The Geometry of Musical Logarithms

Pythagoras developed an eight-note musical scale in which ratios of small whole numbers were used to define the length of the strings. While Pythagoras committed nothing to writing, later Pythagoreans wrote a good deal about these matters, and these principles were used to tune musical instruments until fairly modern times. In the seventeenth century, equal-tempered chromatic scales were introduced in which the frequencies of successive notes differ by a common ratio. In the early days of Rosicrucian activity in America, H. Spencer Lewis published a system of " natural harmonics " in which the notes of the chromatic scale were correlated with the colors of visible light and certain vowel sounds. Dr. Lewis used a value of middle C (C4), 273 Hz, that was based on a specific wavelength of green light in the middle of the visible spectrum. Modern scales, on the other hand, use values such as 256 Hz and 262 Hz for middle C. As the physiological and psychic effects attributed to the musical notes may be a function of the scale used, this paper examines and compares different musical scales. Based on this examination, it is believed that no one frequency for middle C is best for all occasions. Résumé Pythagore a développé une gamme musicale de huit notes (accord pythagoricien) qui utilisait les rapports de petits nombres entiers pour définir l'accord. Tandis que Pythagore ne cocha guère ses théories par écrit, des pythagoriciens ont ultérieurement écrit de nombreux traités sur cet argument et ses principes ont été utilisés pour accorder les instruments de musique jusqu'à la fin du Moyen-Age. Le XVIIe siècle vit l'introduction des gammes tempérées (tempérament égal), dans lesquelles les fréquences des notes successives sont reliées par une relation commune (intervalles égaux). Dans les premiers temps d'activité de la Rose-Croix en Amérique, le Docteur H. Spencer Lewis publia un système d' « harmoniques naturels », dans lequel les notes de la gamme chromatique sont en corrélation avec les couleurs de la lumière visible et avec certaines voyelles. Le Dr. Lewis utilisa à cet effet une valeur pour le do central de 273 Hz, qui correspond à une longueur d'onde spécifique de la lumière verte du milieu du spectre visible. Les gammes actuelles, en revanche, utilisent des valeurs d'environ 262 Hz. Du moment que les effets physiologiques et psychiques attribués aux notes musicales peuvent différer selon les fréquences utilisées, cet article examine et compare les différentes gammes musicales. On en déduit qu'aucune fréquence unique pour le do central n'est idoine pour toutes les circonstances.

Brepols Publishers eBooks, 2008

Lecture Notes in Computer Science, 2022

Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ratios, they determined geometric number sequences with triangular tables, and they treated the converse problem of dividing musical intervals equally. The collection of mathematical manuscripts Codex Basel F II 33 (ca. 1360) contains treatises by Nicolaus Oresme, Jordanus Nemorarius and others. Images in Nemorarius' treatise combine number triangles into complex spider webs and they display recursive algorithms. Oresme diagrams make use of irrational ratios. These little known images and their relationship to music theory are the focus of this paper.

Pan prim ofi calamos cera conjungere phires Instituit.-\ forming thereby an instrument, to which Isidore, bishop of Seville, gives the name of Pandorium, and others that of Syringa and which is frequently repre-sented in collections of antiquities.:}: ' from the second to the third was a tone ; and from * the third to the fourth was a diatessaron ; so that the ' first with the second, and the third with the fourth, ' contained a diatessaron ; the first with the third, * and the second with the fourth, a diapente or fifth.' Admitting all which, it is clear that the first and fourth strings must have constituted a diapason. It is to be observed that the above diagram is used by Boetius, and is adopted by Zarlino, Kircher, and many other writers ; * but that though the appli-cation of the letters C G F C in one edition of Boetius, is plainly intended to shew that the strings immediately below them were ' The number Seven,' says this learned author, has a wonderful property, for it neither begets nor is begotten, as the rest are, by any of the numbers witliin ten, wherefore philosophers resemble it to the ruler or governor of all things, who neither moves nor is moved. Philolaus the Pythagorean, no ignoble author, testifies thus, and writes that the eternal God is permanent, void of motion, similar to himself, and different from others ; and Boetius has a passage much to the same purpose. The idea of virginity had such a relation to the number Seven, that it was also named Pallas ; and the Py-thagoreans, initiated in her rites, compare the virgin Minerva to that number, seeing she was not born, but sprung from the head of Jupiter. God rested on the Seventh day, wherefore it is named Sabbath, a word signifying rest. The Seventh petition of the Lord's Prayer is, deliver us from evil ; because the number Seven denotes rest, and all evil being removed from man, he rests in good ; and farther, the seventh day or sabbath represents death, or the rest of the soul from worldly labours. In Seven days after Noah entered the ark the flood began : in the Apocalypse Seven trumpets are men-tioned : Job speaks of the visitation of six tribula-tions, which six succeeding days brought on him, but on the Seventh no harm could touch the just : God blessed only the Seventh day, wherefore the number Seven is attributed to the Holy Ghost, without whom there is no blessing. This St. John proves, when in the Apocalypse he calls the Seven horns and the Seven eyes the Seven spirits of God. The fever left the son of Regulas, according to St. John, at the Seventh hour. Elisha breathed Seven times on the dead man. Christ after his resurrection feasted with Seven disciples ; and Seven brothers were sent to baptise Cornelius. The Seven hairs of Sampson ; Seven golden candlesticks : and in Leviticus command was given to sprinkle the blood and oil Seven times. The Seven stars in the bear ; the Seven principal angels who rule the world under God, and have charge of the Seven planets, as namely, Horophiel the spirit of Saturn, Anael the spirit of Venus, Zachainel of Jupiter, Raphael of Mercury, Samael of Mars, Gabriel of the moon, and Michael the spirit of the sun. The moon changes its form Seven times, and completes its course in twenty-eight days, which is the sum of the number Seven, and all the numbers under it. Josephus writes that a certain river in Syria is dry for six days, and full on the Seventh. Farther, the great artist did not only dignify the heavens, but he also adorned with the number Seven his favourite creature man, who has seven inward parts, or bowels, stomach, heart, lungs, milt, liver, reins, and bladder ; and seven exterior, as head, back, belly, two hands, and two feet. There are seven objects of sight, as body, distance, figure, magnitude, colour, motion. and rest : and Seven species of colour, taking in the two extremes of white and Wack, viz., yellow, sky-blue, green, purple, and red. No one can without eating live after the Seventh day. Physicians reckon ten times Seven years to be the period of human life, which Hippocrates divides into Seven stages. The ancient lyre, used both by Orpheus and Amphion, had only Seven chords, answering, as it is said, to the Seven gates of Thebes. Every Seventh daughter, no son coming between, hath, by virtue of the number Seven as I imagine, a great power in easing the pains of child-birth : and every Seventh son, no daughter coming between, has the power of curing the scurvy and leprosy by the bare touch ; so that diseases, incurable by physicians, are curable by the virtue contained in the number Seven. A right-angled triangle is constituted of the sides three, four, five, but three and four contain the right angle, which is ' perfection itself, and therefore their sum seven, ' must as a number be most perfect. Every active ' body has three dimensions, length, breadth, and ' thickness, and these have four extremes, point, line, By such arguments as these do many of the musical writers endeavour to excite a mysterious reverence for that number which is confessedly the limits of a system, as far as it goes, perfect in its kind ; in answer to which it may be said, that this superstitious regard for certain numbers seems to be very deservedly ranked among those vulgar and common errors, which it is professedly the end of a very learned and justly celebrated publication of the last century to refute, wherein it is said, that ' with respect to any extraordinary power or secret But to return from this digression : the rudiments of the present greater musical system are discernible in that of a septenary, adjusted, as we are told, by Terpander, in the form above declared ; and as to the intervals of which it was constitivted, modern authors have not scrupled to assert that they were precisely the same as those contained in a double diatessaron, according to the present practice ; the consequence whereof must be, that each of the two tetrachords, of which the above system is supposed to have been formed, consisted of a hemitone and two tones ; which will be readily conceived by such as reflect, that in the passage either upwards or downwards from any given note to its fourth, in that progression which is most grateful to the ear, those intervals must necessarily occur. Persuaded of the truth of this supposition, succeeding musicians have ventured to apply the modern method of no-tation to the terms of the ancients, and are pretty well agreed that the term Mese answered to a, or la, in our scale. Taking this for granted, the system of Terpander \vill appear in the following form ; -But here it is necessary to observe, that though, as has been said, it was the practice with the ancients to give the grave tones the uppermost, and the more acute the lowermost place in their scale, f wliich they might very properly do, if, as there is the greatest rea-son to believe, their music was solitary, and they were stran2:ers to the art of combining sounds in con-sonance. Yet the moderns, immediately on the making that most important discovery, found it necessary to differ from them, and accordingly we now place the grave tones at the bottom, and the acute at the top of our scale ; | the consequence of this diversity has been, that whenever any of the modern authors have taken occasion to exhibit the whole or any part of the ancient Greek scale, they have done it in their own way, placing Hypate at the bottom of the diagram ; and this wall be the method we shall observe for the future. Great confusion has arisen among the writers on music, in respect to the order of the several additions to the system of Terpander. That it was perfected by Pythagoras will be related in due time ; but the eagerness of most authors to explain the improve-ments made by him, has betrayed them into the error of confounding the two systems together, whereby they have rendered their accounts unintelligible. Boetius has erred in this respect ; and Bontempi, a modern Italian, notwithstanding he professes to have followed the Greek writers, more particularly Nicomachus, has made the same mistake ; for in every one of the representations of the ' Pythagoras being in an intense thought whether ' he might invent any instrumental help to the ear, ' solid and infallible, such as the sight hath by a them these three concords, the diapason, the diapente, and the diatessaron ; but that which was between the diatessaron and the diapente he found to be a discord in itself, thoug!) otherwise useful for the making up of the greater of them, the diapente. Apprehending this came to him from God, as a most happy thing, he hastened into the shop, and by various trials finding the difference of the sounds to be according to the weight of the hammers, and not according to the force of those who struck, nor according to the fashion of the hammers, nor ac-cording to the turning of the iron which was in beating out : having taken exactly the weight of the hammers, he went straightway home, and to one beam fastened to the walls, cross from one corner of the room to the other, lest any difference might arise from thence, or be suspected to arise from the properties of several beams, tying four strings of the same substance, length, and twist, upon each of them he hung a several weight, fastening it at the lower end, and making the length of the strings altogether equal ; then striking the strings by two at a time interchangeably, he found out the afore-said concords, each in its own combination ; for that which was stretched by the greatest weight, in respect of that which was stretched by the least weight, he found to sound a Diapason. The greatest weight was of twelve pounds, the least of six ; thence he determined that the diapason did consist in double proportion, which the weights themselves did shew. Next he found that the greatest to the least but one, which was of eight pounds, sounded a Diapente ; whence he inferred this to consist in the...

The parallel between music and mathematics abounded in sixteenth-century studies. Referring to the concept of proportions, it was then assumed that theorists considered audible and visible proportions to be analogous. However, when approaching the treatises and focusing on how those processes of mensuration took place, we realise that there were differences concerning the very notion of quantifying. In this paper, we would like to identify the roles played by arithmetic and geometry in the musical tradition through the notion of quantification, departing from the definition of the musical interval given by the most important theorist of the sixteenth century, Gioseffo Zarlino, and identifying the core of musical debates within the realm of mathematics.