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[note from Monzo: readers will find it helpful to consult the diagrams in my definitions of mutation and perfect immutable system in connection with Aron's discussion.]

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Division of the monochord by tones, and natural and accidental semitones. Cap. XXXX.

Having reached the end of the promised work, it happened to me, like sometimes it happens to sailors, who, having collected the sails to enter the port: suddenly struck by a different wind are compelled to turn back, and talk, while turning here and there.

Similarly, while wanting to rest of the concluded undertaking, nor anything else being needed, than thanking God, and apologize with the readers, if somewhere (as a man) I was mistaken, I have been attacked by a new thought, that after having shown the things that are the most pertinent on the practical music with the theoretical, without which almost nothing could be done: it will be a good thing showing the division of the monochord per tones, and natural and accidental semitones, along with the temperament, and manner for tuning it, so making me welcome to all players, and scholars of the instrument, that are not expert: considering that others, that have discussed it before, wrote about it in such a very erudite way, that still cannot be intended by the well learned scholars then with a great difficulty.

Then, for the consideration both of myself, as (as Livy wrote) the restless mind is fed by the action, and of each kind person that is being pleased by it, I say so, that in the organic instrument, according to the usual order, a number of 29 natural tones are found, called by the universal use white keys, and a number of 18 accidentals called black keys, or semitones.

We will divide both accidentals, and naturals through their order key by key, showing each interval from the one to the next. And note, that it has been needed that this instrument would exceed the number of twenty ordinary strings, and arranged in [according to] our hand, so that players with more easiness can practise, and accommodate to their intentions, as should those tones, or keys be lacking, perhaps they would be fairly disturbed and hampered.

Then we will say that the first key, or step of the said instrument is located below the Gamma ut by the interval of one tone, where it will be called by the syllable, or word fa so it is concordant with that of F fa ut, that is one diapason apart

.... But from our first string called Gamma ut up to the one that Greeks call proslambanomenos, that is A re by us: there will be only the distance, and interval of a tone similar to the previous called tone.

Similarly it will be again from proslambamenos to hypate hypaton, that is A re, and [natural sign] ... mi: but from [natural sign] ... mi, to C fa ut, or hypate hypaton, and parhypate hypaton ... it will be ... a minor semitone, .... And since from proslambanomenos to hypate hypaton a black key appears in the middle, be acquainted that that key is at a distance of a minor semitone from the white key set at proslambanomenos, and a major semitone from that of hypate hypaton, as you can clearly see by the appearing fifth from [natural sign] ... mi to F fa ut, that is from hypate hypaton to parhypate meson, that is made with two tones, and two minor semitones: so that it remains diminished of one apotome called major semitone, then settling the mentioned black key has been needed for such integration , and perfection, so that that diapente, or fifth be sweet, and pleasant to the hearing.

From the fourth step called parhypate hypaton, or C fa ut to lichanos hypaton called D sol re there will be an interval of ... tone.

Between those parhypate hypaton, and lichanos hypaton the black key in the middle is born, by which parhypate hypaton remains divided from lichanos hypaton in two semitones one major, and one minor, of which the major will be from C fa ut to the black key, and from the black key to the following white, that is D sol re; the minor semitone will stand, as it is seen from A re, and C fa ut, that is a minor third: and since it is need to integrate, and augment to the quantity ... major it will be needed to increase one major semitone, that will properly be that distance, that is from C fa ut to the black key.

From lichanos hypaton to hypate meson, that is D sol re, and E la mi ( as in the past you have intended) the tone drops, and from the mentioned D sol re to the black key a semitone drops, rather bigger then its need, by which it could not be given favour nor augmentation to the minor third, that drops from [natural sign] ... mi to D sol re, wanting it be major, as is convenient for all other divided tones, ... as the experience of the one, and the other shows. Then, wanting to make the major third in the mentioned place, it is needed that that semitone, or black key above the string, or note D sol re, be rather lowered, from which lowering the fifth, and octaves corresponding to the said semitone, or black key, are hampered, from which a big inconvenience would derive, bigger than before.

This ... is placed in such positions, that is between D sol re, and E la mi, and in the octaves, and fifteenths, because the acute, and superacute correspond to the low, that is its first origin, and these drawbacks happen in those places for no other reason, then because of the organists, who want accommodate the C fa ut with its minor third, rather then the [natural sign] mi wit the major, because [natural sign] ... mi is not much used, and then it is impossible, that the above mentioned minor third could be made major, without cutting the mentioned black key, so that a part of it renders the tone lower the previous.

And so the third of that major semitone, that lacks to it, will be helped, and restored. And talking with the black key at the place of E la mi, or hypate meson you will have the quantity of the major semitone, that makes perfect the fifth , or diapente, set between the mentioned E la mi and the acute B fa, diminished and imperfect.

From hypate meson to parhypate meson, that is E la mi, and F fa ut, naturally the minor semitone drops, to which no further division of semitones, or black keys is convenient.

But from parhypate meson to lichanos meson called F fa ut, and G sol re ut naturally drops the tone divided in two semitones by the black key set in the middle, that will be distant by a major semitone from F fa ut, by which the minor sixth placed from A re to low F fa ut, wanting to find its octave, is increased in major, but from that black key to the white placed in lichanos meson, that is G sol re ut: the minor semitone drops, as it is seen from lichanos hypaton to lichanos meson saying ut in lichanos hypaton called D sol re, until the fa of lichanos meson called G sol re ut.

From G sol re ut to A la mi re called mese similarly drops a tone, in which again a semitone or black key is found, from which black key to the position of lichanos meson, that is G sol re ut, the major semitone drops, because from hypate hypaton to lichanos meson, that are [natural sign] ... mi, and G sol re ut: there is a distance of a major sixth, and by adding the mentioned black key it is augmented by the present major semitone, and the minor sixth becomes major.

From that semitone, or black key until A la mi re called mese the quantity of the minor semitone drops, as you will see easily saying ut in E la mi low, and the re at the fourth black key, the note mi at the fifth black key, that are two tones: and the minor semitone follows that fifth black key until the following white, that is A la mi re. This composition, and remark generates a diatessaron [fourth].

From mese, and trite synemmenon [paramese], that are A la mi re, and [natural sign] ... mi high: the natural tone drops, divided by the black key, which black key will be distant from mese, that is A la mi re by one minor semitone, but from it black key until the place of the mentioned [natural sign] mi high the major semitone drops, as clearly all authors show there, especially when they speak of the mutations.

From paramese to trite diezeugmenon, that are [natural sign] ... mi high and C sol fa ut, the minor semitone drops naturally. In which intervals there is nothing in the middle.

From trite diezeugmenon to paranete diezeugmenon, that is C sol fa ut, and D la sol re, a natural tone drops, divided by the black key, which black key is above C sol fa ut for the quantity of the major semitone, and from it black key to the following white ... called paranete diezeugmenon the minor semitone drops.

The mentioned major semitone is placed there, because of the minor sixth made from the E la mi to C sol fa ut, aiming then to make it ... major, it is necessary to touch the above mentioned key, and (as is ... said) from it black key to that of D la sol re there is nothing else then the minor semitone. Because making the third diatessaron, that is ut fa: you will find nothing else then a minor semitone in the last interval saying ut in A la mi re, re in B fa [natural sign] ... mi, mi at the seventh black key, that are two tones, the following semitone will be from the mentioned seventh black key to the following white, that is D la sol re.

From Paranete diezeugmenon to mete diezeugmenon, that is D la sol re, and E la mi, a natural tone drops divided by the semitone, or black key, which black key will be distant in height, as much as (if you well remember) that was, which was from [natural sign] mi, to D sol re because it is clearly seen saying the f of B fa [natural sign] mi set at the sixth black key until the eighth black key a fourth will be,of which is fa mi minor semitone from the mentioned eighth black key to the previous white, and from the following white will be the quantity of the major semitone.

As in the declaration of the black keys alone, compared one another, it will be understood.From nete diezeugmenon to trite hyperboleon called E la mi, and F fa ut the minor semitone drops, with no other division between them.

But from trite hyperboleon called F fa ut to paranete hyperboleon called second G sol re ut the interval of a tone drops naturally, which positions are divided by the semitone, or black key, which semitone will be distant from trite hyperboleon by the quantity of the major semitone, as for the descending sixth from mese called first A la mi re to that position called trite hyperboleon, that is F fa ut, which sixth naturally appears minor then being in need, by the mentioned black key it is augmented in major, which augmentation is useful as well to the descending minor tenth from lichanos hypaton to trite hyperboleon, that is D sol re, and F fa ut high.

Then, being the major semitone in that place, it necessarily remains its minor semitone from the black key to the following white called paranete hyperboleon that is G sol re ut.

From paranete hyperboleon to nete hyperboleon that is G sol re ut, and A la mi re the tone naturally drops, in which the semitone or black key appears in the middle, and it is distant in its upper interval from paranete hyperboleon by a major semitone which semitone augments the descending sixth from [natural sign] ... mi high to second G sol re ut, and the third placed from nete diezeugmenon called E la mi high to paranete hyperboleon called second G sol re ut, and so the descending tenth from hypate meson to paranete hyperboleon ... called E la mi low and G sol re ut high, then it follows that the minor semitone will be from nete hyperboleon called A la mi re to the black key ... above ordered, because should one want to... make the diapason from the sixth and being it in that minor place, in order to ... comply with the rule, it is convenient that the mentioned black key be diminished.

From nete hyperboleon that is A la mi re to third [natural sign] mi the tone naturally drops, in which the semitone or ... black key originates, from which ... black key to nete hyperboleon, the minor semitone drops, and in ascending the major drops, because as, according to the accidental talking, the imperfect fifth is found from the mentioned [natural sign] mi ... above mentioned to the fa that is above at the position of E la, which [fifth] being wished entire and perfect, it must be integrated by that semitone or black key mentioned above, as it was for the preceding [natural sign] ... mi high.

Continuing further on to the position of C sol fa, without any doubt the minor semitone naturally drops from [natural sign] mi superacute to the fa of C sol fa between which nothing in the middle can be found.

From C sol fa to D la sol similarly the interval of tone drops, which is seen divided by the semitone or black key, about which we will say that from C sol fa to the following black semitone there is the quantity of one major semitone, and from the following white key that of the minor, it is shown that the mentioned semitone is major by the minor sixth that originates from nete diezeugmenon called E la mi, to the position of C sol fa, and the same for the minor third made from nete hyperboleon that is A la mi re to C sol fa.

The black key is required, because its descending tenth needs it as well, as without it [the tenth] would remain minor.

From D la sol to the E la last position the tone drops, divided ... by its black key, that is distant from the mentioned E la , by the quantity of the above mentioned semitone, dropping between lichanos hypaton and hypate meson, and from paranete diezeugmenon to nete diezeugmenon, and from it black key to the position called E la the major semitone drops, and that position can correctly be called as the corresponding in the octave below: that is called E la mi ... repeated, because passing above E la, each position, and place, ought to be repeated with that demonstration, order and mode, as it had been made for the first ... low ones, and those that were in the acute, and superacute, you will find the same in the following, continuing the order again above E la mi, like in the hand it is found ordered.

Then we will say that from E la to the following key nothing else follows, than the minor semitone, as ... clearly it is seen without any other division of semitone, or black key.

But from the following repeated F fa ut until the G sol re ut a tone drops, that again remains divided by the black key, which black key is higher above the preceding white key by an interval of a quantity of a major semitone, as is shown in those positions above mentioned, and as the experience more clearly shows.

And the opposite happens on the upper side, as from it black key to the above white there is nothing else then the minor semitone.

From the third replicated G sol re ut to A la mi re there is nothing else, then the quantity of a tone, in which the black semitone shows, being distant from the mentioned G sol re ut by a major semitone, as for the low has been explained, and on the upper side of the mentioned black semitone to the white key the minor semitone drops.

From A la mi re to repeated [natural sign] mi without any doubt the quantity of a tone drops, but from the mentioned A la mi re to the black key that is in the middle, the minor semitone drops, and from the following ... the major.

From the mentioned [natural sign] mi to the following key nothing else drops, then the minor semitone, but from the following replicated place called C sol fa ut to D la sol re nothing else drops, then the quantity of a tone divided by the black semitone, which black semitone remains distant from the white placed behind, for the quantity of the major semitone, and from the following white it remains minor.

Then continuing from the replicated D la sol re to the following key E la mi, nothing else is there, then the quantity of one tone, again divided by the black semitone, which semitone, is higher then the first white key placed behind it by the quantity of the semitone, or interval as it is found in the octaves below in the same position.

But from the following white key to the mentioned black there is only the distance of the minor [major] semitone.

This way in the following the same is found, considering it in an universal manner, because there are some other [instruments], having a higher number of notes, called double instruments, which ratio, and meaning of the added strings, or keys will be understood by the composition, and order shown above.

Because according to the order that the positions described above have had, similarly the strings that were added, and increased in the low part, or below will correspond to the acute parts.

But those, that in the upper part will be put, will be concordant with those, that below have been said.

But now you will understand, that from the first string, or note, or semitone, or black key of the instrument ordered by us to the second following black, there will be the distance of one tone, and major semitone, that make the composition of one semiditone, and comma. because from it black semitone to the following second ... mentioned accidentals, in the middle remains the natural minor semitone.

As well as from the third, fourth, sixth, and seventh, eighth, and ninth, eleventh, and twelfth, thirteenth, and fourteenth, sixteenth and seventeenth you will find the same interval.

Therefore from the mentioned second black semitone to the following third you will have the quantity of the tone rather increased.

From the mentioned third to the needed fourth accidental the distance of the semiditone is almost in excess, and from the fourth to the fifth the tone is seen.

From the fifth to the sixth two minor semitones drop, but from the sixth to the following seventh one tone with a major semitone drops.

From the seventh to the eighth the tone rather increased.

But from the eighth to the ninth an interval of trihemitone corresponds.

The ninth, and tenth are distant by an interval of the major, and minor semitone, as it was for the fourth and fifth black semitone.

From the tenth to the eleventh there is the distance, that was from the fifth to the sixth for two minor semitones.

The eleventh, and twelfth will have the distance of the tone, and major semitone, again as similarly lies between the sixth, and the seventh.

From the twelfth to the thirteenth there is the one, that could be seen only between the seventh and the eighth.

And going on from the thirteenth to the following fourteenth three semitones will be found, as it was for the eighth and the ninth key.

The fourteenth, and fifteenth black keys correspond to the same, that was between the ninth key, and the tenth, that was the tone.

But from the fifteenth, and sixteenth there is the distance, that was between the tenth, and eleventh of two minor semitones.

Then ahead moving from the sixteenth to the seventeenth you will have the quantity, that was between the eleventh, and the twelfth, and also sixth, and seventh, that is the tone, and major semitone.

And from the seventeenth to the eighteenth last semitone, there will be only that, that originated from the twelfth and thirteenth, seventh and eighth.

That was the quantity of the tone rather major. from which, compared each other, few sweetness could be heard, except from the second, and fifth the consonance of fifth drops, and similarly from the third to the sixth is comprised, again from the fourth, and the seventh the fifth originates: from the seventh, and tenth another fifth.

From the first to the sixth the octave originates, and , and from the second to the seventh you will find the same.

The same will be from the third to the eighth, as well as from the fourth to the ninth only one octave is seen to stand.

From the eighth to the eleventh the fifth originates, as from tenth to the twelfth is seen.

But from the eighth to the thirteenth there is an octave, as is seen from the ninth to the fourteenth.

The same will be from the tenth to the fifteenth again, and the same from the eleventh to the sixteenth you'll have.

From the twelfth to the fifteenth the fifth you will have, as from the thirteenth to the seventeenth could be seen.

From the twelfth to the seventeenth a diapason as well as from the thirteenth, and the last is seen.

From the fourteenth, and the seventeenth the fifth, as the fifteenth to the last demonstrates.

And this be sufficient to you for the declaration of the monochord.

On the participation [temperament] and method for tuning the instrument. Cap. XXXXI

It follows that with that fluency, that will be possible to me: briefly I explain, all that ... is necessary ... to the player about the temperament, and union of the notes, because it happens to many, that with no, or very few knowledge, and even less practice are able to that exercise.

Then, you will have care, that we will make our tuning, and temperament in three parts, then if you, who don't know how to do it: want to tune and temperate your instrument, first you must consider the string, or position called C fa ut with that intonation [pitch], that you like, and when you are ready take the octave above C fa ut, and make it such that it is well united all the time.

Then the major third above, that is E la mi: wants to be sonorous, and just, that is as united as possible, and this done take the fifth in the middle, that is G sol re ut, and make it rather a bit lacking, so you will go on to next fifth above, that is D la sol re of similar tuning, and same kind, as the mentioned G sol re ut was.

Then tune D sol re octave to D la sol re , and continuing take its fifth above D sol re established in the place of A la mi re, that ought to be as lacking from E la mi, as from D sol re, that is, it should be as equal from the one, as from the other, that are all fifths, which aren't drawn to the perfection, being lacking on the upper edge.

Then the ascending fifths from the mentioned C fa ut, D sol re, and E la mi, that are G sol re ut, A la mi re, B fa [natural sign] mi: all descend, and lack of their perfection.

For the second order, and mode, again you need, on the string of C sol fa ut, that is united, and just [with the C fa ut], to tune F fa ut descending fifth, that ought to be contrary to the others mentioned above, that is, that it has to be tempered and raised so much, that it exceeds the perfect of a certain amount, and from this comes the temperament and tuning just and good.

Because of this temperament the thirds, and sixths remain blunted, or diminished.

And this way you will tune the semitone of B fa [natural sign] mi below F fa ut, and that of E la mi below B fa [natural sign] ... mi, that is a fifth with that same order, and mode, by which you tuned F fa ut with C sol fa ut.

The third and last mode, you will have care of tuning the major semitones between their thirds, as is the semitone of C fa ut touching A re: you will tune it with E la mi fifth, as much as it remains in the middle major third with A re, and minor with E la mi.

And so from D sol re to A la mi re the third in the middle is the semitone of F fa ut, that is the same, as the previous, and so continuing until the end of your instrument you will tune each octave, from which consideration the true temperament of the notes comes.

Updated:

2003.03.17 -- page created

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