Equivalent question: What Latin letters won't equal 1?

From: the homophonic group: a mathematical diversion --> This is an exercise from Michael Artin's Algebra on, well, abstract algebra. In this exercise for the English language, words are equal if they are homophones, kind of like a formalisation of the joke that sin(x)/n=6. So in English:

bee=be --> This implies e=1 by cancellation of b and e.

buy=by --> This implies u=1 by cancellation of b and y.

rase=raze --> This implies s=z by cancellation of r and e.

canvass = canvas --> This implies s=1 by cancellation of c,a,n,v,a and s. By canvass=canvas and rase=raze, we have s=z=1.

Eventually, all 26 English letters will equal 1. Apparently, this was done for French and Czech.

  • 2
    I don't know whether this is on-topic here (I mean it). I saw your question in Spanish.SE. For now, I'm not upvoting nor voting to close. But I admit I find the question interesting. Moreover, it can have more than one answer, depending on the different pronunciation conventions.
    – Rafael
    Commented Aug 17, 2018 at 14:20
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    Out of curiosity, why Latin? Given that it's a phonetic language (like Spanish and Italian), the problem doesn't really apply.
    – brianpck
    Commented Aug 17, 2018 at 14:21
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    @brianpck well, someone actually managed to collapse all of Spanish to a single class, but I think they cheated a bit.
    – Rafael
    Commented Aug 17, 2018 at 14:22
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    @brianpck I'm just asking everywhere much like I did with my square-rectangle question about 5 months ago.
    – BCLC
    Commented Aug 17, 2018 at 14:24
  • 2
    Ahhh, Ghoti strikes back! Commented Aug 17, 2018 at 23:50

3 Answers 3


Much of this depends which stage of the language you're talking about. I'll focus on Classical Latin, with reconstructed pronunciation.

  • h seems to be the identity: arena vs harena, lacrima vs lachrima. (/h/ disappeared from all but the most educated speech in Classical times.)
  • p = f: Daphne vs Dafne, the latter attested in Pompeii. (The pronunciation of the Greek letter phi was shifting, so it was transcribed in different ways in different places and different times.)
  • i = u = y: optimus vs optumus, girus vs gyrus, silva vs sylva. (There was often confusion between short /i/ and /u/ in certain positions, and /y/ was a Greek sound that tended to assimilate to one or the other.)

…and that's about all we can do. Latin orthography was very closely tied to the spelling, so most of these irregularities come from the sounds themselves changing, or from changes in other languages.

EDIT: I clearly stopped looking too soon!

  • A = AA, and same for the other vowels: Old Latin used double letters for long vowels (aara vs āra "altar")
  • I = J: iam vs jam "now"
  • U = V: uel vs vel "or"
  • QV = C: seqvundus vs secundus "second"
  • E = O, R = L: Vertumnus vs Vortumnus vs Voltumnus, the borrowed name of an Etruscan deity

I'd previously been ignoring assimilation because it seemed to reflect variances in pronunciation. But Lewis and Short show that adsum was pronounced assum at least in common speech, as a pun in Plautus depends on it. So we can add assimilations too!

  • S = D = F = R = G = ∅: adsum vs assum "I am here", adfuī vs affuī vs arfuī "I was here", adnoscō vs agnoscō vs adgnoscō "I recognize"
  • N = M: inmortalis vs immortalis "immortal" (thanks to Rafael's comments for this one!)

If you want to include Emperor Claudius's failed attempt at orthography reform:

  • Ж = PS = BS, indicating that P = B, as in princeж "prince" or urж "city" (though this letter has never been found in inscriptions and might also have looked like Ↄ; it doesn't even get its own Unicode codepoint)
  • Ⅎ = V, as in ⅎel "or"
  • Ⱶ = I = U = Y, as in gⱶbernator "governor"

EDIT: One more:

  • G = C, as in the praenomina Gaius ~ Caius or Gnaeus ~ Cnaeus. These names were pronounced with a G, but often spelled with a C for historical reasons; both spellings are attested in Classical times.
  • Thanks Draconis! I won't accept for now to allow for other answers.
    – BCLC
    Commented Aug 18, 2018 at 4:36
  • If prefixed words' pronunciation evolved before orthography, it is reasonable to think that at some point it was OK to write both inmortalis and immortalis, but the pronunciation was only the latter. In that case you could add a few more equalities
    – Rafael
    Commented Aug 18, 2018 at 12:51
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    Or none of these changes happened during Classical times?
    – Rafael
    Commented Aug 18, 2018 at 13:06

As other comments have indicated, in Classical Latin there seem to have been few or no examples of heterographic homophones of the kind that are found in English (with two separate words, e.g. meet and meat, which have different meanings and consistently different spellings, but consistently identical pronunciations in standard dialects).

Latin of any period definitely has homographic homophones, and Classical Latin had a number of heterophonic homographs (mainly because of the practice of writing long and short vowels the same way), but neither of those are relevant to your question.

Draconis's answer provides examples of what I think are the closest we can get in most cases to heterographic homophones: words that had spelling variants that weren't associated with different meanings or different pronunciations (as far as we can tell). As Alex B. said in a comment, these would not actually be classified as "homophones". (I've posted an ELU question asking for a word or term to describe words with spelling variants.)

Certain variant spellings unrelated to pronunciation in Classical Latin

In addition to the spelling variants that Draconis mentioned, here are some others that show up in Classical Latin:

  • C = K: The letter K was infrequent in Classical Latin, but it was used in the word Kalendae and the place name Karthago. These had alternative spellings with C, Calendae and Carthago. Both letters represented the same sound, the velar plosive /k/.

  • M = N: Coda nasal consonants in Latin typically had the same place of articulation as a following plosive (the only reason I say "typically" is because I'm not quite sure about the usual phonological analysis of the consonant cluster found in emptus and related words, where the p is diachronically/morphologically epenthetic, and I believe sometimes not written). Regressive place assimilation of a nasal within a word was sometimes marked in writing, and sometimes not, but I don't think this variation in spelling is believed to have had a perfect correspondence to any analogous variation in pronunciation. An example of a word which had two spelling variants with M and N is quamquam/quanquam (my understanding is that this word is thought to have been pronounced with a velar nasal [ŋ], regardless of the spelling).

  • X = XS → S = 1: When a word starting with S was prefixed with ex-, the S might or might not be omitted. Even if there was some variation in pronunciation between the use of /ks.s/ and /k.s/ in such words (I don't know if there was, and I'm not sure how we could determine this), I doubt that the spelling variation had any kind of consistent correspondence to a possible difference in pronunciation. An example is exsisto/existo.

More old-fashioned (?) examples

There are a number of other spelling variants in Latin that I can think of, but that I think are less often seen. I have the impression that they came to be old-fashioned even in the time of Classical Latin.

  • II = I → I = 1: in medial position between vowels, it is thought that the single letter I could represent the geminate palatal approximant [j.j] in Classical Latin. An alternative representation of this sound was II. In Vox Latina, W. Sidney Allen says that inscriptions show spellings like PompeIius, cuiIus, eiius, maiIorem for Pompeius, cuius, eius, maiorem, and says that "Quintilian and other grammarians [...] tell us that Cicero and Caesar used in fact to spell such words [as aio, maior, peior, Troia] with ii" (p. 39).

  • EI = I → E = 1: I know that in some time periods, the digraph EI was used as a variant of I (I think only when I was a long vowel). In Old Latin, EI is thought to have represented a distinct sound from both E and I, but Old Latin EI ended up merging with long I, which caused both spellings to be used for the same sound. This led to variation between I and EI in the spellings of various words pronounced with the long I sound [iː].

If we accept both of these, then we have S H E I all equal to 1.

Less certain examples

If we look at post-Classical periods, there are many more identities that could probably be established (although I haven't looked for examples of homophones). The value of the digraphs ae and oe merged early on with the value of the single letter e (although this also contributed to variability and instability in the spelling of individual words). In certain varieties of Latin, ti and ci came to have the same value (something like [tsj]) in certain contexts.

For some spellings, it's a bit unclear when they first came to represent identical pronunciations. Word-final m in Latin is known to have been subject to elision from early on (especially when it came before a vowel); some scholars believe that Classical Latin had distinctive nasalization of a vowel preceding word-final M, even if the nasal consonant was lost, but as far as I know we don't have that much evidence of this. Except for in monosyllables, all nasality has been lost in descendants of Latin. I guess this could be used as a basis for assigning the value "1" to M, based on multi-word equivalences like libra aut = libram aut.

Latin also had a number of assimilations that might or might not be written. For example, N at the end of a prefix, as in con- and in-, could assimilate to a following L or R, and D at the end of the prefix ad- could assimilate to most other consonants.

I found a relevant passage in Reading in Medieval St. Gall, by Anna A. Grotans (2006). Grotans says that in Isidore's Etymologiae (from the late sixth and seventh centuries), the words ad (preposition) and at (conjunction) are given as an example of heterographic homophones (p. 295).

  • To what period/s do assimilations belong? With those you have n(=m?)=r=l=d=most other consonants.
    – Rafael
    Commented Aug 18, 2018 at 22:28
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    @Rafael: I have the impression that assimilations with ad- were fairly old, and existed already in Classical Latin, but I don't think they were systematically applied in writing. And I'm not sure to what extent the variation between assimilated and unassimilated spellings might correspond to variation between assimilated and unassimilated pronunciations.
    – Asteroides
    Commented Aug 18, 2018 at 22:31
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    Variation between adsum and assum shows up in Plautus, so we could get S = D = F = R from that word alone! Given your comment about XS, they all end up equal to the identity.
    – Draconis
    Commented Aug 18, 2018 at 23:20
  • Without any formal knowledge on the subject, it seems logical to me that writing changes come after sound changes, and that they do not become universal at once. Or are there examples of different processes? If not, I don't think it is far fetched at all to say that while the writing change was ongoing, both spellings are an example pair of heterographic homophones. Or am I overseeing something?
    – Rafael
    Commented Aug 18, 2018 at 23:31
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    @Rafael: I don't know of a word for words like that, which is why I used the rather awkward phrase "words that had spelling variants that weren't associated with different meanings or different pronunciations" in this answer.
    – Asteroides
    Commented Aug 18, 2018 at 23:46

Draconis and Asteroides have done most of the work, but there are some new relations. Let me analyze the structure of the group itself and collect all the simplifying relations. It is worth noting that the relations are based solely on spelling variants, not homophones.

Unfortunately the resulting group has no relations — it is free. This is because all the relations are eventually of the form A=1 or A=B rather than AB=C. The relations only remove generators from the original group generated by all Latin letters.

The following letters are identity (with very brief explanations):

  • H: harena~arena
  • S: exsistere~existere
  • D: adsum~assum (with S=1)
  • F: adferre~afferre (with D=1)
  • G: agnoscere~adnoscere (with D=1)
  • I: eiius~eius
  • J: jam~iam (with I=1)
  • Y: sylva~silva (with I=1)
  • U: optumus~optimus (with I=1)
  • V: vel~uel (with U=1)
  • R: arfui~affui (with F=1)
  • P: Daphne~Dafne (with F=H=1)
  • L: Voltumnus~Vortumnus (with R=1)
  • N: conloquium~colloquium (with L=1)
  • M: immortalis~inmortalis (with M=1)
  • B: urbs~urps (with P=1)
  • T: attuli~adtuli (with D=1)
  • Z: zona~sona (with S=1; mentioned in Vox Latina)
  • A, E, O: long vowels can be spelled as double or single

This leaves us with four letters: C, K, Q, X. Between these we have relations:

  • C=K: Carthago~Karthago
  • C=Q: secundus~sequundus (with U=1)
  • C=X: vixit~vicsit (with S=1, mentioned in Vox Latina)

After identifying all these, we are left with a single generator, C. If there are no further relations, the homophonic group of Latin is isomorphic to the infinite cyclic group (the additive group of integers). Perhaps it sounds most grandiose to call it "the free cyclic group"; there are many possible ways to characterize the group up to isomorphism.

If we also accept the relation C=G due to names like Caius~Gaius and Cnaeus~Gnaeus (and archaic orthography in general), then even the final generator is lost. In this case the homophonic group is trivial.

Of course, if you disagree with some of the relations, you may have more generators. The exact set of valid relations is a matter of opinion. However, it seems that under most interpretations the resulting group will be free.

  • One could get to G=C through archaic orthography (though arguably cheating), right?
    – Rafael
    Commented Aug 20, 2018 at 11:52
  • Sorry, no. That would be the exact opposite of what we are looking for
    – Rafael
    Commented Aug 20, 2018 at 12:44
  • In fact, I would say that it's the trivial group! G = C by the praenomina Gaius ~ Caius or Gnaeus ~ Cnaeus, which used both spellings with the same pronunciation in classical times.
    – Draconis
    Commented Aug 20, 2018 at 15:41
  • (Also, just a nitpick: your example for G doesn't quite work, since both forms have the same number of Gs in them. The attested variant adnoscō does work.)
    – Draconis
    Commented Aug 20, 2018 at 15:44
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    @Rafael By the classical time the abbreviations for Gnaeus and Gaius are Cn. and C., and in this context one can argue that the C and G sound alike. I agree that this is fishy (unless someone finds a better way to put it), so perhaps it's indeed best left as an additional remark as it is now.
    – Joonas Ilmavirta
    Commented Aug 20, 2018 at 19:29

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