Partition is the rule of means in which one executes executes the temperament across a compass of an octave and a half around a given note taken toward the middle of the keyboard.
The fifth is best suited of all the intervals to execute this partition because the ear easily senses the different nuances one wishes to impart it.
My partition is formed of a non-interrupted series of twelve descending fifths, flat and equally tempered, where the final is rejoined with the first, and which thereby forms the harmonic circle. Example:
(1) In this sequence of fifths, which must be learned by heart to begin with, it is apparent that one often moves by an octave, until the fifths lie one aside the other in the middle of the keyboard, where the strings are less susceptible to derangement.
The continuous succession of descending fifths is preferable the any of the adopted manners up to this time in obtaining the same degree of just-ness in each note, for the durations of chords, and for the implementation of temperament. The obvious checks guide the course of this operation. Each fifth is flattened by raising the lower note without a change in the movement of the tuning key; this permits a perception of the just fifth as a comparison which is followed by alteration with surety and facility. The string retains its tune infinitely better, and one masters it far easier by raising rather than lowering. If in a D-A fifth the D is too low, the D would be raised to just, and to flatten it is raised further by turning the key still in the same direction until the fifth is troubled by a weak alteration. Whereas in the ordinary partitions the ascending fifths are tuned by their higher note, one can but temper by a continuous experimenting by raising and lowering the string bit by bit, finally to leave it slightly above the just fifth, which prevents it from holding tune, rendering the operation very difficult and even impossible for amateurs. Also this vicious practice gives birth to the false idea that, since a newly tuned piano doesn't hold tune that the tuner didn't push enough, but to the contrary it is the result of the manner of tempering and controlling the strings.
The checks that I use to guide the ear in the partition are the major third, which is very sensitive to alteration, the fourth, the major chord in three parts - and above all its second inversion, the chord of the fourth and sixth, in which the third, being highest, dominates the other parts and allows a perception of that adjustment one has given it.
I divide the partition into three parts composed of four fifths each, which I will illustrate below, with the tests and descriptions and the instruction of their alteration. The first staff presents the fifths to tune, the second reaffirms the fourths serving as tests, the third the major thirds, the fourth the chords and the fifth the four successions of three major thirds completing the octave, which are infallible tests of a well encompassed temperament. The same number is repeated above each staff to designate the interval of the partition and its tests which are found below, establishing their relations clearly.
To execute the partition with the tests contained in this illustration, one must proceed in the following manner
1º Place the mute above the two strings that sound the A of the diapason marked with the number 1 in the partition, and which corresponds with the fourth A of the keyboard proceeding from low to high; then tune the diapason, or at your will, the string which remains free; then remove the mute and tune the second string to the first;
2º Place the mute above the two strings of the inferior A marked as 2; then tune, rigorously just, the string which is free to its octave which already is tuned, by striking both well and several times together; then remove the mute and tune the second string to the first, and play this A again with its octave, to assure yourself that nothing is deranged.
3º Place the mute above the two strings of the D marked 3, which forms a descending fifth with the A of the diapason; then tune this fifth rigorously just, after which you flatten it minutely by raising the D, and only in the manner of troubling the purity of the fifth by a very slow beat. Sound the sharp fourth A-D to see whether it presents the same degree of harshness as the fifth D-A; then remove the mute and tune the second string to that which was just tuned. Alternately sound the fifth and its inversion the fourth, to assure yourself again if the two intervals are tempered equally.1
1 Hereon, to abridge the following instructions, I will not express the three detailed operations to tune each unison, know this: place the mute above the two strings, then tune the string which is free; then remove the mute and tune the second string to the one which just was tuned; I will only say: tune this note, which will suppose these three operations.
4º Tune the fifth G-D first just, then by raising the G, flatten it to the same degree as the D-A fifth.
5º Tune the octave G-G sound the sharp fourth D-G alternately with the G-D fifth, to see if they both present the same degree of falseness.
6º Tune the fifth C-G just, then flatten it to the same degree as D-A; sound the fourth G-C, alternately with the C-G fifth to convince yourself that they are equally tempered;
7º Sound the F-C first just, then temper it as with D-A; sound the F-A major third which must not be just or too sharp, but supportable, which is to say sharp to the same degree as one of the three in the octave - then the perfect chord F-A-C, to judge them together, and which must be a little harsh. This F-A third is an important test to verify if up to this point one has operated well, as it is a product of the four first fifths. If it is too just, the fifths have been tempered too much; if on the contrary it is too sharp, the fifths haven't been enough. It is necessary in either case to start again; but as before sounding in order the four first fifths in the following order: D-A, C-G, G-D, F-C, to see if they are tempered equally, and if the error results from too much or little alteration in the four fifths , or only from one;
8º Tune the octave F-F, sound the fourth C-F which must offer the same degree of sharpness as the F-C fifth, then strike the chord of the fourth and sixth C-F-A, which must be false but supportable, and in which the major third being highest, the ear judges better its true degree of alteration than in the chord F-A-C.
Here ends the first section of the partition. If up to this point one has operated correctly, the greatest difficulties will have been overcome; one has but to continue, and one is better assured to complete it well.
9º Tune the flat fifth A# or Bb and F natural; sound the sharp fourth F-Bb, the sharp third Bb-D and the chord of the fourth and sixth F-Bb-D, which must offer the same degree of falseness as the chord C-F-A from 8;
10º Tune the octave A#-A#, sound the sharp fourth F-Bb;
11º Tune the flat fifth D#-A#, sound the sharp fourth Bb-Eb, the major third Eb-G, and the chord of the fourth and sixth Bb-Eb-G which must be false as with those preceding;
12º Tune the flat fifth G#-D#, sound the sharp third Ab-C and the major chord Ab-C-E which must be harsh and reminiscent to the first chord obtained, namely F-A-C;
13º Tune the octave G#-G#, sound the sharp fourth Eb-Ab;
14º Tune the flat fifth C#-G#, sound the sharp fourth Ab-Eb, the major third Db-F and the chord of the fourth and sixth Ab-Db-F, false like those preceding;
Here ends the second portion of the partition, that which gives an infallible means, and also to say it is mathematical, in assuring whether one has operated well. This is the three major thirds in 14, F-A, A-C#, and Db-F, which in forming the F-F octave must be equally sharp, and which, struck one after the other, must produce the same effect to the ear;
15º Tune the flat fifth F#-C#, the major chord F#-A#-C# which must produce the same effect as the F-A-C chord and as a second verification the sixth chord F#-Ax-D which must be a little sharp, but still tolerable;
16º Tune the octave F#-F#, sound the sharp fourth C#-F#, the sharp third D-F#, and the sixth chord A-D-F# which must be sharp like the preceding, and bear resemblance to the other sixth chord C#-F#-A#; finally, sound, as a more general test, the three sharp major thirds F#-A#, Bb-D, and D-F# which form the octave F#-F#;
17º Tune the flat fifth B-F#, sound the sharp fourth F#-B, the sharp major third B-D#, and the sixth chord F#-B-D#, resembling those preceding, and as a further general test sound the three sharp major thirds G-B, B-D#, and Eb-G, which form the octave G-G;
18º Tune the octave B-B, sound the sharp fourth F#-B, the sharp major third G-B, and the sixth chord D-G-B;
19º Tune the flat fifth E-B, sound the sharp fourth B-E, the sharp major third C-E, the sixth chord G-C-E, resembling those preceding and the other sixth chord B-E-G#, sound the fifth A-E, which being formed by the octave of the first note of the partition, A 2, and the final E 19, find itself flat like the others, and which must offer exactly the same degree of tempering as them. If this fifth is good, it is the most certain proof that the partition is well formed, all while having been executed as prescribed. Finally, sound the three sharp major thirds Ab-C, C-E, E-G# which form the octave G#-G# or Ab, and which like those preceding must be equally tempered. If the fifth A-E is too flat, the third C-E will be too Just, and the third E-G# too sharp. If this fifth is too sharp, the contrary will be found: the third C-E will be too sharp and the third E-G# too Just. One must then visit the last preceding fifths, making use of the indicated tests, to try to discover the cause of this error. If one doesn't happen upon it, one will be obliged to employ a sure and certain means to discover it which I call the counter partition.
The counter partition is a succession of twelve ascending flat fifths A-E-E-B, etc., serving to review their positions in the partition, to fall again on the committed error while correcting in each fifth the defect until the point where one has found this error.
To execute the counter partition found in this table, one must proceed in the following manner:
1º Touch up the A of the diapason, which is susceptible to derangement, to retune it with its octave A;
2º Touch up the fifth A-E, by tuning the E to the A, which is to say you will raise it if it is too low, or you lower it if it is too high, until the point where you obtain this fifth flat to the same degree as that of the partition, while leaving here the E nearly imperceptibly below the perfect justness; sound the sharp fourth E-A, which must be as harsh as the fifth A-E;
3º Touch up the fifth E-B, if it is deficient by tuning the B to the E, in the manner that it is flat like the preceding fifth A-E;
4º Tune the just octave B-B, sound the sharp fourth B-E, which must be as harsh as the fifth E-B;
5º Touch up the fifth B-F# if it still is deficient, by tuning as with the preceding fifth the high note to the low note, in the manner that it becomes flat like the others; sound the sharp fourth F#-B;
6º Tune the octave F#-F#, sound the sharp fourth F#-B;
7º Touch up the fifth F#-C#, flat like the others; sound the sharp fourth C#-F#; sound the major third A-C#, which must be sharp like those one had obtained in the partition; sound the chord A-C#-E, supportable and resembling in harshness the first chord of the partition F-A-C;
8º Correct in the fifth C#-G# any deficiency that might be present, by tuning it flat as with those preceding, and continue to operate as it has been indicated, by following the table of the counter partition step by step until the error has disappeared completely. Sometimes one will be obliged to continue these steps up to the first fifth D-A of the partition, and if by hazard by a string of errors committed in the counter partition, this fifth D-A is furtive, it will be necessary to commence the partition anew to dispense with this new error.
Nothing remains but to render the piano in tune, which is the object of the following article.