Dialogue of Galilei

Three rectangular diagrams illustrating musical interval proportions through nested lines and text.
The first diagram displays intervals based on the numbers 1, 2, 3, and 4, labeling "Unison" between 1 and 1, "Octave" between 1 and 2, "Fifth" between 1 and 2 and 3, "Twelfth" between 1 and 3, "Fourth" between 3 and 4, and "Fifteenth" (double octave) between 1 and 4.
The second diagram uses the numbers 4, 5, and 6 to show the "Major Third", "Fifteenth", "Seventeenth", "Minor Third", "Major Seventeenth", and "Nineteenth".
The third diagram uses numbers 3, 5, and 8 to show "Major Sixth", "Fifth", "Major Sixth", "Minor Sixth", "Major Sixth", and "Eleventh".

STR. I have understood the cause of everything very well; therefore, you may continue as you please.

BAR. Within the aforementioned terms, then, those who seek to persuade us have shown that the Diatonic Diatonic: A scale consisting of seven notes, moving primarily by whole tones and semitones. in which one composes and sings today is the Syntonic of Ptolemy, and that every singable interval is found therein. I want us to prove, using those same principles, that this particular species is by no means what they claim it to be, and that it consists of a greater number and variety of intervals than proposed. They have refused to mention these, not out of a lack of care, but because they saw that it did not suit their purpose and would hinder their designs.

We shall first reveal that in the Syntonic of Ptolemy, the string for D-la-sol-re The note D. for B-natural is sharper than that for B-flat; that from the natural strings of F-faut F and C-solfaut C to the same notes altered by this sign X A sharp sign.; and likewise from B-flat to B-natural, the interval is not the same as from E-la-mi E to its flat, or from G-solreut G to its sharp X. I say further, that between D-solre D and E-la-mi E, one does not find the same distance that exists between A-lamire A and B-natural; and thus between C-solfaut and D-solre, it is not the same interval found between G-solreut and A-lamire. That in ascending, the interval between D-solre and F-faut is not the same as between E-la-mi and G-solreut. Nor is such an interval between F-faut and C-solfaut altered by the sign X, or between A-lamire and E-lami. That from A-lamire to C-solfaut altered by this same sharp sign X is not the same space as from C-solfaut to E-lami. That from A-lamire to D-la-sol-re is not the same interval as from G-solreut to C-solfaut. That from the interval between F-faut and B-natural, considered as a Fourth and a minor Semitone, one cannot obtain a similar interval by joining together the Semidiapente Semidiapente: A diminished fifth. and the minor Semitone of the minor Tone, which is what most have understood until this day. That from G-solreut altered by the sharp sign X to E-lami is not the same interval as between E-lami and C-solfaut. That between F-faut and D-la-sol-re, and C-solfaut and A-lamire, it is not a Major Sixth. That between D-solre and C-solfaut, it is not a Minor Seventh. That the interval found between E-lami and D-la-sol-re, considered however in two Sesquiterzas Sesquiterza: The ratio 4:3, a perfect fourth., is not a similar interval. That between G-solreut and F-faut it is not even such an interval, considered however as a Fifth in the lower part and a Minor Third in the upper. That between D-solre and D-la-sol-re, and between C-solfaut and C-solfaut, it is not an Octave. That it is the same distance between B-flat and B-natural as for B-flat and B-natural Original: "b fa & h mi per b molle, che per h duro.". That the same distance is from B-natural to C-solfaut as from C-solfaut to B-flat altered by either of these signs ♮, X; but its proper sign is this: ♮. And finally, that each Tone does not have semitones of the same proportion and measure, but diverse ones.

STR. This lofty principle of yours represents to me, instead of a safe harbor, that part of the earth which under the southern pole is called unknown Original: "terra... incognita.". For the things you have just said are so new to my ears that I would despair of reaching the port if I did not have such a faithful Pilot as my guide.

BAR. Be of good heart, and let these few rocks not frighten you; we shall pass through them very safely in our little vessel. And to reassure you and make the way easier, here in practice is the example of everything that has been said; which I will go on explaining to you step by step with ease. Nor do I wish to leave off before telling you that in this second numbering, done with such diligence regarding the deformity that the first intervals have with the second, I did not mean to say that such intervals are not found among the strings of the Syntonic; but rather that they are not all its own proper and particular ones, just as the first ones were not; and that they are not taken into any consideration by modern practitioners. And further, I want to tell you that those they do have knowledge of are not actually sung in the shown proportions, as you will see hereafter. I will prove to you then, according to what I have proposed, that the two notes placed here at the bottom are not unisons.

A The signature "A" indicates the first gathering of the printed book.