[1]    [An exposition of the hand according to Master Johannes Tinctoris,
      licentiate in laws and chaplain to the King of Sicily.]

       Prologue
      To Johannes de Lotinis, a youth consummately adorned with the finest character
[5]   and numerous noble skills, Johannes Tinctoris, least among music teachers, sends
      fraternal good wishes.

       The first thing, O youth of the most shining talent, that a well-organized instructor
      in any skill delivers to young men keen to learn, is the milk – that is, the sweetness –
      of certain straightforward principles, lest, if he should offer them the gall – that is,
      the bitterness – of difficulty right from the beginning, he put them off through loss
[10]  of confidence. Thus it was that a learned musician from Italy, a man of considerable
      and lofty abilities, with great wisdom put together the principle of the hand, to provide
      a starting-point in the form of a simple set of instructions, handed down, as it were,
      for the use of anyone intending to apply himself to the art of sound. Encouraged by
      these same motives, I have resolved to offer a simple explanation of this hand at the
      outset, hoping to deal with more difficult matters at a later stage. And I have decided
      that this exposition should itself be dedicated to your most gracious and noble name,
[15]  not as a singer ignorant of his own hand, since I know you to be highly proficient
      in this skill – and there is no more dreadful insult with which to charge a musician
      than the claim that he does not know his hand – but as my dearest friend and colleague,
      beseeching you most earnestly that you might deign to accept this humble work as a
      gift, and that you might read it thoroughly in the same spirit of goodwill as that in which
      I offer it to you for your studies.

[20]   Chapter 1: On the definition of the hand and its distinguishing features
      The hand is a concise and useful teaching method, demonstrating comprehensively the
      qualities of musical pitches. In this context, moreover, it is called the hand as from the
      container rather than the content, for every hand – the outermost recognized member of
      the human body, according to the physicians, located on the forearm – contains that
      teaching in the tips and joints of its fingers. For indeed on this bodily hand there are
[25]  five digits, that is to say the thumb; index finger; middle, which is commonly called the
      large finger; medical [ring] finger; and the ear finger, commonly known as the little
      finger. Of these the first, that is, the thumb, has one tip and two joints; each of the others,
      however, has one tip and three joints. Since there are four of these latter, and since four
      times four make sixteen, together with the three previously mentioned this makes a total
[30]  of nineteen. These nineteen, sharing an equal status, are ascribed to nineteen musical
      positions through intrinsic visual association; but the final joint of the middle finger
      is assigned to the final position, which is the twentieth, through extrinsic relationship, as
      will become clearer below. And although this teaching method can be set up using either
      hand, it is nevertheless universal standard practice to use the left hand, because it is
      more convenient to indicate the musical positions on this left hand with the index finger
[35]  of the right. Having said this, there are some who find it most convenient to indicate
      the positions on the left thumb with the index finger of the same, and the positions on the
      remaining fingers similarly with the thumb of the same. As a result, they use only the
      one hand, that is, the left, in this particular method of instruction.

       Furthermore, this teaching hand is also known by another name, the gamma, from this
      letter gamma which is called gamma by the Greeks. And this for good reason, since naming
      takes place after that which is the more worthy, but that which precedes is seen to be
[40]  the more worthy; so, since on the hand gamma, that is G, comes first, it is proper
      that the hand be named the gamma after it. And in my opinion the creator of this
      method, wishing it to be called by this name, adopted the name of the Greek letter by
      itself, so that he could properly honour the Greeks as the greatest originators of the art
      of music, from whom the Latins received this same art.

[45]   In this system of hand-teaching, then, there are seven topics to be considered, which
      is to say: positions, clefs, pitch-syllables, properties, hexachords, mutations, and
      intervals; and I have decided to treat each of these under its own heading for ease of
      reference.

       Chapter 2: On positions
      With respect to the first topic: a position is the location of musical pitches. Furthermore,
      there are twenty such positions on our hand, which are most conveniently set out on
[50]  the tips and joints of the digits in the following way:

      The first is gamma ut on the tip of the thumb.
      The second is A re on the second joint of the thumb.
      The third is bquadrum_large mi on the first joint of the thumb.
      The fourth is C fa ut on the first joint of the index finger.
[55]  The fifth is D sol re on the first joint of the middle finger.
      The sixth is low E la mi on the first joint of the ring finger.
      The seventh is low F fa ut on the first joint of the little finger.
      The eighth is low G sol re ut on the second joint of the little finger.
      The ninth is high A la mi re on the third joint of the little finger.
[60]  The tenth is high brotundum_small fa bquadrum_small mi on the tip of the little finger.
      The eleventh is C sol fa ut on the tip of the ring finger.
      The twelfth is D la sol re on the tip of the middle finger.
      The thirteenth is high E la mi on the tip of the index finger.
      The fourteenth is high F fa ut on the third joint of the index finger.
[65]  The fifteenth is high G sol re ut on the second joint of the index finger.
      The sixteenth is highest a la mi re on the second joint of the middle finger.
      The seventeenth is highest brotundum_small fa bquadrum_small mi on the second joint of the ring finger.
      The eighteenth is c sol fa on the third joint of the ring finger.
      The nineteenth is d la sol on the third joint of the middle finger.
[70]  The twentieth is e la above the same joint, that is, the third of the middle finger,
      on the outside,

      as is shown in the following diagram:

       [Figure 1]

hand

 

       Moreover, of these twenty positions described above ten are lines and ten are spaces,
      organized alternately. A line, then, is a position produced by a straight protraction
      drawn in some colour, which in this context is more often called a rule, because it
      is ruled in a straight direction. A space is a position remaining above or below a line.
[75]  Hence there are some who call gamma ut 'on the line', A re 'in the space', and so on
      alternately with the others. But it is the greatest error to speak in these terms, since
      gamma ut is the line itself, and A re is the space itself, and so on alternately with the rest;
      they cannot, therefore, be said to be positioned 'on' the line or 'in' the space. And so
      we should say:

      gamma ut is a line
[80]  A re is a space
      bquadrum_large mi line
      C fa ut space
      D sol re line
      low E la mi space
[85]  low F fa ut line
      low G sol re ut space
      high A la mi re line
      high brotundum_small fa bquadrum_small mi space
      C sol fa ut line
[90]  D la sol re space
      high E la mi line
      high F fa ut space
      high G sol re ut line
      highest a la mi re space
[95]  highest brotundum_small fa bquadrum_small mi line
      c sol fa space
      d la sol line
      e la space

       Of these twenty positions, however, only one is the lowest, that is, gamma ut, since in that
[100] position resides the lowest pitch. There are seven low positions, that is, those contained
      within the first complete ordering of cleffing letters, which is to say from A re
      inclusive through to the first A la mi re exclusive; and these are so called because they
      contain the low pitches. There are seven high positions, that is, those contained within
      the second complete ordering of cleffing letters, which is to say from the first
[105] A la mi re inclusive through to the second exclusive; and these are called 'high' since
      their pitches are high. There are five highest positions, that is, those which are
      contained within the third, albeit incomplete, ordering of cleffing letters, which is
      to say from the second a la mi re through to e la inclusive; and these are called 'highest'
      because in them are positioned the highest pitches. But some of these positions
      without their qualifying adjectives have the same name, such as low F fa ut and high
[110] F fa ut, low G sol re ut and high G sol re ut, high A la mi re and highest a la mi re,
      high brotundum_small fa bquadrum_small mi and highest brotundum_small fa bquadrum_small mi; and so, in order that they may be generally
      distinguished by those unaware of the differences between low, high and highest, low
      E la mi, F fa ut and G sol re ut, high A la mi re, and high brotundum_small fa bquadrum_small mi are commonly
[115] known as 'bottom' notes; and again, high E la mi, F fa ut and G sol re ut, highest
      a la mi re, and highest brotundum_small fa bquadrum_small mi are commonly called 'top' notes, as is shown in
      the following diagram:

       [Figure 2]

 em_fig2_tr 

       Chapter 3: On clefs
      With respect to the second topic: a clef is the sign of a line- or space-position. For
      each one of the positions on our hand has its own clef, distinguished from the others
[120] by its name, position, or form.

       There are, then, just seven letters of the alphabet that make up clefs of this kind, which
      is to say A, B, C, D, E, F and G. Hence, since we have twenty positions, so that there
      may also be twenty clefs, these seven letters are all repeated once in order, and then
      five of them once again. The last of them, however, that is, G, albeit in a different form
      and under a different name, is placed in front of all of these, for reasons to be explained
[125] below. As a result, since twice seven plus five plus one make twenty, these seven letters
      in twenty positions, by means of the stated repetitions, make twenty clefs.

       To the first position, therefore, namely gamma ut, is assigned this letter gamma, which
      differs from the rest in both name and form, because it is Greek; and it is called the
[130] lowest clef, taken up, as it were, by the lowest position. To the second position, that is
      to say A re, is assigned A. To the third, namely bquadrum_large mi, bquadrum_large [B]. To the fourth, that is
      to say C fa ut, C. To the fifth, namely D sol re, D. To the sixth, that is to say low
      E la mi, E. To the seventh, namely low F fa ut, F. To the eighth, that is to say low
      G sol re ut, G. And these seven letters are upper-case, distinguished from each other
      by name; the clefs are termed 'low', serving, as it were, the low positions.

[135]  Next, to the ninth position, namely high A la mi re, through repetition is allocated
      A. To the tenth, that is to say high brotundum_small fa bquadrum_small mi, is allocated brotundum_small bquadrum_small [B], in two-fold
      form on account of the two-fold property of the pitches occurring together in it. To the
      eleventh, namely C sol fa ut, C. To the twelfth, that is to say D la sol re, D. To the
      thirteenth, namely high E la mi, E. To the fourteenth, that is to say high F fa ut, F.
      To the fifteenth, namely high G sol re ut, G. And these seven letters repeated for the
[140] first time are also distinguished from each other by name, but distinguished from the
      seven aforementioned by position and not form, since they are upper-case just like the
      first ones, except that where previously they were applied to lines they are here applied
      to spaces, and vice versa; and the clefs are termed 'high', assigned, as it were, to the
      high positions.

       Then to the sixteenth position, that is to say highest a la mi re, through further repetition
      is assigned a. To the seventeenth, namely highest brotundum_small fa bquadrum_small mi, brotundum_small bquadrum_small also in two-fold
[145] form on account of the two-fold property of the pitches coinciding in it. To the
      eighteenth, that is to say c sol fa, c. To the nineteenth, namely d la sol, d. To the
      twentieth, that is to say e la, e. And these five letters repeated for the second time, like
      the others, are distinguished from each other by name, and are distinguished from the
      first five of the same name in form also, since these first are upper-case, whereas the last
      five are lower-case. This distinction of form is necessary, since those letters which are
[150] applied in the first set to lines and spaces have also been applied here to lines and spaces
      in a similar ordering. These last five, however, are distinguished from the second five of
      the same name in both form and position, for the second set, like the first, uses upper-case
      letters, whereas the last set uses lower-case; and those which are found in the second set
      on lines are here in spaces, and vice versa. Again, the 'highest' clefs are so called because
      they are assigned to the highest positions, as is shown in this diagram:

       [Figure 3]

     em_fig3_tr 

[155] This said, however, as far as notation is concerned not all of these clefs are in use; for
      in order to understand all the positions, no matter many there may be, it is sufficient to
      apply only one clef, since, having learned one position by means of its sign, it is
      entirely straightforward to learn the rest, both above and below, because the progression
      from one to another must follow its fixed and organized scheme. And although any
      composer could adopt whichever of these twenty clefs he preferred in his notation, I
[160] have nevertheless found only six in use.

       The first is gamma for gamma ut, whose name, just as its form, is Greek, in order to bestow
      due honour on the Greeks, as has been explained.

       The second is bquadrum_small for bquadrum_large mi and both positions of brotundum_small fa bquadrum_small mi whenever mi is sung
      there and this is called 'square' from its form, because it is square-shaped at the bottom.
[165] There are many, however, who notate this clef thus sharp_crosshair , but incorrectly; for this, indeed,
      is the proper sign of the chromatic semitone.

       The third is F for low F fa ut: previous generations once used this clef adopting the form
      of the proper letter itself, as is shown in ancient manuscripts; but – for what reason I know
      not – modern musicians, departing from the footsteps of their ancestors, notate this clef
      thus fclef1 or thus fclef2 , these being common in plainchant; or thus fclef3 , also in
[170] plainchant; or thus fclef4 , especially in composed music, although more frequently it is
      written void, like this fclef5 . It is, however, of no consequence either way whether it is
      void or filled, as in the latter or former case; or even half one and half the other, as
      here fclef6 fclef7 .

       The fourth is brotundum_small for both positions of  brotundum_small fa bquadrum_small mi whenever fa is sung there; and this
      is called 'round' from its form, because it is round at the bottom. There is a distinction,
[175] therefore, between round brotundum_small and square bquadrum_small both in form and name. Nor should we pass
      over the fact that we also use this fourth clef, that is to say round brotundum_small , in all positions
      where fa is irregularly sung, as is shown in virtually all the works of composers.

       The fifth is C for C sol fa ut, whose strict letter form has been altered for I know not
      what reason; for in all music, especially plainchant, if it is filled it is notated like
[180] this cclef1 ; but if it is void, in composed music, it is like this cclef2 , although it is of no
      consequence if it is filled or void in either the latter or former case.

       The sixth is G for high G sol re ut; the use of this clef, however, is rare in composed
      music, and even rarer in plainchant, since this particular position is not indicated except
      in cases where C sol fa ut is missing, and this occurs only very rarely.

[185]  And so that the function of these cleffing letters can be understood concisely, they may be
      defined in their correct order as follows:

      gamma is the clef of gamma ut.

      A is the clef of A re and both positions of A la mi re.

      bquadrum_small is the clef of bquadrum_large mi and both positions of  brotundum_small fa bquadrum_small mi, which is two-fold, namely
[190] square and round. Square bquadrum_small is the clef of bquadrum_large mi and both positions of  brotundum_small fa bquadrum_small mi,
      indicating that in that place mi should be sung through hard bquadrum_small . Round brotundum_small is the clef
      of both positions of  brotundum_small fa bquadrum_small mi, indicating that in that place fa should be sung through
      soft brotundum_small .

      C is the clef of C fa ut, C sol fa ut, and c sol fa.

      D is the clef of D sol re, D la sol, and d la sol.

[195] E is the clef of both positions of E la mi, and e la.

      F is the clef of both positions of F fa ut.

      G is the clef of both positions of G sol re ut.

       Chapter 4: On pitch names
      With respect to the third topic: pitch is the sound formed from either natural or
[200] artificial instruments. Moreover, there are six universally applicable pitch names,
      that is to say ut, re, mi, fa, sol, and la; and we can define these in their correct order
      as follows:

      Ut is the first pitch name, standing a tone away from the second.
      Re is the second pitch name, standing a tone away from the first, and the same distance
      from the third.
      Mi is the third pitch name, standing a tone away from the second, and a minor semitone
      from the fourth.
[205] Fa is the fourth pitch name, standing a minor semitone from the third, and a tone from
      the fifth.
      Sol is the fifth pitch name, standing a tone away from the fourth, and the same distance
      from the sixth.
      La is the sixth and final pitch name, standing a tone away from the fifth.

       And although, as I have just said, there are only six universally applicable pitch names,
      nevertheless, since in many of the positions on our hand a number of different pitch names
[210] are located through repetition, it turns out that on this hand forty-two pitch names are
      found:

      One in gamma ut, namely ut.
      One in A re, namely re.
      One in bquadrum_large mi, namely mi.
      Two in C fa ut, namely fa and ut.
[215] Two in D sol re, namely sol and re.
      Two in low E la mi, namely la and mi.
      Two in low F fa ut, namely fa and ut.
      Three in low G sol re ut, namely sol, re and ut.
      Three in high A la mi re, namely la, mi and re.
[220] Two in high  brotundum_small fa bquadrum_small mi, namely fa and mi.
      Three in C sol fa ut, namely sol, fa and ut.
      Three in D la sol re, namely la, sol and re.
      Two in high E la mi, namely la and mi.
      Two in high F fa ut, namely fa and ut.
[225] Three in high G sol re ut, namely sol, re and ut.
      Three in highest a la mi re, namely la, mi and re.
      Two in highest  brotundum_small fa bquadrum_small mi, namely fa and mi.
      Two in c sol fa, namely sol and fa.
      Two in d la sol, namely la and sol.
[230] One in e la, namely la.

       Again, of these forty-two pitch names only one is the lowest, that is to say ut in gamma ut,
      because, relative to the others above, it sounds the lowest. The rest, then, are low,
      high, or highest.

       The low pitch names are all those that are contained in this hand of ours from A re
[235] inclusive through to high A la mi re exclusive; and they are so called because, relative
      to the others above, they sound low.

       The high pitch names are all those that are contained in this hand of ours from high
      A la mi inclusive through to highest a la mi re exclusive; and they are so called because,
      relative to the others below, they sound high.

[240]  The highest pitch names are all those that are contained in this hand of ours from highest
      a la mi re through to e la inclusive; and they are so called because they sound higher
      than the high ones, or above the high ones, as is shown in the following diagram:

       [Figure 4]

 em_fig4 _tr 

       Chapter 5: On properties
      With respect to the fourth topic: a property is a certain individual quality possessed
      by pitches which are to be strung together into hexachords. There are, moreover,
[245] three properties, namely hard bquadrum_small , natural, and soft brotundum_small .

       Hard bquadrum_small is the first property, through which ut is sung in all positions whose clef
      is G; and from this note the other five pitches are then derived in their correct order.
      And it is called hard bquadrum_small because through that property mi is sung in any position
      whose clef is square bquadrum_small ; and this mi is hard, that is harsh, in comparison with the
      fa sometimes found in the same position, which is to be sung through soft brotundum_small .

[250]  Natural is the second property, through which ut is sung in all positions whose clef
      is C; and from this note the other five pitches are then derived in their correct order.
      And it is called natural because all the pitches of this particular property remain in
      a fixed and stable scheme, just as with natural matter: hence the saying, 'That which
      nature has given, nobody can take away.'

       Soft brotundum_small is the third property, through which ut is sung in all positions whose clef
      is F; and from this note the other five pitches are derived in their correct order.
[255] And it is called soft brotundum_small because through that property fa is sung in any position
      whose clef is round brotundum_small ; and this fa is soft, that is sweet, in comparison with the mi
      sometimes found in the same position, which is to be sung through hard bquadrum_small .

       From this, so that you may commit the fundamental clefs of these properties more
      firmly to memory, take note of this verse: 'C gives natural, F soft brotundum_small , and G hard.'

[260]  And and this point it should be noted that there is a great difference between square
      bquadrum_small and hard bquadrum_small , and between round brotundum_small and soft brotundum_small : for square bquadrum_small and round brotundum_small 
      are the names of clefs, so called from their form, as has been shown above in
      Chapter 3; but hard bquadrum_small and soft brotundum_small  are the names of properties, so called from the
      quality of the pitch names fa and mi that are to be sung in the positions of the
      aforementioned clefs. The common form of this letter 'b', that is to say with a rounded
[265] rounded bottom, remains in both round brotundum_small  and soft brotundum_small  for two reasons: firstly because
      that which is soft, by which we understand sweet, is more worthy than that which is
      hard, that is, harsh; and secondly because, since round brotundum_small  and square bquadrum_small occur together
      in one and the same position – that is, in both positions of brotundum_small fa bquadrum_small mi – round brotundum_small 
      comes first. And it is certainly the most fitting reasoning that the more worthy should
      take precedence, since it takes the primary form. In order to differentiate from
[270] this common form of the letter in question, another was invented, that is to say with
      a squared bottom, so that, through their different forms, different positions and
      different properties, indicated by means of this same letter, could be clearly
      recognized.

       [Figure 5]

 em_fig5_tr 

       Chapter 6: On hexachords
      With respect to the fifth topic: a hexachord is an ordered string of pitch names,
      deriving from one position and proceeding to another through any one of the
      properties. And since, as I have said above, there are three properties, namely
[275] hard bquadrum_small , whose fundamental clef is G, natural, whose fundamental clef is C,
      and soft brotundum_small , whose fundamental clef is F, and since there are three Gs in our
      hand, namely  gamma , which is G in Latin, the G of low G sol re ut, and the G of
      high G sol re ut, and two Cs containing ut, namely the C of C fa ut and the C of
      C sol fa ut, and two Fs, namely the F of low F fa ut and the F of high F fa ut,
[280] given that three plus twice two make seven, it is inevitable that there are seven
      hexachords in this hand of ours, that is to say three of hard bquadrum_small , two natural, and
      and two of soft brotundum_small .

       The first hexachord, then, is from the ut of gamma ut through to the la of low E la mi
      inclusive; and this is the first hexachord of hard  bquadrum_small .

       The second hexachord is from the ut of C fa ut through to the la of high A la mi re
[285] inclusive; and this is the first natural hexachord.

       The third hexachord is from the ut of low F fa ut through to the la of D la sol re
      inclusive; and this is the first hexachord of soft brotundum_small .

       The fourth hexachord is from the ut of low G sol re ut through to the la of high
      E la mi inclusive; and this is the second hexachord of hard  bquadrum_small .

[290]  The fifth hexachord is from the ut of C sol fa ut through to the la of highest
      a la mi re; and this is the second natural hexachord.

       The sixth hexachord is from the ut of high F fa ut through to the la of D la sol
      inclusive; and this is the second hexachord of soft brotundum_small .

       The seventh hexachord is from the ut of high G sol re ut through to the la of e la
[295] inclusive; and this is the third hexachord of hard  bquadrum_small .

      As is shown here in the following diagram:

       [Figure 6]

 em_fig6_tr 

       Furthermore, all those pitches that are grouped, as has been explained, into a
      hexachord through the property of hard bquadrum_small are said to be sung 'through hard bquadrum_small ';
      those grouped through the natural property are said to be sung 'through natural';
      and those grouped through the property of soft brotundum_small are said to be sung 'through
      soft brotundum_small ', the root pitch names of every hexachord nevertheless conforming to their
[300] own proper positions, and the other five following on from the positions of these
      root notes.

       At this point, since in the preceding material I have dealt separately with positions,
      clefs, pitch names, properties, and hexachord groupings, so that we can have a
      comprehensive understanding of all of these together, we may define the positions
      of the hand in their correct order thus:

[305]  gamma ut is a line whose clef is gamma and in which a single pitch name, that is to say ut,
      is sung through hard bquadrum_small , starting from its own position.

       A re is a space whose clef is A and in which a single pitch name, that is to say re,
      is sung through hard bquadrum_small , starting from the position gamma ut.

       bquadrum_large mi is a line whose clef is square bquadrum_small and in which a single pitch name, that is to say
[310] mi, is sung through hard bquadrum_small , starting from the position gamma ut.

       C fa ut is a space wholse clef is C and in which two pitch names, that is to say fa
      and ut, are sung: fa through hard bquadrum_small , starting from the position gamma ut, and ut
      through natural, starting from its own position.

       D sol re is a line whose clef is D and in which two pitch names, that is to say sol
      and re, are sung: sol through hard bquadrum_small , starting from the position gamma ut, and re
      through natural, starting from the position C fa ut.

[315]  Low E la mi is a space whose clef is E and in which two pitch names, that is to say
      la and mi, are sung: la through hard bquadrum_small , starting from the position gamma ut, and
      mi through natural, starting from the position C fa ut.

       Low F fa ut is a line whose clef is F and in which two pitch names, that is to say
      fa and ut, are sung: fa through natural, starting from the position C fa ut, and ut
      through soft brotundum_small , starting from its own position.

       Low G sol re ut is a space whose clef is G and in which three pitch names, that is to
[320] say sol, re and ut, are sung: sol through natural, starting from the position C fa ut;
      re through soft brotundum_small , starting from the position low F fa ut; and ut through hard bquadrum_small ,
      starting from its own position.

       High A la mi re is a line whose clef is A and in which three pitch names, that is to
      say la, mi and re, are sung: la through natural, starting from the position C fa ut;
      mi through soft brotundum_small , starting from the position low F fa ut; and re through hard bquadrum_small ,
      starting from the position low G sol re ut.

[325]  High brotundum_small fa bquadrum_small mi is a space, one of whose clefs is round brotundum_small , the other square bquadrum_small ,
      and in which two pitch names, that is to say fa and mi, are sung: fa through soft brotundum_small ,
      starting from the position low F fa ut, and mi through hard bquadrum_small , starting from the
      position low G sol re ut.

       C sol fa ut is a line whose clef is C and in which three pitch names, that is to say
      sol, fa and ut, are sung: sol through soft brotundum_small , starting from the position low F fa ut;
[330] fa through hard bquadrum_small , starting from the position low G sol re ut; and ut through natural,
      starting from its own position.

       D la sol re is a space whose clef is D and in which three pitch names, that is to say
      la, sol and re, are sung: la through soft brotundum_small , starting from the position low F fa ut;
      sol through hard bquadrum_small , starting from the position low G sol re ut; and re through natural,
      starting from the position C sol fa ut.

       High E la mi is a line whose clef is E and in which two pitch names, that is to say
[335] la and mi, are sung: la through hard bquadrum_small , starting from the position low G sol re ut,
      and mi through natural, starting from the position C sol fa ut.

       High F fa ut is a space whose clef is F and in which two pitch names, that is to say
      fa and ut, are sung: fa through natural, starting from the position C sol fa ut, and
      ut through soft brotundum_small , starting from its own position.

       High G sol re ut is a line whose clef is G and in which three pitch names, that is to
[340] say sol, re and ut, are sung: sol through natural, starting from the position C sol fa ut;
      re through soft brotundum_small , starting from the position high F fa ut; and ut through hard bquadrum_small ,
      starting from its own position.

       Highest a la mi re is a space whose clef is a and in which three pitch names, that is
      to say la, mi and re, are sung: la through natural, starting from the position C sol fa ut;
      mi through soft brotundum_small , starting from the position high F fa ut; and re through hard bquadrum_small ,
      starting from the position high G sol re ut.

[345]  Highest brotundum_small fa bquadrum_small mi is a line, one of whose clefs is round brotundum_small , the other square bquadrum_small ,
      and in which two pitch names, that is to say fa and mi, are sung: fa through soft brotundum_small ,
      starting from the position high F fa ut, and mi through hard bquadrum_small , starting from the
      position high G sol re ut.

       c sol fa is a space whose clef is c and in which two pitch names, that is to say sol
      and fa, are sung: sol through soft brotundum_small , starting from the position high F fa ut, and
      fa through hard bquadrum_small , starting from the position high G sol re ut.

[350]  d la sol is a line whose clef is d and in which two pitch names, that is to say la and sol,
      are sung: la through soft brotundum_small , starting from the position high F fa ut, and sol through
      hard bquadrum_small , starting from the position high G sol re ut.

       e la is a space whose clef is e and in which a single pitch name, that is to say la, is
      sung through hard bquadrum_small , starting from the position high G sol re ut.

       Chapter 7: On mutations
[355] With respect to the sixth topic: mutation is the changing of one pitch name into
      another. All pitch names, moreover, are mutable, but some more so, others less:

      Ut, then, is mutated into three other pitch names, that is to say into re, fa and sol.
      Re into four others, that is to say into ut, mi, sol and la.
      Mi into two others, that is to say re and la.
[360] Fa into two others, that is to say ut and sol.
      Sol into four others, that is to say ut, re, fa and la.
      La into three others, that is to say re, mi and sol.

       As is clear, therefore, to an attentive observer, there are eighteen universally
      applicable mutations, namely ut–re, ut–fa, ut–sol; re–ut, re–mi, re–sol, re–la; mi–re,
[365] mi–la; fa–ut, fa–sol; sol–ut, sol–re, sol–fa, sol–la; la–re, la–mi, and la–sol. Of these
      eighteen mutations, nine take place in order to ascend from one property into
      another, and nine in order to descend from one property into another. Whence
      the verses:

       To ascend
      'Ut–re, re–ut, re–mi with mi–re, and fa–ut and sol–ut,
[370] And sol–re, la–re, la–mi enable you to rise.'

       To descend:
      'Ut–fa, ut–sol, re–sol with re–la, and mi–la, fa–sol,
      And sol–fa, sol–la, la–sol head for the bottom when you sing.'

       Furthermore, every ascent takes place either from hard bquadrum_small to natural, or from natural
[375] to soft brotundum_small , or from natural to hard bquadrum_small , or from soft brotundum_small to hard bquadrum_small , or from hard bquadrum_small 
      to soft brotundum_small , or from soft brotundum_small to natural. And every descent takes place either from natural
      to hard bquadrum_small , or from soft brotundum_small to natural, or from hard bquadrum_small to natural, or from hard bquadrum_small 
      to soft brotundum_small , or from soft brotundum_small to hard bquadrum_small , or from natural to soft brotundum_small .

       And although, as I have said above, there are only eighteen universally applicable
      mutations, nevertheless, because all of the pitch names and hexachords of our hand
[380] (albeit some more than others) are repeated, on account of the large number of
      positions, there are fifty-two mutations in all found in this hand of ours:

       Two on C fa ut, which is the first position of mutation, namely fa–ut and ut–fa:
      fa–ut to ascend from hard bquadrum_small to natural; and ut–fa to descend from natural to hard bquadrum_small ,
[385] as here:

      [Example 1]

      em_example1_transcr

       Two on D sol re, namely sol–re and re–sol: sol-re to ascend from hard bquadrum_small to natural;
      and re–sol to descend from natural to hard bquadrum_small , as here:

      [Example 2]

      em_example2_transcr

       Two on low E la mi, namely la–mi and mi–la: la–mi to ascend from hard bquadrum_small to natural;
      and mi–la to descend from natural to hard bquadrum_small , as here:

      [Example 3]

      em_example3_transcr

[390]  Two on low F fa ut, namely fa–ut and ut–fa: fa-ut to ascend from natural to soft brotundum_small ;
      and ut–fa to descend from soft brotundum_small to natural, as here:

      [Example 4]

      em_example4_transcr

       Six on low G sol re ut, namely sol–re and re–sol; sol–ut, ut–sol; re–ut, ut–re: sol–re to
      ascend from natural to soft brotundum_small ; re–sol to descend from soft brotundum_small to natural; sol–ut to
      ascend from natural to hard bquadrum_small ; ut–sol to descend from hard bquadrum_small to natural; re–ut to
[395] ascend from soft brotundum_small to hard bquadrum_small ; and ut–re to ascend from hard bquadrum_small to soft brotundum_small , as here:

      [Example 5]

      em_example5_transcr

       Six on high A la mi re, namely la–mi, mi–la; la–re, re–la; mi–re and re–mi: la–mi
      to ascend from natural to soft brotundum_small ; mi–la to descend from soft brotundum_small to natural; la–re to
      ascend from natural to hard bquadrum_small ; re–la to descend from hard bquadrum_small to natural; mi–re to
[400] ascend from soft brotundum_small to hard bquadrum_small ; and re–mi to ascend from hard bquadrum_small to soft brotundum_small , as is
      shown here:

      [Example 6]

      em_example6_transcr

       Six on C sol fa ut, namely sol–fa, fa–sol; sol–ut, ut–sol; fa–ut, ut–fa: sol–fa to descend
      from soft brotundum_small to hard bquadrum_small ; fa–sol to descend from hard bquadrum_small to soft brotundum_small ; sol–ut to ascend
[405] from soft brotundum_small to natural; ut–sol to descend from natural to soft brotundum_small ; fa–ut to ascend
      from hard bquadrum_small to natural; and ut–fa to descend from natural to hard bquadrum_small , as is shown here:

      [Example 7]

      em_example7_transcr

       Six on D la sol re, namely la–sol, sol–la; la–re, re–la; sol–re et re–sol: la–sol to descend
      from soft brotundum_small to hard bquadrum_small ; sol–la to descend from hard bquadrum_small to soft brotundum_small ; la–re to ascend
[410] from soft brotundum_small to natural; re–la to descend from natural to soft brotundum_small ; sol–re to ascend from
      hard bquadrum_small to natural; and re–sol to descend from natural to hard bquadrum_small , as here:

      [Example 8]

      em_example8_transcr

       Two on high E la mi, just like low E la mi, namely la–mi and mi–la: la-mi to ascend
      from hard bquadrum_small to natural; and mi–la to descend from natural to hard bquadrum_small , as here:

      [Example 9]

      em_example9_transcr

[415]  Two on high F fa ut, just like low F fa ut, namely fa–ut and ut–fa: fa–ut to ascend from
      natural to soft brotundum_small ; and ut–fa to descend from soft brotundum_small to natural, as is shown here:

      [Example 10]

      em_example10_transcr

       Six on high G sol re ut, just like low G sol re ut, namely sol–re, re–sol; sol–ut, ut–sol;
      re–ut, ut–re: sol–re to ascend from natural to soft brotundum_small ; re–sol to descend from soft brotundum_small 
[420] to natural; sol–ut to ascend from natural to hard bquadrum_small ; ut–sol to descend from hard bquadrum_small 
      to natural; re–ut to ascend from soft brotundum_small to hard bquadrum_small ; and ut–re to ascend from hard bquadrum_small 
      to soft brotundum_small , as is shown here:

      [Example 11]

      em_example11_transcr

       Six on highest a la mi re, just like high A la mi re, namely la–mi, mi–la; la–re, re–la;
      mi–re, re–mi: la–mi to ascend from natural to soft brotundum_small ; mi–la to descend from soft brotundum_small to
[425] natural; la–re to ascend from natural to hard bquadrum_small ; re–la to descend from hard bquadrum_small to
      natural; mi–re to ascend from soft brotundum_small to hard bquadrum_small ; and re–mi to ascend from hard bquadrum_small to
      soft brotundum_small , as is shown here:

      [Example 12]

      em_example12_transcr

       Two on c sol fa, namely sol–fa and fa–sol: sol–fa to descend from soft brotundum_small to hard bquadrum_small ;
      and fa–sol to descend from hard bquadrum_small to soft brotundum_small , as is shown here:

      [Example 13]

      em_example13_transcr

[430]  Two on d la sol, namely la–sol and sol–la: la–sol to descend from soft brotundum_small to hard bquadrum_small ;
      and sol–la to descend from hard bquadrum_small to soft brotundum_small , as is shown here:

      [Example 14]

      em_example14_transcr

       On gamma ut, A re, bquadrum_large mi, and e la, however, no mutation takes place, because in each
      of these positions there is only a single pitch name; but where there is only a single
      pitch name no mutation can occur, since where all mutation is to take place two pitch
      names are required, that is to say one which is mutated and the other which is taken
[435] up through the process of mutation itself.

       In addition, no mutation takes place on high and highest brotundum_small fa bquadrum_small mi, because mutation
      has necessarily to occur between two pitch names coinciding on a unison: that is to
      say, that the pitch name which is mutated and the other which is taken up through the
      process of mutation itself must be of one and the same sound, such as the fa and ut
      of C fa ut, or the sol and re of D sol re, and so on. Hence, since fa and mi are never
[440] of one and the same sound in any position at all, but rather they stand at a distance of
      a major semitone from one another, it is impossible for either to be mutated into the
      other.

       Nor should I neglect to mention that mutations were invented to account for the
      movement across from one property to another. Hence, after we have entered one
      particular property, we must never mutate before its last available pitch name; and
[445] so we understand by this that mutation should occur as rarely and as late as possible.

       Again, mutation on any note does not alter its sound, but only its name. Hence,
      when we solmize, we mutate only because at that particular time we are performing
      the notes by name, for solmization is indeed the sung performance of notes by means
      of their names.

       From the foregoing, so that we can understand comprehensively the function of each
[450] mutation, let us define them in their correct order as follows:

       Ut–re is the mutation that takes place in both positions of G sol re ut in order to ascend
      from hard bquadrum_small to soft brotundum_small .

       Ut–fa is the mutation that takes place on C fa ut and C sol fa ut in order to descend
      from natural to hard bquadrum_small , and in both positions of F fa ut in order to descend from
      soft brotundum_small to natural.

       Ut–sol is the mutation that takes place in both positions of G sol re ut in order to descend
[455] from hard bquadrum_small to natural, and on C sol fa ut in order to descend from natural to soft brotundum_small .

       Re–ut is the mutation that takes place in both positions of G sol re ut in order to ascend
      from soft brotundum_small to hard bquadrum_small .

       Re–mi is the mutation that takes place in both positions of A la mi re in order to ascend
      from hard bquadrum_small to soft brotundum_small .

       Re–sol is the mutation that takes place on D sol re and D la sol re in order to descend
      from natural to hard bquadrum_small , and in both positions of G sol re ut in order to descend from
      soft brotundum_small to natural.

[460]  Re–la is the mutation that takes place in both positions of A la mi re in order to descend
      from hard bquadrum_small to natural, and on D la sol re in order to descend from natural to soft brotundum_small .

       Mi–re is the mutation that takes place in both positions of A la mi re in order to ascend
      from soft brotundum_small to hard bquadrum_small .

       Mi–la is the mutation that takes place in both positions of E la mi in order to descend
      from natural to hard bquadrum_small , and in both positions of A la mi re in order to descend from
      soft brotundum_small to natural.

[465]  Fa–ut is the mutation that takes place on C fa ut and C sol fa ut in order to ascend from
      hard bquadrum_small to natural, and in both positions of F fa ut in order to ascend from natural to
      soft brotundum_small .

       Fa–sol is the mutation that takes place on C sol fa ut and on c sol fa in order to descend
      from hard bquadrum_small to soft brotundum_small .

       Sol–ut is the mutation that takes place in both positions of G sol re ut in order to ascend
[470] from natural to hard bquadrum_small , and on C sol fa ut in order to ascend from soft brotundum_small to natural.

       Sol–re is the mutation that takes place on D sol re and D la sol re in order to ascend
      from hard bquadrum_small to natural, and in both positions of G sol re ut in order to ascend from
      natural to soft brotundum_small .

       Sol–fa is the mutation that takes place on C sol fa ut and c sol fa in order to descend
      from soft brotundum_small to hard bquadrum_small .

[475]  Sol–la is the mutation that takes place on D la sol re and d la sol in order to descend
      from hard bquadrum_small to soft brotundum_small .

       La–re is the mutation that takes place in both positions of A la mi re in order to ascend
      from natural to hard bquadrum_small , and on D la sol re in order to ascend from soft brotundum_small to natural.

       La–mi is the mutation that takes place in both positions of E la mi in order to ascend
[480] from hard bquadrum_small to natural, and in both positions of A la mi re in order to ascend from
      natural to soft brotundum_small .

       La–sol is the mutation that takes place on D la sol re and on d la sol in order to descend
      from soft brotundum_small to hard bquadrum_small .

       Finally, it should be noted that in the layout of these mutations a certain divine pattern
      is held, which can be grasped most easily by means of the following diagram:

      [Figure 7]

em_fig7_tr

 

[485]  Chapter 8: On intervals
      With respect to the seventh topic: an interval is the joining of one note to the next,
      with nothing else in between. Moreover, every interval can be made by arsis, that is
      to say through ascent, or by thesis, that is to say through descent. From this, it should
      be known that in each hexachord there are fifteen intervals, which can be produced
      ascending or descending, namely: four tones, one minor semitone, two dytones [major
[490] thirds], two semidytones [minor thirds], three diatessarons [perfect fourths], two
      diapentes [perfect fifths], and one diapente plus tone [major sixth], as is shown here:

      [Example 15]

      em_example15_transcr

       At this point, so that the essential qualities of these intervals may be grasped, they
      may be defined in their correct order as follows:

       The tone is the interval formed from the span of two minor semitones plus one
[495] comma: of this type are ut–re, re–ut; re–mi, mi–re; fa–sol, sol–fa; sol–la and la–sol.

       The minor semitone is the interval formed from the span of two diaschismata: of this
      type are mi–fa and fa–mi; and it is called a semitone from 'semus, -a, -um', that is
      to say 'imperfect', and from the noun 'tone', as is were an imperfect tone. And the
      word 'minor' is added to differentiate it from the major semitone, which is composed
[500] of two diaschismata and one comma.

       The dytone [major third] is the interval formed from the span of two tones: of this
      type are ut–mi, mi–ut; fa–la and la–fa; and it is called a dytone from 'dy' with Greek
      'y', which is 'two', and 'tone', as it were an interval composed of two tones.

       The semidytone [minor third] is the interval formed from the span of a tone and a
[505] semitone, or vice versa: of this type are re–fa, fa–re; mi–sol and sol–mi; and it is called
      a semidytone from 'semus', which, as has been said above, means the same as imperfect,
      and 'dytone', as it were an imperfect dytone.

       The diatessaron [perfect fourth] is the interval formed from the span of a tone and a
      semidytone [major third], or vice versa: of this type are ut–fa, fa–ut; re–sol, sol–re; mi–la
[510] and la–mi; and it is called a diatessaron from 'dia' with Latin 'i', which is 'through', and
      'tessaron', which is 'four', as it were an interval made up of four, because it takes up
      four positions.

       The diapente [perfect fifth] is the interval formed from the span of a diatessaron
      [perfect fourth] plus a tone, or else a tritone and a semitone; of this type of ut–sol,
      sol–ut; re–la, la–re; mi–mi and fa–fa both through ascent and descent. The notes,
[515] however, of each of these last two species of diapente, namely mi–mi and fa–fa, whether
      through ascent or descent, can never occur in one and the same hexachord. Hence they
      necessarily belong to different properties, as here:

      [Example 16]

      em_example16_transcr

      And it is called a diapente from 'dia' with Latin 'i', which is 'through', and 'pente',
      which is 'five', as it were an interval made up of five, because it takes up five positions.

[520]  The diapente plus tone [major sixth] is the interval formed from the span of a diapente
      plus a tone: of this type are ut–la and la–ut; and it is called a diapente plus tone because
      in this interval a diapente is placed along with a tone.

       There are, indeed, many other genera and yet more species of interval to be found in
      our hand, which are explained in the greatest detail, along with those described here,
[525] in my Speculum musices. But since these hold not inconsiderable difficulty, and since
      it has been my wish here to proceed with ease, I would refer readers desiring to know
      about them to this same Speculum.

       Chapter 9: Conclusion of the work
      Finally, then, may this exposition of the hand be sufficient for the needs of young men.
      I, Tinctoris, would strongly urge them to study it most earnestly, as being the very
[530] foundation of music. For, as all the best reasoning teaches us: where there is no
      foundation, there no building can be done above; and the result of this is that without a
      proper knowledge of the hand nobody can emerge outstanding in this art of music.

      The end.