It is clear from the preceding that it would be simple to tune the piano by just intervals. Unfortunately, it cannot be so - instead it is necessary to use but twelve of the thirty five notes of the physical scale, in turn which is the source of difficulty in piano tuning. This physical scale comprising thirty five notes is employed by voices and those instruments with more flexible pitch such as violins and bass.
Natural pitches or notes. . . . . . . . . . . . . 7
The same, 1� altered by a sharp [di�se] . . . . . 7#
- - - - - 2� altered by a double ssharp. . . . . . 7[x]
- - - - - 3� altered by a flat. . . . . . . . . . 7b
- - - - - 4� altered by a double fflat . . . . . . 7bb
_______
35
The chromatic scale comprises but twelve discreet pitches which are presented on fixed pitch instruments such as the piano as seven white and five black keys. The difference between diatonic and chromatic semitones becomes inaudible because the arrangement necessitates average semitones composed of four and a half commas each. The average chromatic semitone between C and C# has four and a half commas rather than five, while instead of four there are four and a half commas for the average diatonic semitone between C# and D, continuing throughout the scale.
This "Average Temperament" is a manner of altering intervals by the same amount in order to divide the octave into twelve like semitones. This renders each tone equally just, or, more critically, equally false since not one interval is exactly just but only passable.
The distribution of this adjustment need only bring attention to the three consonances, namely the major and minor third, and the fifth. All the other intervals naturally are tempered, and only the octave actually is tuned just: the major third is strong [sharp],1 and the minor third and fifth are flat. Their inversions obviously are altered in the opposite manner, where the minor sixth is flat, the major sixth and fourth are sharp.
1 Here, a sharp [fort] interval is one in which the two comprising pitches are moved apart slightly, or the higher pitch is raised against the lower, or the lower pitch lowered in relation to the higher. Through the same reasoning, a flat [faible] interval consists of two pitches brought closer together, lowering the higher pitch or raising the lower.
The alteration of semitones and the dissonant intervals is completely subordinated to that of consonances, a point that shall not occupy us.
The following example shows that major thirds must be raised. Tuning a sequence of three just thirds - C-E, E-G#/Ab, Ab-C. Example:
the final C is short of the octave of the first. Octaves are defined as just, but the alignment of the final C with the first renders an Ab-C third so sharp that it is impossible to use. Example:
In order to render a reasonable interval, this alteration must be lessened by raising the Ab, yet only to a tolerable point rather than just. Now, the Ab, spelled G# sounds a third above E - again which is too sharp. Example:
It follows that the same operation must be made for the E, which is to say to raise it until the E-G# third is tolerable, and the three thirds comprising the just octave are equally sharp while tolerable. Example:
The proper degree of falseness for major thirds is arrived at by these means, which imparts the best possible temperament.
It can be demonstrated through similar reasoning that the minor thirds must be flat, by tuning a sequence of four just minor thirds - C-Eb, Eb-Gb/F#, F#-A, and A-C. Example:
The final C is sharp of the octave of the first, but lowering it to just leaves a very flat A-C minor third. Example:
The A must be lowered in turn, flatter than just but tolerable, which flattens the F#-A third. Example:
This necessitates lowering the F#, as well as the Eb to the end that the four minor thirds making up this just octave each are equally flat and supportable. Example:
It is necessary to investigate tempered fifths, as mentioned above which must be flat. Tuning an ascending sequence of five just fifths - C-G, G-D, D-A, A-E - Example:
the E two just octaves below the E of the final fifth (through tuning which does not change its quality) renders a major third with the initial C, yet one not only sharper than a just third but also sharper than the demonstrated tempered thirds. Example:
This excess must be distributed between all four of the fifths by flattening the higher note of each interval to the point where the C-E major third is sharp to the same degree as any one of the three tempered to the octave. Example:
Beginning with this final E, in the same manner one flattens the four fifths E-B, B-F#, F#-C#, C#-G#, which then is tuned two octaves below to obtain a supportable E-G# major third which resembles the C-E third; one also flattens the G#-D#, D#-A#, A#-E#, E#-B# fifths, the final which is tuned two octaves below to have the G#-B# third, or Ab-C sharp to the same degree as the two others, and that this C, the higher note of the third third and the twelfth fifth is a just octave above the C from which we began.
In the same manner, if one tunes a sequence of descending fifths, A-D, D-G, G-C, C-F, Example:
one finds the F of the final fifth being raised two octaves will be, with the initial A, a major third F-A, far too sharp and easily resembling the discordance of the C-E third produced by the former four ascending just fifths. Example:
Therefore, it is necessary conveniently to temper the F-A third to render it similar to one of the three in the octave by flattening the descending fifths, not by lowering the higher notes but on the contrary by raising the lower notes of each of the fifths. One raises the D of the first closer to A, the G of the second to D, the C of the third to G, the F of the fourth to C, in the manner that they all are equally false, and one obtains an F-A third which is conveniently tempered. Example:
Starting from this last F, one flattens equally, by raising the lower note, the four other descending fifths, F-Bb, Bb-Eb, Eb-Ab, Ab-Db, and this Db is retuned two octave higher to obtain a major third Db-F resembling the first third F-A; one always continues flattening, by raising the lower note the four final descending fifths, Db-Gb, Gb-Cb, Cb-Fb, Fb-Bbb, whereby the Bbb obviously is tuned two octaves higher for the Bbb-Db (or A-C#) third sharp to the same degree as its two precedents, and the A, the lower pitch of this third third and the twelfth descending fifth makes a just octave to the point of departure.
The alteration of the major third, of the minor third and of the fifth has been comprised, one concedes the octaves have remained just, and their inversions are tempered contrarily, as one has seen. I repeat this in finishing this article: the major third being sharp, the minor sixth will be flat: the minor third being flat, the major sixth will be sharp; and the fifth being flat, the fourth will be sharp, end vice versa. I engage students, after having read this article to exercise their ear to appreciate the convenient alteration of the major third, fifths and fourths before passing onward.
The ear being used to appreciating tempered consonances, it is necessary to exercise one's perception of a perfect major chord, conveniently tempered, and compare it with the just chord. After having tuned the perfect just chord C-E-G, one will raise the C in the manner of tempering the C-G fifth, after which one will raise the E in the manner of rendering the C-E third sharp to the convenient degree; then one will strike the entire C-E-G chord which will be tolerable and hardly false enough to shock the ear. Example:
Next, one will tune the lower octave of G, and one will obtain the chord of tempered fourth and sixth, which is to say in which the G-C fourth and the C-E third are both sharp, chords which are very useful in facilitating the execution of a partition, its nature will permit an appreciation with more precision than the other chords the degree of alteration. Example:
One will tune equally the octave above C, which with E-G will give the sixth chord E-G-C conveniently tempered. Example:
One will repeat the operation in the other useful keys until the ear is familiar with their alteration.