Diagram introduced by Morley to explain the early hexachordal system, used by singers in the middle ages as a means to sight-read and learn melodies.
Author: Thomas Morley
Title: A plaine and easie introduction to practicall musicke, 2nd ed. (London, 1608)
Shelfmark: D.2.30
D.2.30
![Entitled simply <i>Harmonics</i>, ideas from this treatise by the Greek astronomer Claudius Ptolemy (100–c.170 AD) played an essential part in medieval and Renaissance education. This important translation from Greek into Latin was undertaken by the famous mathematician and onetime Queens’ Fellow, John Wallis (1616-1703). The illustrated frontispiece shows Ptolemy, as conceived in the seventeenth century, with his monochord. (An instrument used in antiquity and the middle ages to measure musical intervals in terms of ratios, and thus to quantify music in terms of numbers.) Although Ptolemy agreed with Pythagoreans that musical structures should be analysed intellectually through numbers, he also argued that such rationally derived systems must be assessed by the sense of hearing. His interest in practical music makes this treatise all the more significant as a rare insight into the practical music and instruments of ancient Greece. Like the Pythagoreans, Ptolemy related his harmonic analyses to the universal structures of all perfect beings, the soul and the heavens. <br>
With the development of harmony and music in parts (polyphony), Ptolemy became of particular interest in the Renaissance. This was because, whereas the Pythagorean scale had served well for the singing of medieval chants, it proved less suitable for polyphony due to the harshness of its thirds and sixths. From the sixteenth century, Ptolemy’s ‘syntonic scale’, with its mathematically perfect thirds and sixths, better reflected musical practice and provided the means for theorists seeking to use music as a foundation upon which to speculate about nature and the world.
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<b>Author:</b> Claudius Ptolemy<br>
<b>Title:</b><i> Harmonicorum libri tres</i> [Harmonics] (Oxford, 1682)
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<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586145160003606"> H.5.33</a>](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-Rfwshjm/0/73e2cbde/Ti/H-005-033-005-Ti.jpg)
![Illustrated frontispiece showing Ptolemy as conceived in the seventeenth century, with his monochord. <br><br>
<b>Author:</b> Claudius Ptolemy<br>
<b>Title:</b><i> Harmonicorum libri tres</i> [Harmonics] (Oxford, 1682)
<br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586145160003606"> H.5.33</a>](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-2KHHGPF/0/5406b4ed/Ti/H-005-033-002-Ti.jpg)
![Woodcut illustration of a monochord, an instrument used in antiquity and the middle ages to measure musical intervals in terms of ratios, and thus to quantify music in terms of numbers.<br><br>
<b>Author:</b> Claudius Ptolemy<br>
<b>Title:</b><i> Harmonicorum libri tres</i> [Harmonics] (Oxford, 1682)
<br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586145160003606"> H.5.33</a>](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-Dbs7Dmd/0/631fc207/Ti/H-005-033-006-Ti.jpg)
![In an appendix to his translation of Ptolemy's <i>Harmonics</i>, Wallis set out his own thoughts on music and mathematics. It includes this diagram that seeks to relate the ancient Greek scale with the more recent hexachordal system and the modern scale.<br><br>
<b>Author:</b> Claudius Ptolemy<br>
<b>Title:</b><i> Harmonicorum libri tres</i> [Harmonics] (Oxford, 1682)
<br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586145160003606"> H.5.33</a>](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-5Gmk6F9/0/383ce20a/Ti/H-005-033-008-Ti.jpg)
![Descartes’ <i>Compendium Musicae</i> is one of many books originally from the collection of Queens’ member and ‘Cambridge Platonist’, John Smith (1618–52), that invoke music as a kind of philosophy through which to explore nature. As one of the earliest in Cambridge to have read the works of the great French philosopher Descartes, Smith was an early proponent of the new ways of thinking now associated with Descartes and the seventeenth-century ‘Scientific Revolution’. Smith’s understanding of divine musical harmony as a means to resolve inner and outer conflict in religion and society is reflected in his library’s preponderance of metaphysically ambitious works that adopt music as an organising principle. One such was Descartes’ <i>Musicae</i> compendium. Written in 1618 when Descartes was twenty-two the <i>Compendium</i> represents the young philosopher’s first attempts at articulating modes of thought that later came to great prominence in the form of his mechanistic philosophy. <br><br>
<b>Author:</b> René Descartes<br>
<b>Title:</b><i>Musicae compendium </i> [Compendium of music] (Utrecht, 1650) <br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586146700003606"> D.20.52</a><br>
<b>Provenance:</b> Bequeathed to Queens’ College by John Smith](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-WDNKnXJ/1/38c85347/Ti/D-020-052-012-Ti.jpg)
![Descartes’ approach to music theory was indebted to the Pythagorean discovery that musical consonances and scales are the product of simple whole-number ratios. Unlike Pythagoreans, however, Descartes did not consider the number ratios themselves to be the mystical cause of musical consonance and harmony. For him the basis of music was in sound rather than in numbers. Although Descartes continued to examine musical intervals in terms of ratios (as Pythagoreans had done), the numbers became merely a description whilst the string lengths they represented became the true foundation of sound. In crucial ways this conceptual shift informs similar mechanistic explanations he would propose for the entire cosmos later in the century.
<br><br>
<b>Author:</b> René Descartes<br>
<b>Title:</b><i>Musicae compendium </i> [Compendium of music] (Utrecht, 1650) <br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586146700003606"> D.20.52</a><br>
<b>Provenance:</b> Bequeathed to Queens’ College by John Smith](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-8K7NW9T/0/f4c8e575/Ti/D-020-052-001-Ti.jpg)
![Descartes' table of consonances. <br><br>
<b>Author:</b> René Descartes<br>
<b>Title:</b><i>Musicae compendium </i> [Compendium of music] (Utrecht, 1650) <br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586146700003606"> D.20.52</a><br>
<b>Provenance:</b> Bequeathed to Queens’ College by John Smith](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-qrBBR7g/0/1f7cbcf1/Ti/D-020-052-010-Ti.jpg)
![Descartes' demonstration of how simple consonances are derived from the octave.<br><br>
<b>Author:</b> René Descartes<br>
<b>Title:</b><i>Musicae compendium </i> [Compendium of music] (Utrecht, 1650) <br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586146700003606"> D.20.52</a><br>
<b>Provenance:</b> Bequeathed to Queens’ College by John Smith](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-P8fKxr2/0/f994999d/Ti/D-020-052-002-Ti.jpg)
![Circular pitch diagrams devised by Descartes to demonstrate the division of an octave into consonances: as pitch ascends (clockwise), the numbers around the circumference of the circle measure the length of a piece of string needed to sound each note, thereby revealing mathematical relationships between the different intervals. <br><br>
<b>Author:</b> René Descartes<br>
<b>Title:</b><i> Musicae compendium </i> [Compendium of music] (Utrecht, 1650) <br>
<b>Shelfmark:</b> <a href="http://idiscover.lib.cam.ac.uk/permalink/f/1nnjft8/44CAM_ALMA21586146700003606"> D.20.52</a><br>
<b>Provenance:</b> Bequeathed to Queens’ College by John Smith](https://photos.smugmug.com/Exhibitions/Heavenly-Harmony/Civil-War-Cambridge-Platonism-Gallery/i-ZjrZCTP/0/b50591cb/Ti/D-020-052-011-Ti.jpg)