In the Republic, Socrates names the musical modes and he correlates them with the different constitutions of state and soul (399a–c). Since Socrates ranks the different constitutions (580a) and he correlates them with his categories of modes (398e–399d), he ranks his categories of musical modes. Glaucon gives the relations between the Socratic categories of modes and the modes as they are commonly called (398e–399c). While the constitutions are ranked explicitly, the correlations among the Socratic categories, the common categories, and the constitutions of state and soul are not explicit. Interpreters leave most of these correlations tacit and implicit.
In what follows, I argue that Socrates says enough for us to chart the correlations and to rank the modes. I begin with the ranking of the constitutions. Then, I correlate the constitutions with the Socratic modes and with the common modes. These correlations show that two commonly named modes can fall within the same Socratic mode. So, two common modes can receive the same ranking. To refine the ranking of the modes, I consider the internal anatomy of the common modes. I offer a set of Socratic preferences concerning the modes. These preferences function to generate a complete ranking of the common modes. I conclude by charting the correlations and the rankings.
Socrates distinguishes the different constitutions of state and soul, and he ranks them from best to worst (580–588). In particular, he holds that there are three parts of the soul, reason, spirit, and appetite (441a–b). He holds that there are three parts of society, the rulers, the auxiliaries, and the tradesmen. The three parts of the soul fit with the three parts of the state, reason with the rulers, spirit with the auxiliaries, appetite with the tradesmen (441c). Each constitution of soul and state sets one part as ruler of the others. Concerning the constitutions of soul, reason rules in the aristocrat. Spirit rules in the timocrat. Socrates divides the appetite into three parts, the necessary appetites, the unnecessary appetites (558d–559a), and the lawless appetites (572b). The necessary appetites rule in the oligarch. The unnecessary appetites rule in the democrat, and the lawless appetite’s rule in the tyrant. To put things together:
|Constitution of State and Soul||Ruling part of the State||Ruling Part of the Soul||Primary Good, or Primary Virtue|
|Timocratic||The military||Spirit||Honor and Courage|
|Oligarchic||The wealthy||Necessary appetites||Wealth|
|Democratic||The majority||Unnecessary appetites||Freedom|
|Tyrannical||The tyrant||Base appetites||Lawlessness|
F. Mountford offers the most detailed account of the musical modes named in the Republic. He notices that Socrates correlates the different musical modes with the different constitutions of state and soul. In this passage, Mountford correlates some constitutions of soul with the common modes:
“As the Dorian scale resulted from the best tuning or ‘fitting’ of the lyre-strings (ἁρμονíα), so the best man was the result of the best fitting of the constituents of this soul. A man might be amatory, warlike or effeminate, but it was not his soul as a whole which was more of less highly strung; the difference lay in the altered relations of the parts of the soul to one another. In the same way a scale might dispose a man to courage, like the Dorian, or have a relaxing effect like the Lydian or Ionian. The intervals were the same in each case but the alteration in their collocation gave rise to the different effects.”
Mountford tells us that Socrates ranks the Dorian mode as the best mode and that this mode is the courageous and “warlike” mode. Mountford holds that the Dorian mode is the warrior’s mode. He describes other modes as “amatory,” or “effeminate.” Since Socrates takes the constitutions of soul as analogous to the constitutions of state, the correlations concerning the constitutions of soul also apply to the constitutions of state. Mountford does not tell us with which constitutions of state and soul the remaining musical modes correlate.
Consider Socrates’ description of the modes that are accepted in the kalipolis. He names two Socratic categories, the voluntary (ἑκоὐσιον) and the enforced (βἰαιον). Concerning the voluntary modes he says:
“Leave me also another mode, that of someone engaged in a peaceful, unforced, voluntary (ἑκоὐσιον) action persuading someone of asking a favor of a god in prayer or of a human being through teaching and exhortation, or on the other hand, of someone submitting to the supplications of another who is teaching him in trying to get him to change his mind, and who, in all these circumstances, is acting with moderation and self-control, not with arrogance but with understanding (σωϕρόνωϛ), and is content with the outcome.” (399b–c)
This passage describes a person of “understanding.” He beseeches gods and men through teaching and exhortation. He submits to teaching and exhortation of others.
Socrates describes the aristocratic constitutions of state and soul with the term “understanding” (σωϕροσὐνην) (430c, 431e).” Subsequently, Socrates describes the Aristocratic constitution of soul with the same term, “He binds together those parts and any other there may be in between, and from having been many things he becomes entirely one, moderate (σὠϕρονα) and harmonious” (443d). The term “σὠϕρονα” translated as “moderate,” is cognate with the term (σωϕρόνωϛ) that was translated as “understanding” at (399c). Socrates description of the voluntary modes, therefore, correlates with his description of the aristocratic constitutions of state and soul.
The enforced mode is the only other mode allowed in the kalipolis. Concerning it, Socrates says:
“Just leave me the mode that would suitably imitate the tone and rhythm of a courageous (ἀνδρεíων) person who is active in battle or doing other enforced (βἰαιον) deeds, or who is failing and facing wounds, death, or some other misfortune, and who, in all these circumstances, is fighting off his fate steadily and with self-control.” (399a)
This passage describes vividly a person suffering failure in battle, but fighting courageously. This person faces wounds, or death, steadily and with self-control. The enforced mode is the warrior’s mode.
Socrates describes the timocratic state as “inclined toward . . . people, who are more naturally suited for war than peace” (547e). The timocracy will “value the tricks and stratagems of war and spend all its time making war. Aren’t most of these qualities peculiar to it?” (547e). The timocratic soul loves victory and honor. His most prized “abilities and exploits” concern “warfare and warlike activities” (549a). With the timocratic love of war comes its high esteem for courage (ἀνδρεíων). The timocrat is not “educated by persuasion but by force (ὐπὸ βἰαϛ) (548b). The term force described the timocrat’s education and it names a Socratic category of modes. Given Socrates’ description we can see that the enforced modes correlate with the timocratic constitutions of state and soul.
So, we have correlations between the aristocratic constitutions and the voluntary modes, and between the timocratic constitutions and the enforced modes. We still need to determine the correlate common modes. In the text surrounding the description of the voluntary and enforced modes (above), Socrates feigns ignorance of the common names for the modes (399a3). He never uses the common names. Instead, Glaucon gives the common names. Before Socrates describes the voluntary and enforced modes, Glaucon tells Socrates that he has only the Dorian and Phrygian modes left to discuss (399a3–4). After Socrates describes the voluntary and enforced modes, Glaucon tells him that he has just described the Dorian and Phrygian modes (399c5). He does not, however, tell Socrates whether the voluntary mode corresponds to the Dorian mode, or the Phrygian mode. He does not tell him which mode, Dorian or Phrygian, correlates with the enforced modes.
When we notice that Glaucon lists the modes as Dorian:Phrigian and Socrates responds with his modes listed as enforced: voluntary, it is natural in Greek to take the two lists as a chiasma (399a). In which case, the Dorian mode correlates with the voluntary modes and with the aristocratic constitutions of state and soul. A glance outside of the Republic provides convincing evidence for holding that the Dorian mode correlates with the voluntary mode. In the Laches, the Dorian mode is the “most beautiful” mode and it is the only “genuinely Greek mode” (188c). In Laches, we are told that the Dorian mode is preferable to the Phrygian mode. Though both modes have their proper function, the voluntary modes are clearly preferable to the enforced modes. So, in the Republic, the Dorian mode correlates with the voluntary modes and with the Aristocratic constitution. We saw that Mountford correlates the Dorian mode with the warrior’s mode and with the enforced modes. We now see that the Phrygian mode correlates with the warrior’s mode, the enforced modes, and with the timocratic constitution.
Though Socrates allows two modes in the kalipolis, he correlates one mode (Dorian) with the voluntary modes and with the aristocratic constitution. Socrates has good reason to allow the modes from two different constitutions in the kalipolis, but to correlate just one of the modes with the aristocratic constitutions. There is a natural relationship, Socrates argues, between the part of the soul that rules the aristocrat and the part of the soul that rules the timocrat, between reason and spirit (440). At 440, Socrates holds that the spirited part of the soul is the natural helper of the rational part. In the aristocratic state, the military class is the natural helper of the rulers. Thus, Socrates includes the timocratic mode, with the aristocratic mode, in the kalipolis.
Since the kalipolis includes the modes of both the aristocratic and timocratic constitutions, the remaining constitutions probably correlate with the remaining modes. So, it is left to correlate the democratic, oligarchic, and tyrannical constitutions with their respective mode(s). Socrates offers two more Socratic categories, the lamenting modes and the soft modes:
[A] What are the lamenting modes, then? You tell me, since you’re musical.
The mixo-Lydian, the syntono-Lydian, and some others of that sort.
Aren’t they to be excluded then? They’re useless even to decent women, let alone to men.
[B] Drunkenness, softness, and idleness are also most inappropriate for our guardians.
How could they not be?
What then are the soft modes suitable for drinking parties?
The Ionian and the Lydian modes that are said to be relaxed.
Could you ever use these to make people warriors?
Glaucon tells Socrates that the Mixolydian and Syntonolydian modes are the lamenting modes. Also, he tells him that the Ionian and Lydian modes are the soft modes. In this passage, Socrates does not explicitly correlate either the relaxed modes, or the lamenting modes, with any of the remaining constitutions of state and soul. The passage does not explicitly determine the oligarchic, democratic, or tyrannical modes.
If we look closely at the contrasts that Socrates draws, when distinguishing the lamenting and the relaxed modes, we may correlate the two sets of modes with two types of constitutions. The lamenting modes are described as “useless for decent women, let alone men” (398e1–3). The lamenting modes are indecent. Socrates applies the term “lament” (θρῆνοϛ, θρηνεὠ) not only to musical modes, but also to a certain constitution of state and soul:
What about fear? Aren’t a tyrannical city and man full of it?
And do you think that you’ll find more wailing, groaning, lamenting (qr»nouj), and grieving in any other city?
Then, are such things more common in anyone besides a tyrannical man . . . ? (578a)
Here Socrates identifies the tyrannical constitution as the lamenting constitution, among other things. It is not much of a stretch to connect the lamenting constitution with the lamenting modes. So, the Mixolydian and the Syntonolydian modes correlate with the tyrannical constitutions of state and soul.
Turning to the soft modes, Socrates contrasts these modes with the modes that are fit for warriors. He decries that we could never use these modes to “make people warriors” (399a). The Phrygian mode is the warrior’s mode, the enforced mode, and anthem of the timocratic constitutions. The soft modes are set below the warrior’s mode. The soft modes correlate with the constitution that is set below the timocratic constitutions. The oligarchic constitutions are set below the timocratic. So, the soft modes, commonly called the Ionian and Lydian modes, correlate with the oligarchic constitutions.
Just one constitution of soul and state remains, the democratic. Yet, we have correlated the constitutions with all the common modes in the Republic. Immediately after Socrates finishes discussing the modes, he turns to discuss musical instruments:
Well then, we’ll have no need for polyharmonic or multistringed instruments to accompany our odes and songs.
It doesn’t seem so to me at least.
Then we won’t need the craftsman who make triangular lutes (τριγὠνων), harps, and all other such multistringed and polyharmonic instruments.
What about flute-makers and flute-players? Will you allow them into the city? Or isn’t the flute the most “many-stringed” of all? And aren’t the panharmonic instruments all imitations of it? (399c–d)
The term translated “harmonic” is the same term that is translated as “mode” (ἁρμονíα). Socrates rejects “polyharmonic,” and “panharmonic” instruments. When he distinguishes some instruments as “multi-stringed” he is not referring to the strings. The flute, we are told, is the most many-stringed instrument. He means that the flute is poly- and pan-harmonic. He also says that the triangular lute is poly-modal. On this lute a musician can play every mode without retuning the instrument. Socrates banishes the instruments that can play multiple modes.
It is important to notice that Socrates distinguishes the category of instruments according to the modes that can be played on them. So the category of instruments implies a category of modes. We saw that Socrates names the voluntary, enforced, soft, and lamenting modes. When he distinguishes the multi-modal musical instruments, he distinguishes a new category of modes. He describes the category of instruments as “poly-harmonic” (poly-modal), and “pan-harmonic” (pan-modal) (399d). I will call this Socratic category the poly-harmonic category of modes. The poly-harmonic category of modes includes all the modes. It includes the voluntary mode, or the Dorian; the enforced mode, or the Phrygian; the soft modes, or the Ioinian and Lydian; and the lamenting modes, or the Syntonolydian and Mixolydian modes.
The poly-harmonic modes are similar to the soft and lamenting modes, since all three Socratic categories include more than one commonly named mode. This category of modes differs from the other categories. Unlike all the other categories, the poly-harmonic category does not have a unique commonly named mode. It is a superset of the Socratic categories and a superset of the commonly named modes.
With the poly-harmonic modes in hand, I turn to the democratic constitutions of state and soul. Concerning the democratic constitution of state, Socrates says:
It is also a convenient place to look for a constitution.
Because it contains all kinds of constitutions on account of the license it gives to its citizens. So, it looks as though anyone who wants to put a city in order, as we were doing, should probably go to a democracy, as to a supermarket of constitutions, pick out whatever pleases him, and establish that. (557d)
The democratic constitution of state contains, in some sense, all the other constitutions of state. It borrows the other constitutions to make a constitution. In this way, the democratic constitution is a constitution of constitutions. It is a poly-constitutional constitution of state.
Concerning a person with a democratic constitution of soul, Socrates says:
And so he lives, always surrendering rule over himself to whichever desire comes along, as if it were chosen by lot. And when that is satisfied, he surrenders the rule to another, not disdaining any but satisfying them all equally.
He . . . declares that all pleasures are equal and must be valued equally. (561b)
Here we see that the democratic soul rules by allowing all the parts of the soul to rule. This soul does not rank desires. For the democrat, all desires are worthy of being indulged. In this way, the democratic constitution of soul is a poly-desiderative constitution of soul.
Democratic constitutions preserve freedom, the freedom to indulge every kind of state and every kind of soul. Socrates tells us that the democrat sometimes “drinks heavily while listening to the flute; at other times he drinks only water” (561d). In this passage, he associates the flute with the democrat. Since the flute is one of the poly-harmonic instruments and it implies the poly-harmonic modes, the democratic constitution of soul correlates with the poly-harmonic modes. Since the democratic constitution of soul correlates with this category of modes, the democratic constitution of state correlates with the same category. This correlation fits well with Socrates’ description of the democratic constitutions. In the democratic state every class rules and in the democratic soul every desire rules. In the poly-harmonic instruments every mode rules. Moreover, Socrates implies that there are other modes than the ones he names. He describes the “dense-interval” in music (531a). Dividing the whole tone into smaller and smaller segments generates unlimited possibilities. These unlimited intervals allow for unlimited modes. The democratic constitution would seek to allow every, and any, possibility. So, the democratic modes are the poly-harmonic modes and all other possible modes.
Aristides Quintilianus is the only classical authority on Greek music to state the intervals among the notes in the modes of the Republic. Since his is our only classical source on the intervals, I will adopt the intervals as Aristides charts them. Though some interpreters reject Aristides’ account of the intervals, Mountford accepts the account and applies it to the Republic. Since Mountford has worked out the technical details in terms of Aristides’ intervals, I will adopt Mountford’s technical account of the intervals and the notes in the modes.
Mountford argues that the modes are various scales in different tonal ranges. The following chart contains the notes in each mode and the tonal relations among the modes:
|[ Meson ]||[ Diezeugmenon ]|
|[ Hypaton ]|
A “+” indicates the quartertone, and the “ ′ ” indicates the octave. A tetrachord is a set of four notes that span no more than a perfect fourth. The terms “hypaton” (farthest), “meson” (middle), and “diezeugmenon” (disjoined) name three tonal ranges for the tetrachords. The tetrachord diezeugmenon is highest in pitch. The meson is in the middle. The hypaton is the lowest pitch. The lowest note in the lowest tetrachord is the hypate hypaton (B). The mese meson is the highest note in the middle tetrachord (A). The nete diezeugmenon is the highest note in the disjoined tetrachord (E).
Mountford instructs us to distinguish the modes by the internal relations of their notes, not by pitch alone. When we chart the internal tonal relations of the modes we get the following:
Each mode contains a unique arrangement of tonal relations. One set of intervals occurs in every mode. From high pitch to low, right to left on Chart 3, we find the intervals 2 – ¼ – ¼ in every mode. I will name this set of intervals T-alpha. To better reveal T-alpha in the chart, I italicized its occurrence in each mode. T-alpha occurs among the notes E, C, B+, and B in the tetrachords hypaton and diezeugmenon; and among the notes A, F, E+, and E in the tetrachord meson. T-alpha is a sets a baseline among the modes.
We may chart the modes by locating the tonal register of the tetrachord in which T-alpha occurs. We saw that T-alpha can occur in all three tonal ranges, tetrachord hypaton, meson, and diezeugmenon:
The Dorian mode is the only mode with T-alpha twice. All other modes have T-alpha once. The Dorian and Lydian modes have T-alpha in the tetrachord diezeugmenon. The Dorian and Phrygian modes have T-alpha in the tetrachord meson. The Ionian, Mixolydian, and Syntonolydian modes have T-alpha in the tetrachord hypaton.
In practice, Socrates prefers one mode over another based on certain general features; the intervals among the notes in the modes, and the tonal range of the modes. He takes T-alpha as a preferred feature of a mode and he prefers multiple occurrences of it. There are other desirable features in T-alpha. It spans no more than a perfect fourth. The largest interval between two notes in T-alpha is the largest interval possible in a tetrachord that spans no more than a perfect fourth. Socrates prefers modes that have no more than two whole steps between any two notes. Also, T-alpha has one of three notes (E, A, or B) as its limits. He prefers modes that have these three limiting notes. Concerning the tonal range of the modes, he prefers the middle tones over the others, and he prefers the high tones over the low tones. Pulling these desiderata together yields the preferences that Socrates employs in practice:
1. Socrates prefers that T-alpha occur in a mode.
2. Socrates prefers that no more than two steps separate two notes.
3. Socrates prefers that the three limiting notes, B, E, and A occur in a mode.
4. Socrates prefers the tetrachord meson to all other tetrachords, and he prefers the tetrachord diezeugmenon to the tertachord hypaton.
These preferences need not be taken in the order that they are offered, since their order does not alter their results. I will go through them as offered.
(1) To apply the first preference to the modes, the Dorian mode contains T-alpha twice and all other modes contain T-alpha once. So, the Dorian mode is preferred over all others. (2) To apply the second preference, the Mixolydian mode has an interval of three whole steps between two of its notes. This is the only mode that has an interval greater than two whole steps between two notes. So, all modes are preferred over the Mixolydian mode. We have now identified the most and the least preferable modes.
(3) To apply the third preference, the Mixolydian and Syntonolydian modes contain two of the three limits, E and B. Every other mode has all three limits. So, all modes are preferred over the Mixolydian and Syntonolydian modes. Since we know from the second preference that the Mixolydian mode is the least preferred mode, the Syntonolydian mode must be above the Mixolydian mode and below all the other modes. We have determined that the Syntonolydian mode is the second least preferable mode.
(4) To apply the fourth preference, the Dorian and Phrygian modes have T-alpha in the tetrachord meson, so they are preferred over all the other modes. Since we know from the first preference that the Dorian mode is the most preferred, we now know that the Phrygian mode must come below the Dorian and above all the other modes. So the Phrygian mode is the second most preferred mode. To apply the fourth preference again, the Lydian mode has T-alpha in the tetrachord diezeugmenon. The Ionian, Syntonolydian, and Mixolydian modes have T-alpha in the tetrachord hypaton. So, the Lydian mode is preferred over the Ionian, Syntonolydian, and Mixolydian modes. From the first and third preferences we learned that the Dorian mode is most preferred and the Phrygian is second in preference. The fourth preference already showed that the Lydian mode is preferable to the remaining modes. So, the Lydian mode is the third most preferred mode. From the second and third preferences we learned that the Mixolydian mode is least preferred and the Syntonolydian mode is second least preferred. The fourth preference shows that the Ionian mode is the third least preferred mode. So we have the three most preferred modes in order and the three least preferred modes in order:
This is a complete list and ranking of the commonly named modes that are mentioned in the Republic.
We find confirmation for the top rankings on this list outside of the Republic. In the Laches, Laches tells us that the best man renders his “own life harmonious by fitting his deeds to his words in a truly Dorian mode, not in the Ionian, or even, I think, in the Phrygian or Lydian . . .” (188c). Laches names the top four modes, according to the Socratic preferences. One way to interpret Laches’ partitions is to put the Dorian (1) at the top and the Ionian (4) at the bottom of his list. On this interpretation, Laches completes his list by filling the gap between the top and the bottom with the Phrygian (2) and the Lydian (3) modes. If this interpretation is correct, then Laches ranks the Lydian mode above the Ionian mode. These results are echoed in the rankings produced by the Socratic preferences.
We may now fit preceding sections of this paper together; the constitutions of state and soul, the Socratic categories, the commonly named modes, the results of the preferences, and the rankings. The best constitution of state and soul is the aristocratic. It has the voluntary mode, or the Dorian mode, as its anthem. The second best constitution is the timocratic constitution. It has the enforced mode, or the Phrygian mode, as its anthem. The oligarchic constitution ranks third. It has the soft modes, the Lydian and Ionian modes, as its anthems. The democratic constitution ranks fourth. It has the pan-harmonic modes, all the modes, as its anthems. Finally, the tyrannical constitution ranks last. It has the lamenting modes, or the Syntonolydian and the Mixolydian modes, is its anthems. The following chart displays the results:
|Ranking||Constitution of Soul and State||Socratic Categories||Common Categories|
The Socratic preferences serve to refine the ranking of the modes beyond the results of the ranking according to the Socratic categories. The preferences show that the Lydian mode is preferred over the Ionian mode. So, these modes are on the chart as “3.1” and “3.2.” The preferences also show that the Syntonolydian mode is preferred over the Mixolydian. So, these modes are on the chart as 5.1 and 5.2. We now have the complete results of Socrates’ categories, correlations, and rankings.
 The Socratic modes are the voluntary, enforced, soft, lamenting, and poly-harmonic modes. The common modes are the Dorian, Lydian, Ionian, Syntonolydian, and Mixolydian modes.
 Since there are six modes named in the Republic and interpreters correlate two modes, I claim that most of the correlations remain tacit. I have not found any interpretation that specifies the correlations between more than two modes, the Dorian and Phrygian. Interpreters of the Republic have little to say about the relations between the modes and the various constitutions. Some interpreters discuss the modes: Mountford offers an authoritative interpretation of the modes, in “The Musical Modes in Plato’s Republic,” The Classical Quarterly, vol. 17, No. 3/4 (Jul–Oct., 1923), pp. 125–136. Harap offers an incomplete correlation in “Some Hellenic Ideas on Music and Character,” Louis Harap, The Musical Quarterly, vol. 24, no. 2 (Apr., 1938), 157. Other interpreters discuss the constitutions, without the modes. See, White, A Companion to Plato’s Republic, Indianapolis: Hackett, 1979, 97–98. This paper fits the topics together.
 “Greek Music and its Relation to Modern Times,” The Journal of Hellenistic Studies, Vol. 40, Part I, (1920), 21. Mountford seems to get the identification of the Dorian mode as the warrior’s mode from Plutarch (On Music, cap.17, 1136e), which he cites in his work on the Republic (1923, 128).
 The indented quote is from Mountford’s 1920 article on the relation between Greek music and modern music. The 1920 article does not specify the correlations. In his 1923 article Mountford focuses on the Republic. The 1923 article adds nothing new concerning the correlations.
 See Laches 188c, and 193e. Laws 670b implies the same top ranking of the Dorian mode.
 For a discussion of the “triangular lute” see, “The Hendecachord of Ion of Chios,” Flora Levin, Transactions and Proceedings of the American Philological Association, Vol. 92, (1961), pp.295–307.
 Socrates is clearly describing the Greek aulos.
 Some of the modes require the use of a “stop.” See, Lavin (1961), 306.
 Socrates uses the terms “many-chords” (πоλὐχоρδα) and “many-harmonic” (πоλυαρμόνἰα) (399d1). We should take the term “chords” as referring to strings, not to sets of notes played at one time.
 Not surprisingly, Socrates holds that constitutions of soul and state admit of “intermediate” types. The “Dynasties and purchased kingships and other constitutions of that sort, which one finds no less among the barbarians than among the Greeks, are somewhere intermediate between these four” (544c). “These four” refers to the four constitutions below the aristocratic constitution. Since there are intermediate constitutions of state and the constitutions of state have corresponding constitutions of soul, he implies that there are intermediate constitutions of soul. The correlation between modes and constitutions thereby implies that there are intermediate modes.
 Mountford makes a similar claim (1923), 125.
 I will discuss the enharmonic modes and not the diatonic modes. See Mountford (1920), 25. Henderson argues, unlike Mountford, that Plato refers to the diatonic modes and not the enharmonic modes (1942) 95. Henderson thinks that Republic 531a–c forbids the ¼ tone and that it forbids all enharmonic music (1942), 95. Yet, we need not take “πὐκνωμα” to mean “quartertone”. Shorey’s translation of the term to “minims” is preferable, Plato Republic Books VI-X, Shorey trans., Loeb Classical Library, Suffolk: St., Edmundsbury Press, 1994, 191, note f. Also, we need not embellish “σὐμϕωνоι ἀ ριθυоí” (perfect consonance) by insisting that it forbids the ¼ interval, Henderson (1942), 95. Henderson’s interpretation would have the passage forbid an interval that is in all the enharmonic modes. Anderson notices, however, that the music of the spheres is in the enharmonic genus, “The Importance of Daimonian Theory in Plato’s Thought,” W. Anderson, Transactions and Proceedings of the American Philological Association, Vol. 85, (1955) 99. Anderson directs us to Aristides claim as found in Ancilla to the Pre-Socratic Philosophers, 5th ed., K. Freeman, Oxford (1984) 149 Meibom.
 See Mountford, (1923).
 R.P. Winnington-Ingram agrees with Mountford about the intervals, in “Ancient Greek Music: A Survey,” Music and Letters, vol. 10, no.4 (Oct, 1929), 333, 338. M.I. Henderson also accepts these as the enharmonic intervals, in “The Growth of Greek armoniai,” The Classical Quarterly, vol. 36, no 3/4 (Oct., 1942), 94. Moreover, E. Clemens accepts these as the enharmonic intervals, “The Interpretation of Greek Music, An Addendum,” E. Clemens, The Journal of Hellenic Studies, vol. 56, Part 1., (1936), 35.
 The chart is based on a chart in Levin (1961), 305.
 In the C major diatonic scale there are two tetrachords, CDEF and GABC’.
 For an explanation of why “the highest sounds” have the lowest pitch (tetrachord hypaton), see Lavin (1961), 306, fn. 46. For a discussion on the same point see, “The Music and Tone-System of Ancient Greece,” M. Shirlaw, Music and Letters, vol. 32, no.2 (Apr., 1951), pp. 131–139.
 Levine (1961) identifies these three notes as the “three limits” in the soul that Socrates refers to at 443d–e.
 M. Shirlaw (1951) offers an alternative position. He attempts to distinguish the modes primarily by pitch.
 Mountford suggests that we focus on the largest internal ration between two notes, when distinguishing the modes, (1923), 136.
 In the Mixolydian mode T-alpha occurs in the tonal relations 1 – 1 (2) – ¼ – ¼, among the notes E, C, and B.
 For a similar point about import of the tetrachords comprising the Dorian mode see, Winnington-Ingram (1929) 339, sec.3.
 This assumes that the ¼ step is the smallest step allowed in the modes, which is true of the modes under consideration. Though Socrates displays preferences about the smallest permissible steps (Republic, 531a–b), they do not help us rank the modes. So, I do not include them in my list of preferences.
 More completely, “And such a man seems to me to be genuinely musical, producing the most beautiful harmony, not on the lyre or some other pleasurable instrument, but actually rendering his own life harmonious by fitting his deeds to his words in a truly Dorian mode, not in the Ionian, or even, I think, in the Phrygian or Lydian, but in the only harmony that is genuinely Greek” (Laches, 188c).
This was originally published in The Journal of the Utah Academy, Vol. 84 (2007): 104–116.