: 



TEMPERAMKXT AXD Tl 



TEMPKHAMKXT AND Tl 



the Moond, third, and with. Now, the pitch note c> being given, 

 there are eleven note* to be determined, and there are 3 x 12 or 36 

 equations between the value* of thoee note* and the temperament* of 

 the major third*, minor third*, and fifth*. But M equation* between 

 the eleven value* of the note*' should give 85 equations butneeu the 

 values of the temperaments ; and these 25 equation* are contained in 

 our first, fourth, and fifth rule*. 



In every system of temperament which de*eive the name, the fifths 

 must be flattened, and also the minor thirds ; while the major thirds 

 must be sharpened. In any other case, the algebraist might use the 

 preceding rule* by considering a* tiryatitt the temperament of a 

 sharpened fifth or minor third, or of a flattened third. In this sense, 

 these rules are always true, from the instant when the strings of the 

 instrument are put on, and througli.mt it* existence a* a sounding 



;\ 



It is now easy to determine the temperament* of tbe third*, major 

 and minor, from thoee of the fifth*. From a comma subtract the sum 

 of the temperament* of the fifths in any one key and the three 

 following dominant keys, and the remainder will be the temperament 

 of the major third in that key. Again, from a comma subtract the 

 mini of the temperament* of the fifth* in the three lubdominant keys 

 following any given key, and the remainder will be the temperament 

 of the minor third in that key. Hence we may, without any trouble, 

 .vn at pleasure the temperaments of the fifths, and deduce those 

 of the thirds. But we cannot, from the temperament* of the thirds, 

 thoee of the fifths. It must be remembered that a succession 

 of fifths, setting out from a given note, runs through every note of the 

 scale before it reaches that note again ; while the major thirds are 

 brought up, so to speak, by the original note, in three successions, and 

 the minor thirds in four. There are then four distinct parcels of 

 major thirds, and three of minor ones, so that it is impossible to pass 

 out of one into another by thirds alone. It would be possible to tem- 

 per all the major thirds equally, and yet to retain an unlimited number 

 of modes of tempering the fifths, depending upon the manner in which 

 one system of thirds is joined on to the others ; and the same of the 

 minor third*. 



We have, from the scale, shown how to construct the temperament*: 

 we now take the inverse question, namely, from the temperaments to 

 construct the scale. Let the sharps be denoted by accent* placed 

 above the letter* : thus A' represent* A!, and so On. Let the tempera- 

 ments of the fifth*, in the several keys, be denoted by the small letters : 

 thus a represents the temperament of the fifth in the key of A, or is 

 the portion of a mean semitone by which the interval from A to E falls 

 short of a perfect fifth. And, for abbreviation, let simple commas 

 denote addition : thus a, b may mean o + 4. Also let the notes them- 

 selves be descriptive of their intervals from c : thus c means ; o' 

 mean* the interval between ot and c : we have, then, v meaning the 

 number of mean semitones in a perfect fifth, 



c = 

 . o = v e 



D 2 v 12 e,y 



A = 3 v 12 e, y, d 



- 4 v-24-e,y,rf,o 



B = 5 v 24 c, g, d, a, e 



r 1 = 6 v 86 - e, y, d, a, e, b 



0-= 7v-48-e,.7,rf,o,e,6,/' 



tf- 8r - 48 -e.'j,d,a, t,b,f, <? 



Uf = 9v-6Q c,y,d,a,r,b,f,c',y' 



A' = 10 v - 60 - e, g, d, a, e, b,f, e", g', d' 



r = 11 v- 7-2 -f,g,d,a,t, b,f, c'.g'.d'.a! 



That is, the interval from c to B, for example, is found by deducting 

 from the excess of five perfect fifths above two octaves the sum of the 

 temperament* of the fifths in the keys of c, o, D, A, and E. Towards 

 the end, in isolated questions, trouble will be saved by remembering 

 that the sum of all the temperaments of the fifths is '2346 of a mean 

 semitone, and that we thus Lave 



e, g, d, a, e, b,f, e', <j', d', a', = -2346 -/ 



t, y, d, a, f, b, f, c 1 , y', d', = -2346 - /, a', Ac. 



We shall now take an example of this, and our instance shall be the 

 proposal of a system of temperament which we should like much to 

 see tried. We are fur variety in the several keys, and against equal 

 temperament ; but we do not like variety without law. We do not 

 like, for example, to find the greatest temperament hi one key, and 

 the least in an adjacent key, a* that of the dominant or sutxlot 

 Suppose, then, we ask what can be done toward* an ascending and 

 descending temperament, which, proceeding, *ay from the key of c, 

 hall incressr through the keys of o, o, D, A, K, B, and diminish through 

 thoee of r', </, o', r/. A', r. And, a* a first step, let the increment* 

 *nd decrement* of the temperament* of the fifth* be equal, or let c= m, 



a' = 3,/=L'm. Here, a* far as the fifth* are concerned, the effect of 

 modulation into tbe dominant or lubdominant key* is the same every- 

 where, a* much a* hi equal temperament And, from the first rule, 

 we have 48m -= -284, or m- -0048875, and the greatest temperament 

 of a fifth is seven time* this, or -034. Now, if we compute the tern- 



peraments of the thirds, major and minor, from the fourth and fifth 

 role*, we may exhibit the temperament* of all tbe key*, as follows : 



Key. Temperament. Key. Temperament. 



O 

 D 

 A 

 E 



B 



The three columns contain the temperament* of the minor third, 

 major third, aud fifth. The effect* of modulation into adjacent keys 

 are everywhere very small, nowhere amounting to more than about 

 the tenth of a comma, in alteration of temperament ; while the fifths 

 are in different keys so differently tempered, that in c that interval 

 may be called perfect ; while in F J there is nearly twice as much tem- 

 perament as in the equal semitone system. There is then variety 

 without sudden change. In the system of equal semitones, the tem- 

 perament* of the minor third, major third, and fifth, are always 



-156, + '137, - -020. 



Now, to form the scale iu this system. Proceeding by the table 

 given above, of which we take a few steps a* an example, we have 



c = O'OOOOOO 

 v = 7-019550 



7-019550 

 c = m = -001888 



A = 7-014662 

 v - 7-019550 



2-034212' 

 a = 2i = -009775 



D = 2-024437, 4c, 



Proceeding in this way, we find for the intervals of the several semi- 

 tones from the key-note, expressed in mean semitones, the following 

 table : 



C 



c; 



D 



D3 



o-ooo 



1-000 

 2-024 



2-985 



E 

 F 



Fit 

 G 



4-029 

 4-990 

 6-014 

 7-015 



0$ 

 A 



AS 

 B 



7-900 



9-029 



9-985 



11-024 



To carry this or any other system strictly into practice without 

 comparisons with the monochord, or the use of beats, presently de- 

 scribed, would be impossible ; but the following might be suggested a* 

 an approximation. In tuning by fifths, let the intervals c a and p c be 

 made perfect, or all but perfect ; let there be greater temperament in 

 o D, D A, and D$ A J, AS F ; and most of all, decidedly, in the remaining 

 intervals. 



The system of temperament is sometime* described by givinc the 

 number of vibrations made by the several semitones, or numbers pro- 

 portional to them. It is easy enough to deduce the number of mean 

 semitones in each interval from such data, either by the common table* 

 of logarithms, or by that given in SCALE. 



First, by the common table of logarithms. From the logarithm of 

 the number answering to the higher note, subtract that answering to 

 the lower ; from the result take its three hundredth part, and multiply 

 the remainder by 40. The product is the number of mean semitones 

 in the interval, with an excess of very little more than the thousandth 

 of a mean semitone in an octave. For example, to find the interval* 

 in mean semitones, of a fifth and of a comma ; in the former of which 

 the lower note makes two vibrations while the higher makes three 

 in the latter 80, while the higher makes 81 : 



For tbe fifth. 



log. 3 = -47712 

 log. 2 = -30103 



300) -17609 

 00059 



17550 

 40 



Kesult 7-02 

 More exactly 7-01956 



Error t" 



For tbe Comma. 



log. 81 = 1-90849 

 log. 80 = 1-90809 



300) -03540 

 00002 



00538 

 40 



Result -2162 

 More exactly -2151 



Error '0001 



Next, by the table in SOALK. If the numbers be in the table 

 Jnpbr mil .tract the logarithm of the lower number from that of 

 the higher, and the result is the answer required, within about the 

 hundredth of a mean semitone. But if the numbers be not in the 

 table, divide both by any number which will bring them within the 

 table, accurately or approximately, and then nibtract as before : inter- 



* Throw out the twelves ai fart si they arUe. 



