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Mr. Malcolm has made a very induftrious, and learned Inquiry into the Harmonica, or Harmonic Principles of the Antients. — They reduced their Doctrines into feven Parts, viz. of Sounds ; of Intervals ; of Sy ferns $ of the Genera ; of the Tones, or Modes $ of Mutations $ and of the Melopma. See each confider'd under its proper Article Sound, Interval, System, Genera, Mode, Mvta- tion and Melopoeia.
HARMONICAL Arithmetic, is fo much of the Theory and Doctrine of Numbers, as relates to making the Comparifons, Reductions, tfr. of Mufical Intervals, which are expreffed by Numbers, in order to our finding thek- mutual Relations, Compofitions and Rcfolutions. See In- terval, &c.
Harmonical Compofition, in its general Senfe, includes the Composition both of Harmony and Melody, i. e. of Mufic, or Songs, both in a fingle Part, and in feveral Parts. See Composition.
In its more proper and limited Senfe, Harmonical Compofition is reltrain'd to that of Harmony. In which Senfe it may be defined the Art of difpofing and con- certing feveral fingle Parts together, in fuch Manner, as to make one agreeable whole. Sec Song, &c.
The Art of Harmony hss been long known under the Name of Counterpoint. See Counterpoint.
At the Time when Parts were firft introduced, Mufic being then very fimple, there were no different Notes of Time $ and the Parts were in every Note made Concord.
This they afterwards call'd fimple, or plain Counter- point, to diftinguifh it from another Kind, then intro- duced, wherein Notes of different Value were introduced, and Difcords brought it. between the Parts. See Discord,
This they call'd Figurative Counterpoint. See Figu- rative Counterpoint.
Harmonical Interval, is an Interval, or Difference of two Sounds which are agreeable to the Ear, whether in Confonance or Succeffion. See Interval.
Har?nomcal Intervals, therefore, are the fame with Concords. See Concord.
They arc thus call'd, as being the only effential Ingredients ofHarmony. See Harmony.
H/lrmoni cal "Proportion, is a Sort of Proportion be- tween three Quantities, wherein the Difference of the firft and fecond, is to the Difference of the fecond and third ; as the firft is to the third.
Thus, 2 : 3 : 6 are Harmonical, becaufe 2 : 6" : : I : 3 are Geometrical.
So four Numbers arc Harmonical, when the firft is to the fourth, as the Difference of the firft and fecond, to the Difference of the third and fourth.
Thus, 24: 16 : 12 : 9 are Harmonical 5 becaufe 24: x : : 8 : 3 arc Geometrical.
For the Laivs and Rules of Harmonical Propor- tion. See Harmonical Proportion.
Harmonical Series, is a Series of many Numbers in continual Harmonical Proportion. See Series.
If there be four or more Numbers, whereof, every three immediate Terms arc Harmonical $ the whole makes an Harmonical Series, of continual Harmonical Proportionals :
As 30 : 2G : I J : 12 : to.
Or if every 4 immediately next each other are Harmo- nical, 'tis alfo a continual Harmonical Series, but of another Species; as 3, 4, 6, 9, 18, $6, ££c.
Harmonical Sounds, is an Appellation given by M. Sauveur to fuch Sounds as always make a certain deter- minate Number of Vibrations in the Time that fome other fundamental Sound, to which they are referr'd, makes o»e Vibration. See Sound and Vibration.
Harmonical Sounds, are produced by the Parts of Chords, e5V. which vibrate a certain Number of Times while the whole Chord vibrates once. See Chord.
By this they are diftinguifhed from the third, fifth, fifo where the Relations of the Vibrations is 4 to 5, or 5 to 5, or 2 to 3. See Third, &c.
The Relations of Sounds, had only been confider'd in the Series of Numbers 1 : 2, 2 : 3, 3 : 4, 4 : 5, &c. which produced the Intervals call'd OBave, Fifth, Fourth, 'Third) &c. Mr. Sauveur firft confider'd them in the natural Scries, i, 2, 3, 4, &c. and examined the Relations of the Sounds arifing therefrom. — The Refult is, that the firft Interval 1 : 2, is an Octave 5 the fecond 1 : 3, a twelfth ; the third. 1 : 4, a fifteenth, or double Octave ; the fourth 1 : 5, a feventeenth $ the fifth 1:6", a nineteenth, &c.
This new Confederation of the Relations of Sounds, is more natural than the old one $ and docs exprefs and re- prefent the whole of Mufic, and is in Effect all the Mufic that Nature gives without the Affiftance of Art. — The String of a Harpfichord, or a Bell, befide their general Sound, which is proportionate to their Length, Tcnfion, &c. do
alfo at the fame Time yield other fubordinate and acuter Sounds, which a nice Ear, with a good Attention, clearly diftinguifhes.
Thefe fubordinate Sounds arife from the particular Vi- brations of fome of the Parts of the String, or Bell, which are, as it were, detach'd from the reft, and make feparate Vibrations : In Effect, every half, every third, every fourth, ££c. of the Chord performs its Vibrations apart, while a general Vibration is made of the whole Chord. — Now all thefe fubordinate Sounds are Harmo- nical with Regard to the whole Sound : The lcaft acute, which we hear, is Oftave with the whole Sound $ the leaft acute that follows, makes a twelfth with the whoic Sound ; the next a feventeeth, £?c. till they grow too acute for the Ear to perceive them. Now throughout the whole, we hear no iuch Thing as a Sound that makes a fifth, or a third, $$c. with the whole Sound j none in ffiort, but what arc comprized in the Series of Harmonical Sounds.
Add, that if the Breath, or Bellows that blow a Wind Inftrument, be play'd ftronger and ftrongcr, the Tone will be continually rais'd, but this only in the Ratio of the Harmonical Sounds. ■ — So that it appears that Nature, when Jhe makes as it were a Syftem of Mufic her fclf, ufe» no other but this Kind of Sounds 5 and yet they had hitherto remained unknown to the Muficians : Not but that they frequently tell into 'em, but it was inadvertently, and without knowing what they did. — > M. Sauveur Chews that the Structure of the Organ depends entirely on this unknown Principle. See Organ.
HARMONY, in Mufic, the agreeable Refult of an Union of feveral Mufical Sounds, heard at one and the fame Time ; or, the Mixture of divers Sounds, which together have an Effecl agreeable to the Ear. See Sound.
As a continued Succeffion of Mufical Sounds produces Melody ; fo does a continued Combination of thole pro- duce Harmony. See Melody.
Among the Antients, however, as fbmetimes alfo among the Moderns, Harmony is ufed in the ftrict Senfe of Con- fonance ; and fo is equivalent to the Symphony. Sse Consonance and Symphony.
The Words Concord, and Har?no7?y, do really fignify tht fame Thing 5 tho' Cuftom has emade a little Difference between them. Concord is the agieeable Effect of two Sounds in Confonance : And Harmony, the Effect of any greater Number of agreeable Sounds in Confonance. See Concord.
Again, Harmony always implies Confona??CP : But Concord is alfo applied to Sounds in Succeffion; tho' nevw but where the Terms can ftand agreeably in Confonance : The Effcc~t of an agreeable Succeffion of feveral Sounds, is call'd Melody 5 as that of an agreeable Confonance, Harmony.
The Antients feem to have been entirely unacquainted with Harmony 5 the Soul cf the modern Mufic. — In all their Explications of the AfelopQ'ia, they fay not one Word of the Concert, or Harmony of Parts. We have Inftanccs, indeed, of their joyning feveral Voices, or In- struments, in Confonance ; But then thofe Voices, %$c. were not fo joynM, as that each had a diftincf and proper Melody, fo making a Succeffion of various Concords ; but were either Unifom, or Octaves, in every Note 5 and fo all pcrform'd the fame individual Melody, and conftituted one Song. See Song, Synaulia, &c.
When the Parts differ, not in the Tenfion of the whole, but in the different Relations of the fucceffive Notes 3 'tis thus that conttitutcs the modern Art of Harmony. See Music and Melopoeia.
Harmony is well defined the Sum of Concords, arifing from the Combination of two or more Concords, i. e. of three or more fimple Sounds, ftriking the Ear all together : And different Compofitions of Concords make different Harmony.
To underftand^ the Nature, and determine the Number, and Preference of Harmonies 5 it is to be confider'd, that in every compound Sound, where there are no more than three fimple ones, there are three Kinds of Relations, viz. the primary Relation of every fimple Sound to the fun- damental, or graveft, whereby they make different Degrees of Concord with it : The mutual Relations of the Acute founds each with other, whereby they mix either Concord or Difcord into the Compound : And the fecondary Re- lation of the whole, whereby all the Terms unite thwr Vibrations, or coincide more or lefs frequently.
Suppofe, e. gr. four Sounds, A, B, C and D, whereof A is the graveft $ B next 5 then C $ and D the acuteft. — ' Here, A is the fundamental 5 and the Relations of B, C, and D, to A are primary Relations : So, if B be a 3d g above A, that primary Relation is 4 to 5 ; and if C be 5th to A, that primary Relation is 2 to 3 j and if D be
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