Following a significant amount of research I have started the task of typing up the manuscript for the Fellowship of the London College of Music; Target 25k - 30k. This is the first part of the workings for chapter 4.

According to Benito V. Rivera in Theory Ruled by Practise Gioseffo Zarlino stated 'this' and 'that' in part one of l’istitutioni harmoniche .

Well I love old books, old languages, translations and I have a new fascination with language. Added to this I cannot find an English version anywhere.

So regarding chapters 35 to 39 of part one located on IMSLP and the UNT Digital Library I decided to go bac to the primary source and, with the help of Google Translate, see what he actually said.

& so to page 45, chapter 5

Potemo hora dire, che sottrata una sesquiterza da una sesquialtera resta sesquiottaua, & questa esser la diffferenze, che si ritroua tra i'una & l'altra; & esser qualla quantita, per la quale la maggiore supera la minore er questa de qualla e superata. Et che cosi sia il vero si puo prouare: imperoche sommanda infieme nel mudo mostrato la sesquiterza con la sesquiottana, haueremo da tal somma la sesquialtera, che su qualla proportione, che superaua la sesquiterza di una sesquiottaua: Onde da questo potemo ancora vedere, che il sommare delle proportioni e la proua del sottrare; & per il contrario il sottrare la proua del sommare.

We can now say that subtracting a sesquiterza from a sesquialtera remains a sesquioctave, and this is the difference between one and the other; & be that quantity by which the greater exceeds the lesser and this by which it is exceeded. And we can prove that this is true: since by adding the sesquiterza to the sesquioctane in the way shown, we will obtain from this sum the sesquialtera, which on any proportion that exceeds the sesquiterza by a sesquioctave. the addition of proportions and the proof of subtracting; & conversely, subtracting the proof of adding.

## del partire o dividere le proportione; & quello che si proportionalita

### of starting or dividing the proportions; & what is proportionality

si debbe auertire, che per la quarta operatione io non intendo altro, che la diuisione o partimento de qualunque proportione, che si sa la cvllocatione di alcun ritrouato numero, tra lisuoi estremi & e monimato Diuisore: percioche divide quella proportionatamente in due parti; la qual diuisione li mathematici chiamano proportionalita, o progressione.

It must be admitted that by the fourth operation I mean nothing other than the division or partition of any proportion, which is known as the location and location of any number found, between its extremes and the named Divisor: because it divides that proportionately into two parts; which division mathematicians call proportionality, or progression.

Onde mi e paruto offer conueniente dichiarare primieramente quello, che importi questo nome proportionalita, & dipoi venire alle operationioni. La Proportionaliiita adunque, secondo la mente di Euclide, e similitudine delle proportioni, che si ritroua almeno nel mezo di termini, che contengono due proportioni.

Hence it seemed to me convenient to first declare what this name implies proportionality, & then come to the operations. Proportionality therefore, according to Euclid's mind, is the similitude of proportions, which is found at least in the middle of terms which contain two proportions.

Et quantunque appresso li mathematici (come dimostra Boetio) le proportionalita siano Diece; ouero (secondo la mente di Giordano) V ndeci; nondimeno le tre prime, che sono le piu famose, & approuate da gli antichi Filososi Pithagora, Platone & Aristotele, sono confiderate & abbraccate dal musico, come quelle che famno piu al suo propofito che le altre. Di queste la prima e detta Arithmetica, le secondoa Geometrica & la terza Harmonica. Et volendo io ragionaire elcuna cosa di ciascuna di elle prima vederemo quel che fia ciacuma separatamenta.

And although near the mathematicians (as Boethius demonstrates) the proportionalities are Ten; a burden (according to Giordano's mind) ? decide; nevertheless the first three, which are the most famous, & approved by the ancient Philosophers Pythagoras, Plato & Aristotle, are trusted & embraced by the musician, as those that do more for his purpose than the others. Of these the first is called Arithmetica, the second Geometric & the third Harmonica. And if I want to discuss one thing about each of them, we will first see what each of them will do separately.

Incominciando adunque dalla prima dico che la dimifione, o proportionalita Arithmetica e quella, la qual tra due termini di qualunque proportione hauera vn mezano termine accommodato in tal modo, che essendo le differenze de i suoi termini equali, inequali saranno le sue proportioni. Per il contrario, dico che la diuisione, o proportionaliya Geometrica e qualla, le cui proportioni, per virtu del nominato mezano termine, essendo equali, inequali saranno le sue differeze.

Starting then from the first, I say that the dimension, or arithmetic proportionality, is that which between two terms of any proportion has a middle term accommodated in such a way that, being the differences of its terms equal, its proportions will be unequal. On the contrary, I say that the geometric division or proportionality is the one whose proportions, by virtue of the named middle term, being equal, its differences will be unequal.

### The complete paragraph for page 45 Cap. 35 is as follows.

We can now say that subtracting a sesquiterza from a sesquialtera remains a sesquioctave, and this is the difference between one and the other; & be that quantity by which the greater exceeds the lesser and this by which it is exceeded. And we can prove that this is true: since by adding the sesquiterza to the sesquioctane in the way shown, we will obtain from this sum the sesquialtera, which on any proportion that exceeds the sesquiterza by a sesquioctave. the addition of proportions and the proof of subtracting; & conversely, subtracting the proof of adding.

### of starting or dividing the proportions; & what is proportionality

It must be admitted that by the fourth operation I mean nothing other than the division or partition of any proportion, which is known as the location and location of any number found, between its extremes and the named Divisor: because it divides that proportionately into two parts; which division mathematicians call proportionality, or progression. Hence it seemed to me convenient to first declare what this name implies proportionality, & then come to the operations. Proportionality therefore, according to Euclid's mind, is the similitude of proportions, which is found at least in the middle of terms which contain two proportions. And although near the mathematicians (as Boethius demonstrates) the proportionalities are Ten; a burden (according to Giordano's mind) ? decide; nevertheless the first three, which are the most famous, & approved by the ancient Philosophers Pythagoras, Plato & Aristotle, are trusted & embraced by the musician, as those that do more for his purpose than the others. Of these the first is called Arithmetica, the second Geometric & the third Harmonica. And if I want to discuss one thing about each of them, we will first see what each of them will do separately. Starting then from the first, I say that the dimension, or arithmetic proportionality, is that which between two terms of any proportion has a middle term accommodated in such a way that, being the differences of its terms equal, its proportions will be unequal. On the contrary, I say that the geometric division or proportionality is the one whose proportions, by virtue of the named middle term, being equal, its differences will be unequal.

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