Francisco de Salinas, De musica libri septem (Salamanca, 1577)
The Salamanca professor of music who measured tuning instead of arguing about it. His 1577 folio sets out three temperaments of keyboard instruments as exact fractions of the comma, judges the third — the one that leaves the major third in its true form — easier to understand and better suited to instruments, compares two of them by dividing the comma into twenty-eight parts, and reports tuning an organ half tempered and half pure so that the difference could be settled by ear.
Why this wins its question: The standing account of Salinas is a sentence saying that he described third-comma, quarter-comma and two-seventh-comma meantone — the fractions restated in modern terminology he never used, with no page, and with the two things that make the book an experimental document rather than an arithmetical one left out entirely. Both are quoted here from the Salamanca sheets with their printed page numbers: the organ he had tuned with three ranks tempered and three pure and the Cs held in unison, so that legitimate and spurious could be told apart on one instrument, and the two extra keys per octave he asks organ builders for, with his warning that once they exist the fault will be the organist's and not the instrument's. Also here, and also absent from the popular account, is his reason for preferring the third temperament: not that it sounds better, but that building it needs only Euclid VI.9 and no mesolabe.
Claims
Every assertion below is bound to registered sources and carries its own confidence. Weight them; do not treat the page as uniformly authoritative.
The title page reads "FRANCISCI SALINAE BVRGENSIS ABBATIS SANCTI PANCRATII de Rocca Scalegna in regno Neapolitano, & in Academia Salmanticensi Musicae Professoris, de Musica libri Septem, in quibus eius doctrinae veritas tam quae ad Harmoniam, quàm quae ad Rhythmum pertinet, iuxta sensus ac rationis iudicium ostenditur, & demonstratur", with "CVM duplici Indice Capitum & Rerum", the imprint "SALMANTICAE Excudebat Mathias Gastius. M.D.LXXVII" and the tasa "Esta tassado en seyscientos marauedis". The programme is in the title: the truth of the doctrine is to be shown by the judgement of sense and of reason together.
The royal privilege bound into the front matter is in Castilian and dated two years before the imprint. It recites that "el maestro Francisco de Salinas, cathedratico de propriedad de la facultad de Musica, en la uniuersidad de Salamanca" had made and composed "un libro de Musica especulatiua, el qual era muy util y prouechoso", and had asked for licence for thirty years; the Council granted ten, under penalty of thirty thousand maravedís divided between the accuser, the royal treasury and the sentencing judge. It closes "Fecho en Sant Lorenço el real, a cinco dias del mes de Octubre, de mil y quinientos y setenta y cinco años. YO EL REY. Por mandado de su Magestad. Antonio de Erasso."
Book III treats the tuning of imperfect — that is, fixed-pitch — instruments as a quantitative problem, and distinguishes the systems by the fraction of the comma involved. On printed page 154 the first temperament is the one in which all the consonances except the semiditone and the major hexachord "aequaliter minuuntur, & augentur tertia Commatis parte"; in the second they are not all altered equally but "partim vna septima, partim duabus" — the ditone and semiditone diminished by one seventh, the diapente by two, the hexachords augmented by one, the diatessaron by two.
The third temperament is introduced on the same page with a rubric that states its advantage as practical rather than theoretical: "De tertio instrumentorum imperfectorum temperamento, quod multò quàm duo superiora, & intellectu facilius, & instrumentis accommodatius esse videtur. CAP. XXII." Its construction is given in one sentence: "Minuitur ergo in hoc temperamento Tonus maior Commatis dimidio, & tantundem minor augetur, vt vnus sonus fiat medius proportionalis inter duos extremos Ditoni, & ipsum Comma in duo aequè proportionalia diuisum etiam inueniatur. Ex qua positione sequitur, Ditonum in sua vera, ac legitima forma relinqui" — the major third is left untouched and the comma is absorbed by the two tones.
The reason he prefers it is that it can actually be built. The third temperament, he writes, "non eget auxilio Mesolabij ad faciendas Commatis partitiones in duo aequè proportionalia, vt duo superiora egere demonstrata sunt: sed satis est ad hoc faciendum nona sexti Euclidis" — it needs no mesolabe to divide the comma into two proportional parts, Euclid VI.9 being enough. A tuning criterion is being settled by what a craftsman can construct with a ruler and compasses.
The comparison of two temperaments is carried out arithmetically, and the method is stated before the result. Printed page 157: "Quod vt exactiùs fieri possit, primùm Comma diuidendum est in 28 partes, quo suppetat nobis dimidia pars, & quarta & septima, & decimaquarta & vigesimaoctaua, quae sunt omnes eius partes aliquotae … quoniam his partibus nobis opus est ad collationem vtriusque temperamenti faciendam." He then runs the comparison interval by interval — diapente, diatessaron, ditone, semiditone, tone, both semitones, diesis — and states which is larger and by what fraction of the comma.
The book records an experiment on a real instrument, and its motive is an ear. Printed page 166: an earlier arrangement was abandoned because its imperfection in the fourth and fifth ranks "plus nimio non meas solùm, sed omnium etiam aures offendebat"; so, he writes, "in tribus primis ordinibus accommodaui temperamentum instrumenti participati, & in tribus supremis perfecti, sonis C litera signatis fixis manentibus, & ad vnisonantiam temperatis, quo ex vtriusque collatione posset (vt Ptolemaeus ait) legitimum ab spurio, & imperfectum à perfecto discerni". Half the ranks tempered, half pure, the Cs held fixed and tuned in unison so that the two systems could be compared on the same instrument.
He then asks instrument makers for a change to the keyboard itself. On the same page he "omnes instrumentorum artifices vehementer exhortor … saltem in duabus Diapason ex quatuor, quas Organa continent, intermedijs, à C magno ad c paruum, & ab eo ad c geminatum duos sonos in qualibet addere": in the two middle octaves of the organ's four, two extra pitches each — one an enharmonic D sharpened between D diatonic and E chromatic, the other a flat between G chromatic and A diatonic — so that organists may answer the choir when the first mode falls on F or E and the fourth on G, which without them they manage only with the greatest difficulty or never. His closing line puts the burden where the instrument is fixed: "quod nunc instrumentorum defectui meritò ascribitur, tunc ipsius organistae culpae tribuetur".
The same page refuses to make one measure serve for every instrument. In a participated instrument, he argues, no circle can be closed through the same consonances, because the diesis is no common measure of any interval or consonance "vt isti volunt: qui dicunt, Tonum habere quinque, Semiditonum 8, Ditonum 10, Diatessaron 13, Diapente 18, & denique Diapason 31" — the thirty-one-fold division of the octave, which he holds shown to be alien to all harmonic reason. But for the class of instruments "quae Lyrae, aut Violae vulgò vocantur" — the fretted ones — the diesis is to be removed altogether, and there participation is rightly done; the chapter that follows, cap. XXVIII, is devoted to them. Keyboard and fret are given different tunings on principle.
What the four pages read establish and what they do not are different things. They establish, in the author's own words and with the printed page numbers, the three fractional temperaments, the constructional argument for the third, the twenty-eight-part comparison, the split-rank organ and the request to instrument makers. They carry no priority claim against any other theorist, and none is made here: the "isti" whose thirty-one-fold division is rejected are not named on the page read, and a search of this volume's text layer for "Zarlinus" returns nothing, in a layer already shown to be partial. Nothing in these pages bears on how widely his temperaments were adopted, on the author's life, or on the Americas.