Global InformationPythagorean hammers information
According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd-century CE Enchiridion harmonices,[1] Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 8 and 9 pounds. Their ratios with hammer D were (12:8 = 3:2 = perfect fifth) and (12:9 = 4:3 = perfect fourth). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval (ⓘ).
| Hammer | Weight | Interval | Chisel | In C |
|---|---|---|---|---|
| A | 6 | ⓘ | ⓘ | C |
| B | 8 | ⓘ | ⓘ | G |
| C | 9 | ⓘ | ⓘ | F |
| D | 12 | ⓘ | ⓘ | C' |
The legend is, at least with respect to the hammers, demonstrably false. It is probably a Middle Eastern folk tale.[2] These proportions are indeed relevant to string length (e.g. that of a monochord) — using these founding intervals, it is possible to construct the chromatic scale and the basic seven-tone diatonic scale used in modern music, and Pythagoras might well have been influential in the discovery of these proportions (hence, sometimes referred to as Pythagorean tuning) — but the proportions do not have the same relationship to hammer weight and the tones produced by them.[3][4] However, hammer-driven chisels with equal cross-section, show an exact proportion between length or weight and Eigenfrequency.[5]
Earlier sources mention Pythagoras' interest in harmony and ratio. Xenocrates (4th century BCE), while not as far as we know mentioning the blacksmith story, described Pythagoras' interest in general terms: "Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned."[6] Whatever the details of the discovery of the relationship between music and ratio, it is regarded[7] as historically the first empirically secure mathematical description of a physical fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics as nature's modus operandi.[8] As Aristotle was later to write, "the Pythagoreans construct the whole universe out of numbers".[9] The Micrologus of Guido of Arezzo repeats the legend in Chapter XX.[10]
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Dimensions of four Pythagorean hammers.
- ^ Weiss, Piero, and Richard Taruskin, eds. Music in the Western World: A History in Documents. 2nd ed. N.p.: Thomson Schirmer, 1984. 3. ISBN 9780534585990.
- ^ Kenneth Sylvan Guthrie, David R. Fideler (1987). The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy, p.24. Red Wheel/Weiser. ISBN 9780933999510.
- ^ Christensen, Thomas, ed. The Cambridge history of Western music theory. Cambridge: Cambridge University Press, 2002. 143. ISBN 9780521623711.
- ^ Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism, p.375. ISBN 9780674539181. Cited in Christensen 2002, p.143.
- ^ Markus Bautsch: Über die pythagoreischen Wurzeln der gregorianischen Modi, English: About the Pythagorean roots of the Gregorian modes, Mater Dolorosa (Berlin-Lankwitz), March 2012, retrieved on 31 August 2017
- ^ Barker (2004). Andrew (ed.). Greek musical writings (1st pbk. ed.). Cambridge: Cambridge University Press. p. 30. ISBN 978-0-521-61697-3.
- ^ Lucas N.H. Bunt; Phillip S. Jones; Jack D. Bedient (1988). The historical roots of elementary mathematics (Reprint ed.). New York: Dover Publications. p. 72. ISBN 978-0-486-25563-7.
- ^ Christian, James (26 January 2011). Philosophy An Introduction to the Art of Wondering. Wadsworth Pub Co. p. 517. ISBN 978-1-111-29808-1.
- ^ Waterfield, transl. with commentary by Robin (2000). The first philosophers : the Presocratics and Sophists (1. publ. as an Oxford world's classics paperback ed.). Oxford: Oxford Univ. Press. p. 103. ISBN 978-0-19-282454-7.
the pythagoreans construct the whole universe out of numbers.
- ^ "GUIMICB_MBBR2784". chmtl.indiana.edu. Retrieved 2023-05-09.
Erant antiquitus instrumenta incerta. et canentium multitudo sed ceca. Nullus enim hominum uocum differentias. et simphoniae descriptionem poterat aliqua argumentatione colligere. neque posset unquam certum aliquid de hac arte cognoscere. nisi diuina pietas tandem quod sequitur suo nutu disponeret. Cum Phitagoras quidam magnus philosophus forte iter ageret. uentum est ad fabricam. in qua super unam incudem quinque mallei feriebant. Quorum suauem miratus concordiam philosophus accessit. primumque in manuum uarietate sperans [f.20r] uim soni ac modulationis existere. mutauit malleos. Quo facto. sua uis quemque secuta est. Subtracto itaque uno qui dissonus erat a ceteris. alios ponderauit. Mirumque in modum diuino nutu primus .XII. secundus .VIIII. tertius .VIII. quartus .VI. nescio quibus ponderibus appendebant. Cognouit itaque in numerorum proportione et collatione musicae uersari scientiam. Erat enim ea constitutio in quattuor malleis. quae modo est in quattuor litteris .A.D.E.a. Denique si A habeat .XII. et D VIIII. sintque quaternarii in pro passu. habebit .A. in .XII. ternarios .IIII. et D. in .VIIII. ternarios .III. Ecce diatessaron. Rursus cum habeat A .XII. si teneat E. VIII. quaternarios passus tres habebit .A. duos uero E. et diapente patet. Sint iterum .XII in A. et .VI. in alteram a. Senarius medietas est duodenarii. sicut et a acuta alterius A medietate colligitur. Adest ergo diapason. Cum igitur ipsa .a. ad D. diatessaron ad e diapente. alteri uero a diapason reddit. D quoque ad e tonum. ad utrumque A a diatessaron. et diapente sonat. Et E etiam ad D. tonum utrique A.a. diapente et diatessaron mandat. a uero acuta cum A diapason. cum D diapente. cum E diatessaron sonat. quae in supra dictis numeris curiosus perscrutator inuenit. Hinc enim incipiens Boetius panditor huius artis. multam miramque et difficillimam huius artis cum numerorum proportione concordiam demonstrauit. [f.20v] Quid plura? Per supra dictas species uoces ordinans. monocordum primus ille Phitagoras quattuor neruorum composuit. in quo quia non est lasciuia. sed diligenter aperta artis notitia. sapientibus in commune placuit. atque usque in hunc diem ars paulatim crescendo conualuit. ipso doctore semper humanas tenebras illustrante. cuius summa sapientia per cuncta uiget secula.