Dissonances

ALL other Intervalls, except those of which we have now spoken, are called Dissonances; but we will treat of those only, which are necessary found in the newly explicated order of Tones, so as they cannot but be made use of and applyed.

Of these there are 3 kinds:

1. Some are generated from Degrees only, and an Eighth
2. Others from the difference which is betwixt a Tone major and minor, which we have denominated a Schism
3. Others from the Difference, which is between a Tone major, and a Semitone majus

In the First Genus are contained Sevenths and Ninths, or Sixteenths, which are only Ninths compounded, as Ninths are nothing else but Degrees compounded of an Eighth, and Sevenths nothing but the residue of an Eighth, from which one Degree is detracted; whence it is manifest, that there are three divers Ninths, and three Sevenths, because there are three kinds of Degrees; and all these consist betwixt these Numbers:

Among Ninths, two are majors, which arise from two Tones, the First from a major, the Second from a minor, for the distinction of which we have noted one Ninth maxim: on the contrary there are two Sevenths minors, for the same reason, and therefore we have called one Seventh minim.

Now, that these Dissonances cannot be avoided in founds succesively emitted, among divers parts is most clear: yet haply any one may enquire, why they ought not to be admitted in a voyage succesive of the same part equally with Degrees, since it is evident that some of them are explicated in lesser Numbers than the Degrees themselves, and therefore may seem to be more grateful to the Hearing than Degrees.

The solution of which Doubt doth depend on this, which we have before observed, that a voyage [65] doth require so much the more intention of the spirit or breath, by how much the more Acute it is, and therefore Degrees were invented, that they might be Means, betwixt the Termes of Consonances, and that by them we might the more easily ascend from the Grave Terme of any Consonance to the Acute of the same, or vice versa, descend from the Acute to the Grave Term: which cannot be performed by Sevenths or Ninths, as is evident from hence, that the Termes of these are more distant each from other, than the Termes of Consonances are, and therefore they would be emitted with a greater inequality of Contention.

In the Second Genus of Dissonances do consist a Third minor, and a Fifth Deficient by one Schisme; as also a Fourth, and a Sixth major increated by one Schisme. For since (necessarily) there is one moveable Terme by the intervall of a Schisme, in the whole Series of Degrees; it cannot be avoided, but that, from thence, such Dissonances in relation, i.e. in voce succesive emissa a duabus vocibus, will be generated: And that more then these now named cannot arise from thence, may be proved by induction [66]. These consist in these Numbers [67]:

46

Or thus:

{ Third minor defective  { G ad b. 480, 405.
{ by a Schism           { F ad D. 384, 324.
A { Fifth defective by one { G ad D. 480, 324.
  { Schism              { D ad G. 324, 240.
{ Fourth increased by   { b ad G. 405, 240.
{ one Schism            { D ad F. 324, 192.
{ Sixth major increased { 
{ by a Schism           { 

But so great are these Numbers, that such intervals cannot be tollerated of themselves; but, as we have formerly noted, because the interval of a Schisme is so small, as it can hardly bee discerned by the ears, therefore doe they borrow sweetenesse of those Consonances, to which they are neareſt. Nor doe the Terms of Consonances to consist in indivisibili, as that if one of them be a little changed, all the sweetenesse of the Consonance must instantly be lost: and this can only be the reason, why Dissonances of this Second Genus may be, in a voice successive of the same part, admitted in place of Consonances, from which they are divided.

In the Third Genus are contained, a Tritone, and a Fifth false; for in this, a Semitone majus is accounted for a Tone major; but in a Tritone, the Contrary: and they are explicated by these numbers [69]:

Tritone ³²/₄₅.   Fifth false ⁴⁵/₆₄

Or thus [70]:

{ Tritone  { F ad D. 540, 384.
A {         { b ad E. 405, 288.
{ Fifth false { F ad F. 384, 270.
{             { E ad b. 288, 202. vel 576, 405.

Which Numbers are also too great to explicate any intervall which may not be ingrate to the ears; nor have they any Consonances very near, from which they may borrow sweetenesse, as the Precedent ones have. Hence comes it, that these last Dissonances ought to be avoided in relation; at least, when slow and soft Musick is made, and not diminute: for in very diminute Musick and such as is sung swiftly, the hearing is too much imployed to take notice of the defects of such Dissonances: which defect is much more evident from hence, that they are near to a Fifth, with which the Hearing therefore compares them, and, from the conspicuous sweetenesse of this, doth the more clearly discern the imperfection of those.

Here we shall end our explication of all the Affections of a Sound; having first only taken notice, in order to the probation of what we formerly said, that all the Variety of sounds, as to Grave and Acute, doth arise in Musick only from these Numbers 2, 3, and 5. we say that all numbers, by which all well Degrees, as Dissonances are explicated, are compoſed of those three, and by them, division being made, may at length be resolved even to an unity.

[Table 1: Ratios and Musical Intervals (Left page)]

[62.] Viz. Semitonium medium, as before An. 53.

[63.] For (\frac{1}{5} + \frac{1}{6} = \frac{11}{30}); (\frac{1}{4} + \frac{1}{7} = \frac{11}{28}); (\frac{1}{5} + \frac{1}{8} = \frac{13}{40}); (\frac{1}{7} + \frac{1}{8} = \frac{15}{56}); (\frac{1}{2} - \frac{1}{12} = \frac{5}{12}); (\frac{1}{3} - \frac{1}{6} = \frac{1}{6}); (\frac{1}{3} - \frac{1}{8} = \frac{5}{24}); (\frac{1}{2} - \frac{1}{8} = \frac{3}{8}).

[64.] See p. 22.

[65.] Viz. p. 28.

[66.] See Figure An. 61.

[67.] For (\frac{1}{3} - \frac{1}{20} = \frac{17}{60}); (\frac{1}{4} - \frac{1}{20} = \frac{4}{20} = \frac{1}{5}); (\frac{1}{3} - \frac{1}{15} = \frac{4}{15}); (\frac{1}{3} - \frac{1}{13} = \frac{10}{39}); (\frac{1}{3} - \frac{1}{10} = \frac{7}{30}); (\frac{1}{2} - \frac{1}{5} = \frac{3}{10}); (\frac{1}{3} - \frac{1}{4} = \frac{1}{12}).

[68.] 480.

[Table 2: Ratios and Musical Intervals (Right page)] (This table is similar to the left one, with faint fractions for intervals. Visible headers include: Ninth major, Ninth minor, Seventh major, Seventh minor, Sixth major, Sixth minor, Fifth, Fourth, Tone major, etc.)

[68.] 480.405 :: 384.324 :: 32.27. 480.324 :: 40.27. 324.240 :: 27.20. 405.240 :: 324.192 :: 27.16.

[69.] For (\frac{1}{5} + \frac{1}{6} = \frac{11}{30}) ; (\frac{1}{5} + \frac{1}{8} = \frac{13}{40}) ; (\frac{1}{7} + \frac{1}{8} = \frac{15}{56}).

[70.] 540.384 :: 405.288 :: 45.32. 384.270 :: 288.202 (\frac{1}{2}) :: 576.405 :: 64.45.

[71.] viz. the firſt compound Eighth, i. e. a Fifteenth.

[72.] viz. without altering the order of Succeſſion, p. 30, and 41.

Otherwiſe, of Eighths conſidered only as conſiſting of three major Tones, two minor Tones, and two major Semitones. There are 210 ſeveral ſorts, or Moods; and may be found, by the Laws of Combination, as in this Table following; where note a is put for a major Tone; b for a minor Tone; and c for a major Semitone