121 



TEMPERAMENT AND TUNING. 



TEMPERAMENT AND TUNING. 



is.- 



too small in their intervals, the upper note must be flattened when 

 tuned from below, and the lower note sharpened when tuned from 

 above. In the preceding, the octave cc 1 is completely tuned, and 

 also the adjacent interval i\Sc. The rest of the instrument is then 

 to be tuned by octaves. The thirds should all come out a little 

 sharper than perfect, as the several trials are made : when this does 

 not happen, some of the preceding fifths are not equal. The parts 

 which are first tuned by the fifths, and from which all the others are 

 tuned by octaves, are called bearings. 



We shall now show how, by means of the theory of the scale, to 

 examine a system of temperament : the rest of this article is therefore 

 only for those who have some mathematical knowledge of the scale. 

 Everything will be expressed in mean semitones, and the following 

 additions will be convenient : A major tone is 2*039100 mean semi- 

 tones; a minor tone, 1-824037; a diatonic semitone, 1-117313; a 

 comma, '215063 ; the excess of twelve perfect fifths above seven 

 octaves, '234600, a little more than a comma, frequently called a 

 comma; the excess of an octave above three perfect thirds, '410689. 

 Various modes of dividing the octave have been proposed that is, of 

 creating imaginary subdivisions by means of which to express the 

 various intervals required. None is so convenient, in our opinion, as 

 the expression by means of mean semitones and their fractions. 



We prefer to show a complete examination of one system, in such a 

 manner that any one may apply it to another, instead of briefly noting 

 the peculiarities of different systems. We shall take as an example 

 Dr. Young's first system, which is as follows : Tune downwards, from 

 the key-note, six perfect fifths, ascending into the octave interval cc 1 

 when necessary : then tune upwards, from the key-note, six equally 

 imperfect fifths, throwing the whole error of '2346 of a mean semitone 

 equally among them. In the equal temperament the wolf is made to 

 bear twelve cubs : here only six larger ones of course. Now a perfect 

 fifth, being two major tones, a minor tone, and a diatonic semitone, is 

 thus composed . 



Two major tones 



Minor tone 

 Diatonic semitone 



Perfect fifth 



4-078200 mean semitones 



1-824037 



1-117313 



7-019MO 



The imperfect fifth of this temperament is to be flattened by the 

 sixth part of '234600, or '039100, and 7-019550 -'039100 is 6-980450, 

 the imperfect fifth required. We are then to proceed as follows : 



Six fifths downwards 

 perfect. 



C 1 12-00000 

 7-01955 



Six fifth* upwards 

 imperfect. 



C 0-00000 

 6-98045 



1. F 



I-!'-M|.-, 

 16-98045 

 7-01955 



!i MOM 

 7-01955 



3. re! 2-94135 



u: 14-9413J 



7-01955 



1. o 



8-98048 



6-98045 



2. A5 



D l 13-96090 



2. D 1-96090 



6-98045 



!. (.: 



7-92180 

 7-01955 



JS 1 

 4. E 



8-94135 

 6-98045 



15-93180 



3-92180 

 8-98048 



5. LI 090225 



c 1 ; 12-90225 



7-01955 



5. B 10-90225 

 6-98045 



6. n 5-88270 



F'3 1788270 

 F3 5-88270 



There is no doubt, at least in this world,* much surplusage in 

 carrying the results to five decimals, or the hundred-thousandth part 

 of a mean semitone ; but all calculators are aware of the desirableness 

 of using more places than will ultimately be wanted. Collecting the 

 above results, we have, for the interval of every note from C, as far as 

 c 1 , u follows : 



C 0-00000 

 05 0'9022i 

 D 1-96090 

 DJ 2-94135 



K 3-92180 



F 4-98045 



I; .V88270 



G 6-98045 



CJ 7-92180 

 A 8-91135 

 A3 9-96090 

 B 10-9022$ 



We shall now examine the effect upon the several keys. We have 

 remarked [SCALE], that the effect of making an interval too small 

 is to render the consonance of a more plaintive character; while 

 we may suppose that too large an interval has a somewhat contrary 

 effect. As the most important chord of every key is that of the key- 



Mr. Marsh, the author of a treatise on tuning, is seriously of opinion that 

 perfect scale Is one -of the blessings reserved for a future mate, in which " it 

 will be prt of the enjoyment of the blessed to chant the praises of their Creator 

 In eitatio hallelujah*, ichen lyitena of tuning ihall no longer perplex \u, anil 

 temperament thai! lie no more." 



note, its third, and fifth, we must form our idea of the effect of each 

 key from observing the effect of the temperament upon the common 

 chord of the key-note, judging of the character of the key by the 

 amount and direction of the temperament of the third and fifth. 

 Now a major third is made of a major and minor tone, and is there- 

 fore S'86314 mean semitones; while a minor third, or a major tone, 

 and a diatonic semitone, is 3-15641 mean semitones. Hence the 

 principal chord of a key, according as it is major or minor, has the 

 following intervals from the key-note : 



Major 3-86314, 

 Minor 3-15641, 



7-01955 

 7-01955 



To examine any particular key, take out the numbers from the pre- 

 ceding table opposite to the notes of the principal chord (adding 12 to 

 make the octave when necessary) ; subtract the number of the key-note 

 from each of the other two, and the remainders will give the tempered 

 intervals. Compare the tempered intervals with the preceding correct 

 intervals, and the amount and direction of the temperament will be 

 seen. For instance : 



Key of A major. 



A 8-94135 C'J 12-90225 E l 15-92180 



8-94135 8-94135 



Tempered intervals 

 Perfect intervals . 



Temperaments 



3-96090 

 3-863H 



09776 



6-98045 

 7-01955 



-03910 



and + means sharper than perfect, flatter than perfect. We might 

 describe this chord (keeping three decimals, which is more than 

 sufficient) as having a temperament expressed by the following symbol 

 (+ '098 '039); and if we examine all the keys in the same manner, 

 we shall have the following account of this system of temperament. 

 (A person who is used to the subject, and to calculation, might proceed 

 more shortly by considering the law of the system, but the beginner 

 had best take each key by itself. We have preserved the use of sharps 

 only, for the sake of symmetry.) 



Major Key. Temperament. 

 C, D, G (-K059, --OS9) 

 C-.FJ, GJ ( + -215, 0) 



DJ ( + -176, 0) 



E ( + -137, --039) 



f ( + -098, 0) 



A ( + -098, --039) 



At ( + -137, 0) 



B ( + -178, --OS9) 



The rules for the verification of every such process are six in 

 number, and as they express relations which may be made of signal 

 use in searching for systems of temperament, we give them at length. 

 In all these rules it is supposed that the fifths and minor thirds are 

 tempered flat, the major thirds sharp, and that the signs are neglected. 



1. The sum of the temperaments of all the fifths in the twelve keys 

 must be the excess of twelve fifths above seven octaves, or '23460 of a 

 mean semitone. 



2. The sum of the temperaments of all the thirds in the twelve 

 major keys must be the excess of the octave above three major thirds 

 taken four times, or 1-64236 mean semitones; the sum of the tem- 

 peraments of the thirds in any three keys whose tonics are successive 

 thirds being the excess above mentioned, or -41059 of a mean 

 semitone. 



3. The sum of the temperaments of all the thirds in the twelve 

 minor keys is three times the excess of four minor thirds over an 

 octave, or 1'87695 mean semitones; the sum of the temperaments 

 of the thirds in any four minor keys whose tonics are successive minor 

 thirds being the excess above mentioned, or '62565 of a mean semi- 

 tone. 



4. The temperament of the third in any major key, increased by 

 the temperaments of the fifths in that key and the three succeeding 

 dominant keys, makes a comma, or '215063 of a mean semitone. The 

 dominant of a note is the fifth above it ; so that the successive domi- 

 nant keys of c major, for instance, are those of o, D, A. Thus, in the 

 above system, the temperament of the third in At major is '137, and 

 those of the fifths hi At, F, c, o, are 0, 0, -039, '039 : put these together, 

 and we have '215, a comma, as asserted. 



5. The temperament of the minor third in any key, together with 

 the temperaments of the fifths in the three succeeding/ subdominant 

 keys, make a comma, or -21506 of a mean semitone. The subdominant 

 of a note U the fourth above it ; so that the successive subdominant 

 keys of o, for instance, are those of F, AS, 08. Thus, in the above 

 system, the temperament of the third in AJ minor, for instance, is 

 215, and the temperaments of the fifths in D3, 08, 08, are severally : 

 these put together of course give '215, a comma, as asserted. 



6. The temperament of the flat seventh in any key is the difference 

 of those of the fifth in that key and the minor third in the dominant 

 key. 



The algebraist may easily see how these rules are deduced, and also 

 that they are all which can be obtained. They amount altogether to 

 25 equations of condition ; for the first, fourth, and fifth rules contain 



