(1)    A treatise on alterations brought out by master Johannes Tinctoris, licenced
      in laws and chaplain to the King of Greater Sicily.

      To Guillelmus Guinandi, most holy interpreter of the law and most cultivated
(5)   devotee of the muses, head chaplain to the Most Serene Duke of Milan, Johannes
      Tinctoris, lowliest among students of law and the mathematical arts, pays
      undying tribute and honour.

      Not long ago, esteemed Sir, a flitting rumour reached my ears that a certain singer
      under your jurisdiction had stated openly that I had made an error on some point
      of musical alteration, on the grounds that he had found, in a copy in his possession
(10)  of my mass setting based on 'Nos amis' that I brought out some time ago now,
      under perfect tempus between two breves, two semibreves of which the last is not
      altered, in this manner:

      [Example 1]

      In this he shows himself up as somewhat naive, failing to observe that no alteration
      should take place where there is no deficiency in number. Although we should pardon
      his naivety – for this, indeed, is what the law teaches – I have considered,
      nevertheless, that I should not leave this imputation of error uncleared, lest I give
(15)  an impression of agreeing with it through my silence. For this reason I have brought
      out the present short work; and I have placed in it whatever I feel necessary on the
      subject of the alterations of notes, so that, when you have done me the honour of
      reading it thoroughly, you may see that I have been unfairly brought to book, and
      so that, by striving for truth in the face of the accuser – as befits a gentleman of
      strong character – you may watch over me and give me your protection.

(20)   Chapter 1
      On what alteration is
      Alteration is the doubling of the intrinsic value of some note. For indeed to be
      altered, as concerns a note, means quite simply to be made into another of the
      same value, in addition to the note's own intrinsic value. And so, dealing first in
      general terms with the subject of the alteration of notes, I have decided to set out
      eight general rules.

       First General Rule
(25)  The first general rule is that wherever two notes of the same type, lying under
      ternary quantity, are found by themselves, the last of these is altered. Moreover,
      two such notes are said to be found 'by themselves' whevever they do not have any
      other note of the same kind set in continuous juxtaposition with them, or one
      preceding them in syncopation which can be counted in with them. For if some other
      note of the same kind precedes them, either in continuous juxtaposition or through
      syncopation, which can be counted in with them, then in that case all three will be
(30)  counted together; and since if the cause ceases, so does the effect cease, then if the
      imperfection in number ceases, so does the alteration, as is shown here:

      [Example 2]

      Moreover, there are two circumstances in which two notes lying under ternary
      quantity do not have another like note preceding them which can be counted in
      with them: firstly, when there is actually no such note preceding; and secondly,
      when some such note does precede, but it is shown through coloration that it is
(35)  displaced from its own grouping to a different location, or else separated from those
      other notes by a dot, as here:

      [Example 3]


       Second General Rule
      The second general rule is that it does not matter whether two notes found by
      themselves, of which the last is due to be altered, are placed directly next to one
      another or in syncopation, since wherever perfection is necessitated in the latter
      case, just as in the former, alteration occurs as a consequence, as here:

      [Example 4]


(40)   Third General Rule
      The third general rule is that any note which is altered is by necessity altered
      before its next higher-value note, such as a long before a maxima, a breve before
      a long, a semibreve before a breve, and a minim before a semibreve. The reason
      for this is twofold: firstly, because in place of the note to be altered one of the next
(45)  higher value cannot be put, since like before like is not imperfected; hence that
      which is worth only two through alteration would necessarily be worth three, and
      so in order to avoid this a second note of like kind to the first is alterable. The
      second reason is that two smaller notes of the same kind placed before a note of
      next-but-one, or next-but-two, or next-but-three higher value, such as two breves
      before a maxima, two semibreves before a long, or two minims before a breve; or
(50)  else two semibreves before a maxima, or two minims before a long; or else two
      minims before a maxima, can imperfect that larger note, and thus the number
      stands in a state of perfection without alteration, as is shown here:

      [Example 5]


       Fourth General Rule
      The fourth general rule is that, although two smaller notes found by themselves
(55)  before a next-larger note may be preceded by another next-but-one, or next-but-
      two, or next-but-three larger note, nevertheless, since imperfection – being, as it
      were, hateful and so to be avoided more than alteration – this largest note will
      not be imperfected by the smaller notes. Rather, the last of these smaller notes
      will be altered, unless they are either separated from one another, or both from
      the next-larger note, by a dot, as here:

      [Example 6]


(60)   Fifth General Rule
      The fifth general rule is that any note which is available to be altered must
      necessarily be intact as regards form. Its partner note, on the other hand, can be
      divided into its constituent parts, as here:

      [Example 7]


       Sixth General Rule
      The sixth general rule is that, although rests taking the place of the notes
(65)  partnering those which need to be altered, or constituent parts thereof, have the
      power to induce alteration, no rest can itself be altered, as here:

      [Example 8]


      Furthermore, alteration occurs before perfect rests in just the same way as before
      the notes that they represent. Hence it happens that in perfect minor modus a
      breve is altered before a long rest; in perfect tempus a semibreve is altered
(70)  before a breve rest; and in major prolation a minim is altered before a semibreve
      rest, as is shown here:

      [Example 9]


       Seventh General Rule
      The seventh general rule is that a ligature does not create alteration. So, if three
      notes are found before a next-larger note, of which the last two are ligated – or
      indeed ligated to that next-larger note – even if some other next-larger, or next-
(75)  but-one larger, or next-but-two larger note precedes the smaller notes which
      could be imperfected by the first of these, nevertheless, since we should avoid
      imperfection and alteration as far as possible, on account of the impropriety of
      each, the last of the three smaller notes is not altered. Rather, it is counted in
      with the first two of them, unless the first is separated from the last two by a dot,
      as is shown here:

      [Example 10]


      Nevertheless, this manner of ligation, unless it occurs on account of word underlay,
(80)  is to be avoided at all costs. In truth, it is much more convenient if either none
      of three such notes, or all of them, or the first and second, are ligated, as is shown

      [Example 11]


       Eighth General Rule
      The eighth general rule is that any note lying under ternary quantity is alterable.
      So, since the long lies under ternary quantity in perfect major modus, and the
(85)  breve in perfect minor modus, and the semibreve in perfect tempus, and the
      minim in major prolation, these notes are therefore available to be altered in
      these mensurations. So that this may be understood more clearly, I propose to
      deal now with the alteration of each of these notes individually.

       Chapter 2
      On the alteration of the long
(90)  If two longs are found by themselves in front of a maxima in perfect major modus,
      then the last of these is altered, as is shown here:

      [Example 12]


       On the alteration of the breve
      If two breves are found by themselves in front of a long or its rest in perfect
      minor modus, then the last of these is altered, as here:

      [Example 13]


(95)   On the alteration of the semibreve
      If two semibreves are found by themselves in front of a breve or its rest in
      perfect tempus, then the last of these is altered, as here:

      [Example 14]


       On the alteration of the minim
      If two minims are found by themselves in front of a semibreve or its rest in
(100) major prolation, then the last of these is altered, as here:

      [Example 15]


       And so it certainly seems to me that I have dealt sufficiently with the alterations
      of notes, both in general terms and on an individual basis. If, nevertheless, I
      have omitted to mention any necessary points on the subject, or if any of my
      teachings are incorrect, I pray that my accuser take the trouble to add to the
      former whatever is missing, or correct in the latter whatever is false. For only
(105) then will I regard myself as worthy of the charge, and declare him, here and in
      all places, to be the most authoritative master.