour. K can range from 0 to at least 4.

The interval names like 4th, 6th, m3rd, etc in Martin's chart are the generating intervals, I think. From 5/3 we get (5+3)/5, (10+3)/10,(20+3)/20 !, (40+3)/40, and (80+3)/80. 80/83 of 2000 is 1927 & 59/83. The remainder is 162 & 24/83.

From 5/4, one gets 23/20, but 23/20 corresponds to 1739 3/23, which is also in the table tuning, BTW.

So, I think the numbers got mixed up, but 32/31 is what is meant. Vicentino also used this ratio to demonstrate the Greek enharmonic genus.

--John

Updated:

2001.03.03