To be clearer, would Tolkien have been able to point to historical examples within his field of study of languages and history of such a conflict, where warriors resolved conflict with songs instead of swords? Alternatively, are there any examples of such from outside Tolkien's scope?
For reference, here's the passage in question from The Silmarillion Chapter 19, "Of Beren and Luthien":
"For [Finrod] Felagund strove with Sauron in songs of power, and the power of the King was very great; but Sauron had the mastery, as is told in the Lay of Leithian:
He chanted a song of wizardry
Of piercing, opening, of treachery,
Revealing, uncovering, betraying,
Then sudden Felagund there swaying
Sang in answer a song of staying,
Resisting, battling against power,
Of secrets kept, strength like a tower,
And trust unbroken, freedom, escape
Of changing and of shifting shape
Of snares eluded, broken traps,
The prison opening, the chain that snaps
Backwards and forwards swayed their song,
Reeling and foundering, as ever more strong
The chanting swelled, Felagund fought,
And all the magic and might he brought
Of Elvenesse into his words
Softly in the gloom they heard the birds
Singing afar in Nargothrond
The sighing of the Sea beyond,
Beyond the western world, on sand,
On sand of pearls in Elvenland
Then the gloom gathered, darkness growing,
In Valinor, the red blood flowing
Beside the Sea, where the Noldor slew
The Foamriders, and stealing drew
Their white ships with their white sails
From lamplit havens. The wind wails,
The wolf howls. The ravens flee.
The ice mutters in the mouths of the Sea
The captives sad in Angband mourn.
Thunder rumbles, the fires burn -
And Finrod fell before the throne.
Then Sauron stripped from them their disguise, and they stood before him naked and afraid. But though their kinds were revealed, Sauron could not discover their names or their purposes.
He cast them therefore into a deep pit, dark and silent, and threatened to slay them cruelly, unless one would betray the truth to him. From time to time they saw two eyes kindled in the dark, and a werewolf devoured one of the companions, but none betrayed their lord.
1/2 Answer continued in subreply
As always, the clearest place to look when attempting to understand Tolkien is the traditions of the Medieval Northern Arc (a term I use for the lands of Anglo Saxon England/Daneland, Denmark, Norway, Sweden and Iceland/Greenland in the 9-13th centuries). What I believe Tolkien to be drawing on here (there is to know what he actually might be referencing), is the practice of seiðr (sometimes anglicized as seidhr, seidh, seidr, seithr, seith, or seid), a form of spoken spell casting usually foretelling the future, but more generalized to simply spell casting. In this instance I believe there is a fundamental misunderstanding of what is happening in the text. This would not be an example of resolution with words but with spells instead, Sauron breaking down the will of his enemy through spell casting in the form of song. This was a practice frequently associated with Odin, and Freya, and took the form of song/spoken word. As Ynglinga Saga, the origin of the practice occurred with Odin, who spread the practice.
“In all such things he was pre-eminently wise. He taught all these arts in Runes, and songs which are called incantations, and therefore the Asaland people are called incantation-smiths”
Given what we know of Odin, this passage from Hávamál (The Words of Odin the High One) from the Elder or Poetic Edda (Sæmund's Edda) might give a poignant example of why I believe this to be the practice that Tolkien was referencing when he wrote the passage
The Song of Spells
Those songs I know, which nor sons of men
nor queen in a king's court knows;
the first is Help which will bring thee help
in all woes and in sorrow and strife.
A second I know, which the son of men
must sing, who would heal the sick.
A third I know: if sore need should come
of a spell to stay my foes;
when I sing that song, which shall blunt their swords,
nor their weapons nor staves can wound.
A fourth I know: if men make fast
in chains the joints of my limbs,
when I sing that song which shall set me free,
spring the fetters from hands and feet.
A fifth I know: when I see, by foes shot,
speeding a shaft through the host,
flies it never so strongly I still can stay it,
if I get but a glimpse of its flight.
A sixth I know: when some thane would harm me
in runes on a moist tree's root,
on his head alone shall light the ills
of the curse that he called upon mine.
A seventh I know: if I see a hall
high o'er the bench-mates blazing,
flame it ne'er so fiercely I still can save it, --
I know how to sing that song.
An eighth I know: which all can sing
for their weal if they learn it well;
where hate shall wax 'mid the warrior sons,
I can calm it soon with that song.
A ninth I know: when need befalls me
to save my vessel afloat,
I hush the wind on the stormy wave,
and soothe all the sea to rest.
A tenth I know: when at night the witches
ride and sport in the air,
such spells I weave that they wander home
out of skins and wits bewildered.
An eleventh I know: if haply I lead
my old comrades out to war,
I sing 'neath the shields, and they fare forth mightily
safe into battle,
safe out of battle,
and safe return from the strife.
A twelfth I know: if I see in a tree
a corpse from a halter hanging,
such spells I write, and paint in runes,
that the being descends and speaks.
A thirteenth I know: if the new-born son
of a warrior I sprinkle with water,
that youth will not fail when he fares to war,
never slain shall he bow before sword.
A fourteenth I know: if I needs must number
the Powers to the people of men,
I know all the nature of gods and of elves
which none can know untaught.
A fifteenth I know, which Folk-stirrer sang,
the dwarf, at the gates of Dawn;
he sang strength to the gods, and skill to the elves,
and wisdom to Odin who utters.
A sixteenth I know: when all sweetness and love
I would win from some artful wench,
her heart I turn, and the whole mind change
of that fair-armed lady I love.
A seventeenth I know: so that e'en the shy maiden
is slow to shun my love.
These songs, Stray-Singer, which man's son knows not,
long shalt thou lack in life,
though thy weal if thou win'st them, thy boon if thou obey'st them
thy good if haply thou gain'st them.
An eighteenth I know: which I ne'er shall tell
to maiden or wife of man
save alone to my sister, or haply to her
who folds me fast in her arms;
most safe are secrets known to but one-
the songs are sung to an end.
Now the sayings of the High One are uttered in the hall
for the weal of men, for the woe of Jötuns,
Hail, thou who hast spoken! Hail, thou that knowest!
Hail, ye that have hearkened! Use, thou who hast learned!
According to the scholar David Gay, the primary inspiration for this "song contest" was a Finnish poem Kilpalaulanta (The Singing Match) from the 1849 work Kalevala, which itself is based off the most famous kilpalaulanta duel in Finnish mythology.
A young wizard Joukahainen challenges the much older Väinämöinen to a contest of knowledge (no reason is given, other than they come across each other on the road).
Young Joukahainen said:
"Whoever's knowledge is worse
he must move from the road."
The challenge is seen as an insult to the much more experienced Väinämöinen.
I set the sky's pillars,
I placed the sky's arcs,
I set the stars in the sky,
I raised the new moon,
I placed the sun,
I straightened the Great Bear.
Väinämöinen was furious. "He began to sing magic songs" and caused the earth to shake and mountains to tremble.
Joukahainen sinks into the ground, "into the swamp to his waist, into the meadow to his chest". He tries to bargain; all his offers are rejected until he promises to give his sister Aino in marriage, at which point he is released.
Different variants have Joukahainen and Väinämöinen crash their sledges into each other before the contest begins, or Joukahainen's mother happy -- despite the circumstances of Aino's betrothal -- that "a great man" will now become a relative.
You can listen to this in song form here.
..
Drout, M. (2006). J.R.R. Tolkien Encyclopedia: Scholarship and Critical Assessment. Taylor & Francis.
Gay, D. (1997). The Creation of the "Kalevala", 1833-1849. Jahrbuch für Volksliedforschung, 42, 63-77. doi:10.2307/848005
Pentikäinen, J. (1999). Kalevala Mythology, Revised Edition. Indiana University Press.