\documentclass{article}

\title{Regularity in the phonemic subsystem of English spelling: 
A study of the monosyllables}

\author{Brett Kessler\thanks{The author gratefully acknowledges the 
consultation of Rebecca Treiman of Wayne State University, to whom the
idea for many of these measures and analyses is due.}}

\begin{document}

\maketitle

\bigskip

\begin{abstract}

A list of 4,724 English monosyllables was studied in an effort to 
determine the degree of regularity in writing words
according to the standard orthography of American English.  The
spelling and pronunciation of the words were aligned
semi-automatically, and these alignments were used to gather data on
how individual phonemes are spelt in various phonetic environments.
The data were also used to build a set of rules for reducing English
words to writing; the complexity of these rules was then used as
another measure of regularity.

\end{abstract}

\section{Background}

The irregularity of English orthography, when viewed as a system for
mapping phonemes to letters, has been noted for centuries.  Almost all
commentators on English spelling have noted this irregularity and
blamed it on the conservatism of an orthography which has not kept
pace with the phonetic evolution of the language.  The failure of
spelling reformers to convince the public of the need for reform has
in general been explained by non-linguistic factors such as the
unwillingness of already literate adults to inconvenience themselves
for the sake of their struggling children, the expense of retooling
typewriters and printing presses, and the feared inaccessibility of
pre-reform writings to those who have mastered only the revised
spelling rules.  Perhaps the ultimate estimation in this tradition is
that of \cite{Sampson}, who claims that present English orthography
might best be characterized as fundamentally not phonological but
logographic, assigning essentially arbitrary spellings to each
morpheme.  On the other hand, with the advent of generative
linguistics, many prominent researchers in recent decades have taken a
contrarian stance, holding that English spelling is linguistically
quite regular.  \cite{CChomsky}, for example, approvingly quotes
\cite{NChomsky}, who held that it
``corresponds closely to a level of representation that seems to be
optimal for the sound system of a fairly rich version of ... spoken
English,'' the purported irregularities becoming regular when one
considers deeper levels of the phonology and the syntax.
\cite{Venezky} similarly held that the irregularity in English
spelling, if properly considered as a morphophonemic system with
sufficiently complex phonotactic rules, is mostly a matter of unnaturalized
loan words.

I was interested in the degree to which the regularity of English
phoneme to grapheme correspondences could be quantified.  Although
rather comprehensive studies of the writing system have been
published, including the cited work by Venezky, hard figures are rare,
even though Venezky based his work on a machine-readable list of
twenty thousand high frequency words.  A noteworthy exception is
\cite{Hanna}, although that book gives only type, not token counts, and is
restricted to moderately common words.  For the present purposes, it
seemed particularly advantageous to find a way to factor out the
morphophonemic factors that are clearly present in English spelling
(for example, the different pronunciations of the {\em a} in {\em
sane} and {\em sanity}).  Since most of the orthographically levelled
variation is found in polysyllabic derived forms, I worked with a
comprehensive list of the monosyllabic words current in standard
American English, excluding inflections and derivations (notably {\em
th} forms) where the stem does not change.  This tack also had the
advantage in resulting in a smaller (4,724 word) corpus, providing a
reasonable chance for decent quality control in a short time span.

\section{Word list}

The word list consists of all monosyllabic words found in the most
recent un\-abridged Random House dictionary \cite{Random}, provided that
there was no indication that the word was archaic or obsolete, slang,
nonstandard, regional, paralinguistic (interjections like {\em sh!}),
foreign, or not in general use in the United States of America.
Regular inflected or derived forms {(\em calls, called, warmth)} were
omitted, although irregular forms were included {(\em does, did,
health),} counting as ``irregular'' anything not based on
concatenation to the full free form of the base word.\footnote{These
somewhat irregular forms were included because throwing them out would
have resulted in omitting many stems that are clearly spelt
phonemically; it is apparent, for example, that the spelling {\em
taught} is not unduly influenced by the spelling of {\em teach,} and
the same seems to hold true for most irregular past forms.  On the
other hand, this policy resulted in including a certain number of
instances of morphophonemically-spelt morphemes, the only significant
example being those spelt {\em -s.} A replication of this study would
do well to delimit those derivational and inflectional morphemes so
that they can be excluded from the tallies.} Initialisms and words
regularly spelt with any punctuation or accent marks were excluded.
Proper names were included only if they were significantly Anglicised
{(\em Spain),} or common given names {(\em Jane).} This led to the
perhaps surprising result that names of American places and famous
people were omitted, whereas foreign place names like {\em Rhodes}
were included in this list of American words.  The reason for this is
to some extent the uncertain status of the naturalization of American
names, so many of which are of non-English origin, and also the
encyclopedic nature of the source dictionary, which includes many
proper names not likely to be familiar to most writers.  Nevertheless
this policy is to some degree arbitrary, and a replication of the
study omitting proper names might be useful.

\subsection{Details}

\small

As a convenience for me as human editor, a base word list was formed
by combining words found in several other word lists, including a list
supplied by Venezky, and the list published by \cite{SCRL}.  However
I corrected this list by manually adding or deleting words based on the
authority of the Random House dictionary, whose decisions on inclusions,
spelling, pronunciation, and usage were considered absolute.

All monosyllabic words in Random House were included except for the
following:

-- Those having a regional indicator showing chief region of usage
being outside the US: e.g., {\em Scot., Brit.,} even if the term is
qualified by {\em Chiefly.} Example: {\em rax.}

-- Words qualified as being in a different language: {\em ach, German.}

-- Certain classes of words that almost always retain foreign spellings:
terms from non-English monetary systems (coins, monetary units),
weights and measures (liang), calendars (names of months), letters of
the alphabet, peoples and nations.  I include however such terms if
they are manifestly Anglicised, such as by having a different spelling
than in the foreign language.  Terms from Latin or Greek antiquity are
automatically considered Anglicised.

-- Words printed in italics.

-- Initialisms spelt with all caps: A, AID.  I include only
initialisms that now look and are pronounced like normal words {\em
(Wac)}.

-- Words spelt with punctuation marks, including periods, hyphens, and 
apostrophes.

-- Words with accent marks.

-- Prefixes, suffixes: a-, -a

-- Words found as parts of multi-word entries: ad lib

-- Secondary spellings: adze, aesc, Ab.  That is, I exclude words that
are defined exclusively by reference to another spelling, and instead
include that spelling.  However, if a word appears in two different
spellings and each of those spellings has a different main
pronunciation, I consider them to be two different entries and include
both, even if it is obvious that the words have a common origin and
meaning.

-- Words with non-English sounds in their pronunciation: Ain [aN].
These symbols are spelt with caps: A, KH, N, OE, R, Y.

-- Words not containing any indication of the time when they entered
the language.  (These appear universally to be only marginally
English.)  Exceptionally, proper given names are not discarded under
this rule.

-- Words marked {\em Nonstandard, Slang, Obs[olete], Archaic, Rare,
Older Use, Facetious, Baby Talk, Literary, Eye Dialect, Pron.
Spelling.} However, I do include terms with other labels, such as
those indicating usage within a particular subject field (e.g., {\em
Chemistry),} and such labels as {\em Informal, Vulgar, Disparaging} and
{\em Offensive} (if not also {\em Slang,} etc.)

-- Words whose definition indicates that they are used only in non-English
contexts or names, e.g. {\em de.}

-- Compounds of productive morphemes: {\em acts, paid.}  In particular, if
the pronunciation of a plural or verbal 3rd person singular is formed
by adding /s/ or /z/, I omit it; if a past tense or participle is formed
by adding /d/ or /t/, I omit it; if a noun is formed by adding /q/
(e.g., {\em warmth),} I omit it.  This holds even if the spelling seems
irregular {\em (pay, paid).}  However, if the pronunciation of the base
changes in the process, I include the word {\em (catch, caught; have, has;
long, length).}  Also, if a word looks like a plural or past participle
but there is no entry for the base word, I include the word.

-- Most proper names, such as people's surnames, place names.  However
I do include given names that have their own entries as such, e.g.,
{\em Bob,} a male given name.

Words that have the same spelling and pronunciation are considered to
be one entry for the list, even if there are different meanings, or
even if there are multiple entries in the dictionary.  (For example,
the dictionary has 3 entries for {\em bill} in American usage, but
they are all pronounced the same, so I give it one entry.)  However,
if they differ in capitalisation, I include both (e.g., I also include
the given name {\em Bill} as a separate entry), even if the common and
proper words seem to have the same etymology.

On the other hand, a single Dictionary entry may give different
pronunciations for different word senses.  In this case, I make two
different list entries.

If a dictionary has multiple spellings or pronunciations for a word,
the filtering criteria are applied to the first such.  For example, if
the first pronunciation is polysyllabic or includes a foreign sound,
the word is excluded, even if a subsequent pronunciation is monosyllabic
or nativized.

Words whose pronunciations include a stress mark are omitted as polysyllabic,
even if there is only one vowel: {\em button} [but'n].

\normalsize

\subsection{Word list entries}

The word list includes the spelling, pronunciation, and frequency of
each word.  (Its age is also included, but this information was not
made use of in this study.)  

The first spelling of the main entry in Random House was used; other
spellings were included only if a different pronunciation was
associated with it, and then the pair was listed as a separate entry.
The spelling included initial capitalization if appropriate, although
case was not considered distinctive in the actual analyses carried out.
By the rules of exclusion, no spelling had any characters other than
the twenty-six letters of the alphabet, all letters but the initial
being in lower case.

The pronunciation was taken from the first listed in the dictionary.  The
transcription was based on a common scheme for approximating the 
International Phonetic Alphabet in the impoverished ASCII code space.
Lower case symbols are close to the expected IPA value, whereas majuscules
represent arbitrary divergence from the corresponding minuscule.  The
following table describes how the dictionary symbols were transcribed
on the word list.  The sample words will serve to define the phonetic
values of the symbols in my transcription system.

\begin{tabular}{lll}
RHD & Examples & ASCII \\

a & act, bat, marry & a \\
\=a & age, paid, say & e \\
\^a & air, dare, Mary & E (like e, below) \\
\"a & ah, part, balm & A \\
b & back, cabin, cab & b \\
ch & child, teacher & T \\
d & do, madder, bed & d \\
 e & edge, set, merry & E \\
\=e & equal, seat, bee, ear, mere & i \\
f & fit, differ & f \\
g & give, trigger & g \\
h & hit, behave & h \\
hw & which, nowhere & H \\
i & if, big, mirror & y \\
\=i & ice, bite & Y \\
j & just, tragic & D \\
k & keep, token	& k \\
l & low, mellow, all & l \\
m & my, summer, him & m \\
n & now, sinner, on & n \\
ng & sing, Washington & G \\
o & ox, bomb, wasp & O \\
\=o & over, boat, no & o \\
\^o & order, ball, raw & c \\
oi & oil, joint, joy & C \\
oo short & book, tour &	U \\
oo long & ooze, fool, too & u \\
ou & out, loud, cow & W \\
p & pot, supper & p \\
r & read, hurry & r \\
s & see, passing & s \\
sh & shoe, fashion & S \\
t & ten, matter & t \\
th & thin, ether & q \\
th slash & that, either & Q \\
u & up, sun & V \\
\^ur & urge, burn, cur & R  (eg: b\^urn = bRn) \\
v & voice, river & v \\
w & witch, away & w \\
y & yes, onion & j \\
z & zoo, lazy & z \\
zh & treasure, mirage & Z \\

\end{tabular}
\medskip

As can be seen, only quite minor deviations were made from the pattern
employed in the Dictionary; indeed, almost all changes were the direct
substitution of one symbol for another.  The main exceptions were the
omission of the tiny schwa used in diphthong + /r/ sequences, which is
clearly subphonemic; {\em \^ur,} which was probably meant to be
interpreted as two symbols but which I translated to one, /R/ (since
{\em \^u} only appears before {\em r} and the sound is phonetically
unitary); and {\em \^a}, which I treated as a phonetically conditioned
variant of {\em e} (since it only occurs before {\em r,}\footnote{It
also occurs finally, in the word {\em yeah,} but this too is
distinctive, since /E/ never appears there.} where the identical or
similar sound {\em E} never appears).

The pronunciation transcribed is a cross-dialectal generalization,
although there are probably Americans whose phonemic inventory could
be accurately described by it.  Some important qualities of the
``dialect'' described in the word list are listed below.  Most of these
variations in accent are quite regular, and it may be worthwhile
replicating this study with adjustments more closely approximating
more widespread modern dialects.

-- There are two low back vowels, /A/, and /O/, as in {\em spa} and
{\em cot,} although /A/ is fairly marginal (except before /r/, q.v.
infra).  For most Americans, these phonemes fall together into one
phoneme, except before /r/, where another merger takes place.  Thus
they have only one low back vowel.  This is probably the single greatest
caveat to keep in mind when considering the linguistic or psychological
verisimilitude of this transcription for any one speaker.

-- Similarly there is a distinction between /O/ and /A/ on the one hand
and /c/ on the other {\em (cot, caught),} which is rapidly being lost
in America, though this has not yet progressed to the same extent
as the previous merger.  So for many speakers, I am transcribing their
single vowel as three.

-- /r/ is sounded where the letter {\em r} appears, and there is no
intrusive (linking) /r/.  The distinction /o/ vs. /c/ vs. /O/ is neutralized
before /r/, appearing as /c/ ({\em horse = hoarse,} /hcrs/).
Similarly historic /er/ is lowered to /Er/ {\em (care} = /kEr/) and
/ur/ to /Ur/ {\em (boor),} but /ir/ is so transcribed (not /yr/), as in
{\em beer.}  In contrast with Scottish English, the early
Modern English sequences /Er/, /yr/ and /Ur/ (viz. /Vr/) are all
neutralized to one vowel merging with the /r/, giving /R/.

-- The word list recognizes a voicing contrast in /w/ vs. /H/, {\em witch,
which,} which is also being rapidly lost in the recent couple of
generations.

-- Where there is some variation between /a/ and /A/ in such words as
{\em half, can't} and {\em aunt}, this list gives /a/.

-- /ju/ is transcribed for the sound that many speakers pronounce as
centralized /yu/ as in {\em feud.}  Note that it is treated as a sequence
of two phones.  The /j/ of /ju/ is dropped after coronals, as in {\em new}
/nu/ and {\em due.}


Finally, in order to give some idea of how common the words are in
practice, the token count from \cite{Brown} is added.  This is a
frequency analysis of a corpus of approximately one million words of
text published in 1961.  Unlike most computerized word counts, this
has been disambiguated by word class, which usually suffices to
distinguish differently pronounced homographs.  In the very few
instances where such homographs are of the same word class and
therefore have a merged frequency in \cite{Brown}, the frequency was
split between the homographs, evenly if I had no intuition that one
word was overwhelmingly more frequent.  In the word list, I treated
each word as a particular inflected form, not as a lemma; so the
frequency count for {\em cat} gives the frequency only of that base
form, excluding the frequency of {\em cats, cat's, cats'.}

\section{Alignment}

The problem of aligning spelling and pronunciation, that is, indicating
explicitly which letter or letters spell which sounds, is not as
straightforward as it would appear at first consideration.  Everyone
would agree that in the word {\em kit,} the three letters spell the three
sounds /kyt/, in order.  But problems arise in more complex words, 
in the end raising questions even concerning this uncontroversial 
explication.

-- ``Silent {\em e.}'' In words like {\em kite,} one traditional
explication is that the final {\em e} is silent, but makes the vowel
long.  Another is that {\em i} and {\em e} together constitute the /Y/
sound.  Both of these assume therefore that the order of graphemes in
spelling can vary from the order of the phones.  A more historical
account would say that the {\em e} is a relic of the fact that the
syllable was previously open, and that that openness accounts for the
tense vowel; therefore it is a mark that the {\em t} behaves as if it
is in the onset of a following syllable rather than the coda of the
previous.  Or to simplify somewhat, {\em te} spells /t/ and also
conditions a tense pronunciation of the preceding vowel.  These
accounts are equivalent and of little direct interest to the
graphonomist, but make an enormous difference when it comes time to
explicitly align spelling and pronunciation.

-- Conditioning.  In words like {\em cite}, the {\em c} spells /s/ and
the {\em i} /Y/, but this does not happen in a vacuum.  The {\em c}
spelling is licensed by the fact that the succeeding letter is an 
{\em i;} /s/ may not have that spelling except before {\em e, i,} or
{\em y.}  In a very real sense, then, the {\em i} is part of the 
spelling of the /s/.

-- Silent letters versus polygraphs.  Certain spellings such as {\em
sh} for /S/ are commonly explicated as digraphs: both letters together
spell the /S/ sound.  In words like {\em debt} on the other hand, the
{\em b} is consistently explicated as a silent letter, and not part of
an {\em eb} or {\em bt} digraph.  While these particular intuitions
seem clear, multitudinous boundary cases occur.  In {\em night,} the
{\em gh} may be part of an {\em igh} trigraph, or it may be a silent
digraph that nevertheless conditions the {\em i.} In {\em bought,} it
is quite unclear which letters are part of a polygraph and which are
silent.  If {\em o} is /c/ as in {\em soft,} then perhaps {\em au} is
not a digraph in {\em sauce.} If {\em ou} spells /c/, then it is
curious that this never happens except before {\em gh.} If {\em ough}
spells /c/, then perhaps {\em gh} should always be considered part of
polygraphs, even in words like {\em caught} where {\em *caut} would be
a regular spelling.  There do not seem to be any firm criteria for
distinguishing the two cases.  And indeed, there does not seem to be
any firm reason for not saying that in {\em ship,} the {\em h} is
silent but conditions the pronunciation of the {\em s.}

-- Etymology.  Those familiar with a word's etymology are unlikely to
tolerate sound/letter assignments that flout the historical state of
affairs.  In particular, most would agree that recently borrowed
words that retain their original spellings should have assignments
that correspond, within the limits of the nativized pronunciation,
to the presumed sound/letter alignments of the original language.

Confronted with these issues, I adopted the following policies:

-- Spelling is strictly sequential; alignment lines do not cross.
This settles the issue of whether silent letters might not be put to
good use helping spell discontiguous phones in the word.  In
particular, final {\em e} after consonants in syllables with tense
vowels cannot be considered as part of the spelling of the vowel.

-- No many-many alignments.  It may be that one phone is spelt by
multiple letters ({\em she} = /Si/) or that multiple phones are spelt
by one letter ({\em six} = /syks/), but multiple phones cannot be
spelt by multiple letters.  More strictly, a phone may be associated
with any number of contiguous letters, but only if none of those
letters spells another phone in whole or in part.  Thus in cases of
conditioning, a choice must be made as to which letter spells what; we
cannot say that in {\em cite c} and {\em i} both spell /s/ if {\em i} 
also spells /Y/.  Conditioning is considered a secondary factor that
can be explored once the primary associations have been made.

-- No unassigned letters.  All letters must be assigned to one or more
phones; none are analyzed as ``silent.''  Therefore in analyzing words
like {\em bought,} the {\em gh} question reduces to one of whether
those letters should be considered part of the vowel or part of the
following {\em t,} a question on which there are clearer intuitions.
Of course this decision does not literally mean that there are no
silent letters; rather, that all letters in some way contribute to the
phonological spelling.  In particular, final silent {\em e} after a
consonant is considered part of the spelling of the consonant.  In
analyses later on, we can see to what extent {\em e}-less and {\em
e}-ful spellings of consonants condition the spelling of the vowel.

Although it would be nice to write a programme that could align
spellings and pronunciations automatically, given no data beyond the
word lists and these general policies, I abandoned the idea after a
few tentative attempts.  Etymology is one confounding issue; although
a human brings to the task full cognizance of the spelling rules for
Latin, French, and other languages from which words have been borrowed
wholesale, to the computer these are often perturbations whose rarity
causes them to become lost in the forest.  Another factor is that the
spelling system as a whole is a good deal more textured than might be
immediately apparent; it certainly cannot be described by frequency
models.  For example, one programme insisted that in {\em Zeus,} {\em
Ze} spells /z/ as in {\em blaze, doze,} etc., and {\em u} spells /u/ as
in {\em rude, lube,} etc.  Certainly those spellings are more frequent
than any {\em eu} = /u/ correspondence in English.  And yet any human
would prefer the latter.  This is due partly to etymology, partly to
greater sensitivity to context, and partly to a higher understanding
as to the functional utility of the various spellings.  I do not think
the latter two are beyond the scope of a computer programme, but it is
not immediately clear to me how to implement them, at least not in a
short time frame.

Instead I took a semi-automatic approach.  I prepared a list of 292
grapheme to phoneme correspondences operative in the monosyllables list.
See for example the segment reproduced below\footnote{The entire list
is given in the Appendix as file {\tt rules.}}.  The third column, an
example, is a comment for the human reader.  A programme uses these
rules to align the words from left to right, so the ordering of the
rules is significant.  For example, the programme will attempt to
match {\em aye} before attempting to match plain {\em a,} but both
are attempted before {\em ach.} This occasionally resolves some
ambiguity in whether a letter should be assigned as part of a
polygraph to its left or its right.

\begin{verbatim}

aye	Y	aye
aye	E	crayer
aye	e	raye
a	e	a
a	a	abb
a	A	al
a	c	all
a	E	are
a	O	quad
a	V	was
ach	a	drachm
ach	O	yacht

\end{verbatim}

Rather surprisingly, in view of this simple control structure, the
programme produced a satisfactory alignment of all words, without the
need for human post-editing.  Of course, this by no means is to say
that there are not any alignments that different people might disagree
on.  The following snapshot shows part of the monosyllables list with
alignments added.\footnote{The entire file, called \texttt{dictAlign},
is available.} The first three columns are the
pronunciation, spelling, and frequency, as previously discussed.  The
last two columns show the alignment, by padding with dashes when there
is a one-many correspondence.  Again, it is perhaps surprising that
this technique gives a satisfactory alignment, since there is no
indication as to whether the character should form a cluster to the
left or to the right.  The convention used here is that it always
clusters to the left (except of course at the beginning of the word).
That this works perhaps indicates that English orthography, at least
in the purely phonemic module, evolved to minimise or even eliminate
parsing ambiguities, despite the extensive use of digraphs.

\begin{verbatim}

blRt	blurt	0	blurt	blR-t
blVS	blush	2	blush	blVS-
blYQ	Blythe	0	blythe	blYQ--
bcr	boar	1	boar	bc-r
bcrd	board	239	board	bc-rd
bost	boast	8	boast	bo-st
bot	boat	73	boat	bo-t
bOb	bob	0	bob	bOb

\end{verbatim}

This file was used as the data file for all subsequent analyses.  Lisp
programmes read in the entries and generate various reports on what
letters spell which phones in various environments.


\section{Analyses}

The basic method of analysis performed is the {\tt plain-report,}
which is included in full in the Appendix.  Here is the
automatically-generated analysis of the phone /i/:

\small

\begin{verbatim}

      TYPES  TOKENS

  /i/
ea     148    5785 Bea beach bead beak beam bean beard beast beat bleach
ee     144    5492 bee beech beef beep beer beet beeves bleed bleep
e       42   95801 be cede cere cete Cleve deke deme eke eve Eve gene
ie      35     721 bier Brie brief chief fief field fiend fierce frieze
i       21      45 chic clique fine firn gi gigue guige kirsch li lisse
ei       8      46 ceil Keith Neil Reid seize sheik weir weird
ay       1       0 quay
es       1       0 mesne
eigh     1       3 Leigh
eh       1       0 lehr
ey       1      85 key
      ----  ------
TOT    403  107978
H    2.149   0.664

\end{verbatim}

\normalsize

All the different spellings of the /i/ phoneme are listed in the
leftmost column.  The next column tells how many different words they
appeared in.  The heading {\tt TYPES} refers to word types, without
reference to the number of times they appear in text.  The third
column tells how many times the correspondence appears in running
text, in the million-word Brown corpus.  The line is filled out with
as many examples as will fit.  At the bottom these per-type and
per-token counts are totaled, and Shannon's uncertainty statistic H is
given.  This is a probabilistic measure of how easy it is to guess a
spelling; the more spellings possible and the less distinguished they
are by probability, the higher the H measure.  Note here that in the
tokens column, where the spelling {\em e} becomes wildly more probable
than the other spellings (because of the frequency of words like {\em
be, she, he,} and {\em me}), the H measure drops significantly
vis-\`a-vis the types column, which is much smoother and whose top two
entries are virtually indistinguishable probabilistically.

This might be a good opportunity to mention the differential utility of
the types and token counts.  It is of course of great importance to
know the general frequency with which words appear in actual text.  A
spelling that appears in five word types that appear several hundred
times in the Brown corpus is in some sense more important than one
that appears in five hapax legomena, and an educator may get more
mileage out of teaching the totally anomalous spelling {\em ey} as in
{\em key} before teaching {\em i,} which appears in twenty-one
different words.  In general, however, I am concerned with the overall
spelling system, as it appears to the fully literate adult who knows
all the words in the corpus.  One can assume that when the adult comes
to spell a word, s/he brings to the task the assumption that a given
word is always spelt in the same way, and the capability of
associating certain exceptions with the appropriate word.  In this
context, the token count is irrelevant.\footnote{One might observe
that adults in general do not know all the words in the word-list, and
that the frequency might profitably be used as a filter for relegating
some words to a marginal status.  Unfortunately, a one-million word
corpus is not large enough for this task.  Such common words as {\em
ewe} and {\em brooch} are not found at all in the Brown corpus.} In a
rule-based system there is no utility in giving prominence to the /i/
- {\em e} correspondence, even though it works for nine out of ten
word tokens one tries to write, because one needs rules for coping
with words that are not so spelt.  And by the time one finishes the
list of subsidiary rules, which are based on such matters as type of
environment, and exceptions, which must be itemized by word type, one
finds there is no longer any advantage to giving any pride of place to
the {\em e} rule.

It turns out that /i/ is perhaps the most difficult of all phones to spell.
The average values are given below:

\pagebreak

\begin{verbatim}

Phone types: 41
                    WORD TYPES     TOKENS
Phone tokens:          16409      1543854
Ave phone tokens/type:   400        37655

Ave H:                 1.167      0.789
Ave (weighted) H:      1.095      0.681

\end{verbatim}

The average H just sums the H values of each phone and divides by 41, the
number of phones.  The weighted average weights each H by how frequent the
phone is, and so probably gives a more accurate picture of the general
difficulty of spelling.  Of course here the difference is quite tiny.

Taking advantage of the convenience of the uncertainty statistic H, I
computed the uncertainty of spelling phones occurring in various places
in the syllable.  In the table below I present only the weighted H;
full reports on each phoneme are given in the Appendix for
each test.

\medskip

\begin{tabular}{lcc}

 & WORD TYPES & TOKENS \\

Phonemes                            & 1.167  &    0.789 \\

Initial phones  /\#\                & 0.373  &    0.154 \\

Final phones  /\_\#                 & 1.301  &    0.654 \\

VOWELS                              & 1.392  &    0.963 \\

Initial vowels /\#\_                & 1.113  &    0.159 \\

Medial vowels /C\_C                 & 1.139  &    1.080 \\

Final vowels  /\_\#                 & 2.248  &    0.749 \\

CONSONANTS                          & 0.963  &    0.523 \\

Consonant in onset /\#C*\_C*        & 0.372  &    0.177 \\
 
Initial consonant /\#\_             & 0.345  &    0.151 \\

Initial single consonant /\#\_V     & 0.371  &    0.152 \\

Initial clustered consonant /\#\_C  & 0.192  &    0.077 \\

Non-initial, onset consonant /\#CC*\_ & 0.283  &    0.159 \\

Consonant in coda /C*\_C*\#           & 1.191  &    0.619 \\

Non-final, coda consonant /\_CC*\#    & 0.199  &    0.105 \\

Final consonant /\_\#                 & 1.219  &    0.609 \\

Final single consonant /V\_\#         & 1.231  &    0.658 \\

Final clustered consonant /C\_\#      & 0.671  &    0.129 \\

\end{tabular}

To summarize these results, vowels are more unpredictable than
consonants (assuming a purely probabilistic writing algorithm).  Both
vowels and consonants are more uncertain when final.  Consonants are
more certain when in clusters, and especially so when at the beginning
rather than at the end of a cluster, regardless of whether this is
next to the vowel or word-marginal.  It should be noted that a good
deal of the uncertainty in final consonants results from my assigning
final silent {\em e} as a polygraph with the preceding consonant rather
than with the long-distance vowel.  Those who prefer the opposite approach
would transfer much of the uncertainty to medial vowels.

I also generated reports showing how the uncertainty of each phoneme
differs before and after each possible phoneme, as well as word
initially and word finally.  This data is too long to include in this
report, but may be accessed via ftp.\footnote{Files \tt
before-report \rm and \tt after-report, \rm in the previously cited
location.} One interesting observation is that if the following
phoneme is known, a phoneme can be spelt with uncertainty of 0.666
(0.371 over word tokens), a significant reduction over the 1.167
uncertainty of blindly spelling a phoneme.  In the other direction,
spelling a phoneme when the previous one is held constant, the
uncertainty is a comparable 0.621 (0.251 with word tokens).  Thus
knowing the immediate context is valuable, but on the average, the
left neighbour is as good as the right.  Of course the details vary
from phone to phone.

\section{Rules}

Although the uncertainty statistic H is an interesting and convenient
measure of the number of possible spellings for a phone and their
frequency distribution, it has important limitations, in that it
assumes a probabilistic model of spelling.  Therefore I have also
constructed a rule-system for reducing the words in the monosyllable
list to writing, given only accessible linguistic knowledge about the
word.\footnote{Appeal to etymology was ruled out, especially since
people's intuitions about it, shaky at best, seem largely influenced
by knowledge of the spelling.} I then attempted a crude measure of the
complexity required to adequately handle the corpus, and compared it
to the H statistic.

The rule system presented below is ideal, in the sense that it
attempts to explain the regularity in the data, rather than the
technique people actually use to spell.  To be sure, I attempted to
check the growth of the rule system by assigning to rules a ``weight''
equivalent to one exception, and in fact preferring exceptions to the
equal number of rules.  So a rule is not admitted unless it explicates
at least two words more than the number of exceptions it robs from the
productive output of some other rule; in the simplest case, a rule
must explicate at least two forms.  But it seems unlikely that rules
are psychologically so light-weight.  I was certainly surprised by the
majority of the rules here presented.  It also seems likely that a
fairly unproductive rule may be tolerated if it is exceptionless,
whereas rules with exceptions need to have a large ratio of output to
exception.

In general, the rule system appeals to the phonetic environment of a 
pho\-neme.  But in certain cases a great deal of complexity was avoided
by appealing to the spelling.  To avoid circularity in these cases, I
permitted reference only to the spelling of previous phones in the word.
This is because the assumption is that the phones are spelt in strict
sequence.  In particular, the multi-phone spellings such as {\em x} are
treated only under the first phoneme, here /k/, with a notation showing
that the rule also consumes the /s/.

Given that rules were considered to be about as heavy as exceptions,
the weight of a rule system for a phoneme was determined by summing up
the number of rules and exceptions.  But a weight of ten obviously
means one thing if the phone is only found in ten words, and another
if in a thousand, so I introduced a Complexity measure, which is the
ratio of the weight to the number of times the phoneme appears in word
types.  The Complexity ratios discovered vary from 48.88\% for /i/ to
1.27\% for /p/ (all phones must have some Complexity, because they
require at least one rule).

Details for the various phones are given below.  For vowels, an average
Complexity of 21.40\% was found, ranging down to 1.81 for /a/.  The Complexity
measures have a 93\% correlation with the H measures.  The ranked
list of complexities is:

\bigskip

\begin{tabular}{lr}

/i/ &  48.88 \\

/R/  &  47.89 \\

/u/  &  47.30 \\

/e/  &  31.96 \\

/U/  &  31.67 \\

/o/  &  31.58 \\

/c/ &   27.44 \\

/Y/  &  26.39 \\

/W/ &   24.03 \\

/E/ &   13.65 \\

/C/  &  10.71 \\

/V/  &   7.98 \\

/y/  &   6.44 \\

/A/  &   6.36 \\

/O/  &   1.91 \\

/a/  &   1.81 \\

\end{tabular}

It is readily apparent that tense vowels and diphthongs have a higher 
Complexity than lax vowels; the weighted averages are 33.70\% over against
6.87\%.  Since only tense vowels can be final, it might be wondered
whether this difference is related to the fact that final vowels have higher
H measures.  In fact, tense vowels have approximately equal Complexity
regardless of their position in the word.  So it seems likely that, at
least to some extent, the higher uncertainty of vowels in final position
is due to the fact that they are all tense vowels.

For consonants the average weighted Complexity is 6.47\%, and the
correlation between H and Complexity is only 0.605422.  One possible
explanation for the low correlation is that for consonants subject to
final doubling or silent {\em -e,} the variation is quite easily
accounted for by rule, whereas most of the uncertainty of other
consonants is not quite so tractable.  At any rate, the rank order of
the consonants, by rule Complexity, is as follows.  It should be kept
in mind that some phonemes with relatively simple rule systems, such as
/Z/, tend to have a lot of Complexity because so few words have the
sound; others, such as /z/, are complex because there are many rules
and exceptions.  The cases can easily be differentiated by inspecting
the detailed presentation in the Appendix.

\bigskip

\begin{tabular}{lr}

/Z/	 & 42.86 \\

/z/	 & 30.11 \\

/j/	 & 24.42 \\

/S/	 & 12.82 \\

/D/	 & 12.44 \\

/k/	 & 12.06 \\

/Q/	 & 10.00 \\

/s/	 &  9.86 \\

/T/	 &  8.68 \\

/f/	 &  8.26 \\

/n/	 &  7.34 \\

/g/	 &  5.99 \\

/m/	 &  4.91 \\

/w/	 &  4.75 \\

/l/	 &  4.52 \\

/r/	 &  4.41 \\

/v/	 &  4.40 \\

/h/	 &  3.24 \\

/d/	 &  3.07 \\

/t/	 &  2.81 \\

/H/	 &  2.38 \\

/b/	 &  1.72 \\

/q/	 &  1.56 \\

/p/	 &  1.27 \\

\end{tabular}

The average weighted Complexity for spelling any phoneme in particular
was 11.19\%.

The appendix contains a detailed presentation of the proposed rule
system for spelling the English monosyllables.  In addition to the
rules and exceptions, under each phoneme is also given the measure of
Complexity, and the synopsis as discussed above, including exact type
and token counts for each spelling, and both type-based and
token-based H measures.  For an excellent summary of the historical
factors leading to the present multiplicity of spellings for each
phone, see \cite{Cummings}, in particular the {\em Correspondences}
chapters, which are organized by phoneme.  The reader should keep in
mind that the frequencies cited there are for words in general, not
just monosyllables, and so are often radically different from the
figures presented here.

\section{Conclusions and Futures}

The techniques developed in the course of this project are certainly
useful for comparing the degree of regularity across different phones
and/or across different environments in the word.  The computation of
the uncertainty statistic H is particularly easy to use, new
analyses being performed by writing a short, largely stereotyped Lisp
function.  Its basis in statistical science is comforting, and to
those who would rightly argue that it has no necessary relationship to
{\em regularity} per se, since it appeals to no rule system, it can be
answered that we have seen a moderate to strong correlation with rule
complexity, as well as many examples (especially in the Appendix),
where the statistic agrees with our intuitions about the difficulty of
writing a phone by rule.  Studies based on the H statistic could be
used to investigate more closely just where the uncertainty in English
spelling is.  A practical application of this might be in paedagogy,
where statistics could be used to locate the most regular spellings
for early phonics training, and more irregular ones for later study.
The use of the word token frequencies could be particularly useful
here.  Although this study has concentrated on a few environments of
known importance in the paedagogical and psychological literature
(e.g., clusters and the position within the syllable), one could also
imagine a programme that would explore and report on particularly
difficult environments, or pursue environments of decreasing
uncertainty to report on contexts that might be of use in rules.

The work on a rule system for monosyllabic phoneme-oriented spelling
had a few interesting results beyond the wealth of details.  The most
useful spelling rules were based on the values of adjacent phonemes --
some phonemes caring more about the preceding, some the following;
most caring about immediately adjacent phonemes, a few caring about
two or more -- or on the other spellings in the word.  Natural phoneme
classes were only rarely evidenced, with the exception of tense and
lax vowels, which figured vitally in the spelling of following
consonants.  A general principle against mixing uniquely Greek
spellings such as {\em ph} with uniquely English spellings such as
{\em ea} proved disappointingly unprolific, its scope
largely being subsumed by that of more productive rules.  For future work, it
would be interesting not only to attempt to discover potential rules
as described in the paragraph above, but also to implement the rules
in a programme, in order to check their validity and gather more
statistical data.  Unfortunately a real problem here is lack of
clarity as to what constitutes a valid rule.  As already mentioned, it
is highly questionable whether writers actually maintain rules that
generate as few as two words, or rules that have more exceptions than
regular outputs.  Psychological studies, involving for example
requests that subjects spell phonotactically correct nonsense
syllables, might help shed some light on which of these rules are
psychologically real, and ultimately tell what properties rules must
have to be realistic.

Although this study has clearly achieved a measure of success in
exploring and documenting differences in regularity and uncertainty in
the various parts of the phonemic subsystem of English spelling, it is
less certain that it is now possible to give a simple answer to the
question of how regular English orthography is, even for
monosyllables.  The answer that for an average phoneme the ratio of
rules and exceptions to instances in word types (my Complexity ratio)
is 11.19\%, or that the uncertainty statistic H is 1.167 across word
types and 0.789 across tokens, might not satisfy anyone very much.  It
might be more relevant if those statistics were compared to their
counterparts for other alphabetically spelt languages, or if more
confidence could be given to the psychological reality of the rules.
As it stands, I can state that with 180 well-chosen rules, there
remain 1,586 exceptions,\footnote{These exceptions are per phoneme,
not per word.  It may of course be true that spellers tend to memorize
whole words as simple gestalts, instead of potentially many
exceptional phoneme correspondences for the same word.  Of course if
this is the case, and it applies to more regular spellings as well,
the whole question of regularity in spelling takes on much added
complexity.} and any rejection of some of the rules on psychological
criteria will only raise the number of exceptions.  For spelling 4,724
core words of a subsystem with only nominal interference from other
regular linguistic systems such as morphological analysis, that seems
rather complex.


\newpage

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\bibitem[Random House 1987]{Random} 
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\end{thebibliography}

\end{document}

