Means musical ratios and the gift of the Muses

Use in the book

Two kinds of mean are described at 991a. One exceeds the lesser extreme by exactly as much as it falls short of the greater. This is the arithmetic mean. The other exceeds and is exceeded by the same fraction of the respective extremes. This is the harmonic mean. The Greek describes their relations; it does not print those two technical English labels as names.

The extremes given are six and twelve. The arithmetic mean is nine: 9−6 equals 12−9, each difference being three. The harmonic mean is eight: 8−6 is two, one third of six, while 12−8 is four, one third of twelve. Thus eight is not the arithmetic mean and nine is not the harmonic mean. Equal absolute differences define the first; equal relative parts of the extremes define the second.

These terms provide the musical ratios. The pairs 9:6 and 12:8 both yield 3:2, the one-and-a-half ratio associated with the fifth. The pairs 8:6 and 12:9 both yield 4:3, the one-and-a-third ratio associated with the fourth. The endpoints themselves are in the double ratio 12:6. The source's explanatory note works out this arithmetic and compares the Timaeus. Eight and nine as the inserted means are results of that explanation; the Greek sentence explicitly names six and twelve, not a displayed sequence “6, 8, 9, 12.”

The account assigns humankind a concordant and commensurate use for the play of rhythm and harmony, a gift of the Muses' happy dance (991b). Play here is beneficial and divinely grounded. This qualifies any reading of the earlier dismissal of imitative arts as an absolute condemnation of all musical activity. The earlier criticism denied that serious effort in imitation by itself makes its practitioner wise; the present passage studies the numerical relations that help prepare wisdom.

Music thereby mediates between abstract ratios and human practice without exhausting their significance. The curriculum passes from these relations to the divine visible order. Its final demand includes every combination of harmony among the things to be grasped as one. The numerical account must therefore be learned as part of a connected structure, not merely memorized as a set of useful tuning proportions.

Places to read

1. 991aἴσως δὲ τοῦ ἐλάττονος πλέον ἔλαττόν τε τοῦ μείζονος — Equal differences identify the arithmetic mean
2. 991aτῷ αὐτῷ μέρει τῶν ἄκρων αὐτῶν — Equal fractional parts of the extremes identify the harmonic mean
3. 991a–b2ἐν μέσῳ δὲ τοῦ ἓξ πρὸς τὰ δώδεκα — Six and twelve are the stated extremes
4. 991bτό τε ἡμιόλιον καὶ ἐπίτριτον — The one-and-a-half and one-and-a-third relations
5. 991bσύμφωνον χρείαν καὶ σύμμετρον — Concordant and commensurate human use
6. 991bπαιδιᾶς ῥυθμοῦ τε καὶ ἁρμονίας χάριν — The purpose includes the play of rhythm and harmony
7. 991bεὐδαίμονι χορείᾳ Μουσῶν δεδομένη — The gift comes through the Muses' happy dance

Translation in this edition

“One-and-a-half” and “one-and-a-third” keep the Greek ratio names recognizable, while the note gives fifth and fourth and the arithmetic of the means. The edition's “commensurate” at 991b does not assert that the incommensurable side-lengths relevant to geometry at 990d have become commensurable. The same broad proportional vocabulary must be interpreted in each construction.

Mean below the greater — ὑπερέχεσθαι (hyperechesthai)

ὑπερέχεσθαι (hyperechesthai)

Mean below the greater. The complementary passive describes the greater endpoint's excess over the mean, needed to distinguish the harmonic relation from equal differences. Attested: ὑπερεχόμενον (991a). Means musical ratios and the gift of the Muses.

Proportionate musical relation — σύμμετρος (symmetros)

σύμμετρος (symmetros)

Proportionate musical relation. Commensurate qualifies the use bestowed through musical proportions; it does not reverse the geometry note's incommensurability of certain side-lengths. Attested: σύμμετρον (991b). Means musical ratios and the gift of the Muses.

Musical usability — σύμφωνος (symphōnos)

σύμφωνος (symphōnos)

Musical usability. The proportions assign human beings concordant use in rhythm and harmony, linking numerical structure to musical practice. Attested: σύμφωνον (991b). Means musical ratios and the gift of the Muses.

Mean above the lesser — ὑπερέχειν (hyperechein)

ὑπερέχειν (hyperechein)

Mean above the lesser. The middle term exceeds the lesser extreme while being exceeded by the greater in the same fractional relation. Attested: ὑπερέχον (991a). Means musical ratios and the gift of the Muses.

Endpoints of a mean relation — ἄκρον (akron)

ἄκρον (akron)

Endpoints of a mean relation. The extremes are the terms whose equal fractions specify the harmonic-mean relation; six and twelve supply the example. Attested: ἄκρων (991a). Means musical ratios and the gift of the Muses.

Mathematical musical order — ἁρμονία (harmonia)

ἁρμονία (harmonia)

Mathematical musical order. Harmony is the musical field in which the ratios are useful and later one of the structures integrated into the unified studies. Attested: ἁρμονίας (991b). Means musical ratios and the gift of the Muses.

Proportional mean — μέσος (mesos)

μέσος (mesos)

Proportional mean. The mean between doubled extremes is explained through equal differences and equal fractional parts, corresponding to arithmetic and harmonic means. Attested: μέσον (991a). Means musical ratios and the gift of the Muses.

Divine giver of musical relations — μοῦσα (mousa)

μοῦσα (mousa)

Divine giver of musical relations. The Muses' happy dance gives the concordant and commensurate use; no individual Muse is named. Attested: Μουσῶν (991b). Means musical ratios and the gift of the Muses.

Ratio of the fifth — ἡμιόλιος (hēmiolios)

ἡμιόλιος (hēmiolios)

Ratio of the fifth. The 3:2 relation occurs as 9:6 and 12:8 in the note's worked example; the wording names the ratio rather than those inserted terms. Attested: ἡμιόλιον (991b). Means musical ratios and the gift of the Muses.

Ratio of the fourth — ἐπίτριτος (epitritos)

ἐπίτριτος (epitritos)

Ratio of the fourth. The 4:3 relation occurs as 8:6 and 12:9 in the note's example and is distinguished from the 3:2 fifth. Attested: ἐπίτριτον (991b). Means musical ratios and the gift of the Muses.

Relative share defining a mean — μέρος (meros)

μέρος (meros)

Relative share defining a mean. The same part of each extreme means equal fractions of different endpoints, not equal absolute differences. Attested: μέρει (991a). Means musical ratios and the gift of the Muses.

Beneficent musical activity — παιδιά (paidia)

παιδιά (paidia)

Beneficent musical activity. Rhythmic and harmonic play receives the numerical gift, showing that the earlier critique of imitation does not condemn all play. Attested: παιδιᾶς (991b). Means musical ratios and the gift of the Muses.

Ordered musical play — ῥυθμός (rhythmos)

ῥυθμός (rhythmos)

Ordered musical play. Rhythm with harmony is the human practice served by the numerical proportions and the Muses' gift. Attested: ῥυθμοῦ (991b). Means musical ratios and the gift of the Muses.

Lower extreme — ἕξ (hex)

ἕξ (hex)

Lower extreme. Six is explicitly named as the lower endpoint whose double is twelve; the inserted eight and nine are calculated in the note. Attested: ἓξ (991a). Means musical ratios and the gift of the Muses.

Upper extreme — δώδεκα (dōdeka)

δώδεκα (dōdeka)

Upper extreme. Twelve is the stated doubled endpoint from which arithmetic and harmonic means yield the musical relations. Attested: δώδεκα (991b). Means musical ratios and the gift of the Muses.

Benefit of mathematical music — χρεία (chreia)

χρεία (chreia)

Benefit of mathematical music. The concordant and commensurate use assigned to people is for rhythmic and harmonic play, a positive role distinct from mere practical sufficiency for wisdom. Attested: χρείαν (991b). Means musical ratios and the gift of the Muses.

Equal differences around the arithmetic mean (contextual phrase)

ἴσως δὲ τοῦ ἐλάττονος πλέον ἔλαττόν τε τοῦ μείζονος (991a)

The mean exceeds the smaller by as much as it falls short of the larger. Between six and twelve it is nine: each difference is three. Equal absolute differences distinguish this from the following harmonic construction. Means musical ratios and the gift of the Muses.

Equal fractions around the harmonic mean (contextual phrase)

τῷ αὐτῷ μέρει τῶν ἄκρων αὐτῶν (991a)

The same part refers to the same fraction of each endpoint, not the same absolute difference. Eight exceeds six by one third of six and falls short of twelve by one third of twelve. This identifies the harmonic mean, while nine is the arithmetic mean. Means musical ratios and the gift of the Muses.

Six and twelve as the named extremes (contextual phrase)

ἐν μέσῳ δὲ τοῦ ἓξ πρὸς τὰ δώδεκα (991a–b2)

The dialogue explicitly names six and twelve. Eight and nine are the means calculated in the note, not additional numerals printed in this sentence. The resulting ratios give the fourth and fifth as explained in the full article. Means musical ratios and the gift of the Muses.