# Is Morse Code binary, ternary or quinary?

I am reading the book: “Code: The Hidden Language of Computer Hardware and Software” and in Chapter 2 author says:

Morse code is said to be a binary (literally meaning two by two) code
because the components of the code consists of only two things – a dot
and a dash.

Wikipedia on the other hand says:

Strictly speaking it is not binary, as there are five fundamental
elements (see quinary). However, this does not mean Morse code cannot
be represented as a binary code. In an abstract sense, this is the
function that telegraph operators perform when transmitting messages (see quinary).

But then again, another Wikipedia page includes Morse Code in ‘List of binary codes.’

I am very confused because I would think Morse Code actually is ternary. You have 3 different types of ‘possibilities’: a silence, a short beep or a long beep.

It is impossible to represent Morse Code in ‘stirct binary’ isn’t it?

By ‘strict binary’ I mean, think of stream of binary: 1010111101010.. How am I supposed to represent a silence, a short beep and / or a long beep?

Only way I can think of is ‘word size’ a computer implements. If I (and the CPU / the interpreter of the code) know that it will be reading 8 bits every time, then I can represent Morse Code. I can simply represent a short beep with a 1 or a long beep with a 0 and the silences will be implicitly represented by the word length.(Let’s say 8 bits..) So again, I have this 3rd variable/the 3rd asset in my hand: the word size.

My thinking is like this: I can reserve the first 3 bits for how many bits to be read, and last 5 bits for the Morse code in a 8bit word. Like 00110000 will mean ‘A’. And I am still in ‘binary’ BUT I need the word size which makes it ternary isn’t it? The first 3 bits say: Read only 1 bit from the following 5 bits.

Instead of binary, if we use trinary, we can show morse code like: 101021110102110222 etc.. where 1 is: dit 0 is: dah and 2 is silence. By using 222 we can code the long silence, so if you have a signal like *- *— *- you can show it like: 102100022210, but it is not directly possible using only with 1’s and 0’s UNLESS you come up with something like a ‘fixed’ word size as I mentioned, but well this is interpreting, not saving the Morse Code as it is in binary. Imagine something like a piano, you have only the piano buttons. You want to leave a message in Morse Code for someone and you can paint buttons to black. There is no way you can leave a clear message, isn’t it? You need at least one more color so you can put the silences (the ones between characters and words. This is what I mean by trenary.

I am not asking if you can represent Morse Code in 57-ary or anything else.

I have e-mailed the author (Charles Petzold) about this; he says that he demonstrates in Chapter 9 of “Code” that Morse Code can be interpreted as a binary code.

Where am I wrong with my thinking? Is what I am reading in the book, that the Morse Code being a Binary a fact or not? Is it somehow debatable? Why is Morse Code is told be quinary in one Wikipedia page, and it is also listed in List of Binary Codes page?

Edit: I have e-mailed the author and got a reply:

—–Original Message—–

From: Koray Tugay
Sent: Tuesday, March 3, 2015 3:16 PM

To: cp@charlespetzold.com

Subject: Is Morse Code really binary?

Sir, could you take a look at my question here: Is Morse Code binary, ternary or quinary? quinary ?

Regards, Koray Tugay

From: “Charles Petzold”

To: “‘Koray Tugay'”

Subject: RE: Is Morse Code really binary? Date: 3

Mar 2015 23:04:35 EET

Towards the end of Chapter 9 in “Code” I demonstrate that Morse Code can be interpreted as a binary code.

I am not hiding his e-mail address as it is really easy to find on the web anyway.

# Morse code is a prefix ternary code (for encoding 58 characters) on top of a prefix binary code encoding the three symbols.

This was a much shorter answer when accepted. However, considering the
considerable misunderstandings between users, and following a request
from the OP, I wrote this much longer answer. The first “nutshell”
section gives you the gist of it.

Contents

## In a (big) nutshell

When asking “Is Morse Code binary, ternary or quinary?” there is no
comparing possible answers unless one fixes some criteria for an acceptable
answer. Indeed, without proper criteria, one can contrive explanations
for nearly any kind of structure. The criteria I have chosen are the
following:

• it should reflect the three-tiered description of Morse-code
with the dot/dash representation in the second tier;

• it should fit the presentation and mathematical tools developed
for the theoretical analysis of codes, as much as possible;

• it should be as simple as possible;

• it should clearly make apparent the properties of the Morse code.

This is intended to preclude arbitrary hacking, that ignores basic
concepts of code theory as scientifically studied, and which may have
some appeal by giving an illusion of systematic analysis, though
addressed too informally to be conclusive. This site is supposed to be about
computer science, not programming. We should use a minimum of
established science and accepted concepts to answer a technical
question.

A quick analysis of the standard shows that all symbols used in
Morse code are ultimately coded in binary
, since it is transmitted
as a string of units of equal length, whith a signal that can be on or
off for each unit. This indicates that Morse messages are ultimately
coded in a logical alphabet $\Sigma_1=\{0,1\}$.

But that says nothing of the internal structure of the code. The
information to be encoded is a string on an alphabet of 58 symbols
(according to the standard) including 57 characters and a space.
This corresponds to an alphabet
$\Sigma_3=\{A,B,\dots,Z,0,1,\dots,9,?,=,\dots,\times,@,[\;]\}\;$,
the last symbl being the space.

However, the standard specifies that there is an intermediate alphabet
$\Sigma_2$, based on dot and dash and possibly other symbols. It
is quite clear

• that strings in $\Sigma_3^*$ are to be coded as strings
in $\Sigma_2^*$, and

• that strings in $\Sigma_2^*$ are to be coded as strings
in $\Sigma_1^*$

So, given that there is no choice for $\Sigma_1$ and $\Sigma_3$, the
question must be understood as: “What number of symbols should we
consider in the intermediate alphabet $\Sigma_2$ so as to best
axplain the structure and the properties of the whole Morse code,

which also entails specifying the two encodings between the three levels.

Given the fact that the Morse code is a prefix homomorphic (variable length) code that
precludes any ambiguity when decoding a signal, we can explain
simply this essential property with a ternary alphabet
$\Sigma_2=${dot, dash, sep}, and two coding scheme $C_{3\to 2}$ from $\Sigma_3$
to $\Sigma_2$, and $C_{2\to 1}$ from $\Sigma_2$ to $\Sigma_1$, which are both
homomorphic and prefix, thus both unambiguous codes, and thus able to be
composed to give an unambiguous prefix encoding of the 58 symbols into
binary.

Hence Morse code is composed of a prefix ternary code expressed in the
alphabet
$\{$ dot, dash, sep $\}$, with these three symbols themselves
encoded in binary
with the following codewords:

dot $\to 10$, dash $\to 1110$, and sep $\to 00$

Note that what is known as the space between consecutive dot or dash is
actually included in the representation of dot and dash, as this
is the usual mathematical representation for such types of codes, which
are usually defined as string homomorphisms from source symbols to
codewords expressed with target symbols, as I just did.

This departs a little from some of the presentation given in the
standard, which aims more at specifying intuitively the code for
users, rather than at analysing it for its structural properties.
But the encoding is the same in both cases.

Even without the precise timings of the standard, a decoder of the
analog signal could still translate it into the ternary alphabet
we suggest, so that the above understanding of the ternary code would
still be valid.

## Codes: basic points

This answer is based on the Standard ITU-R M.1677-1, dated October
2009 (thanks to Jason C for the reference). I shall use the
terminology dot and dash, rather than dit and dah, as it is
the terminology used by this standard.

Before we start discussing the Morse code, we need to agree on what a
code is. The difficult discussions on this question obviously requires
it.

Fundamentally, information needs to be represented in order to be
transmitted or otherwise processed. A code is a system to translate
information from one system of representation into another
. This is
a very general definition. We must be careful not to confuse the
concept of a representation, and that of a code from one
representation (the source) to another (the target).

A representation can take many forms, such as variable electric
voltage, colored dots on paper, string of characters, numerals, binary
strings of 0’s and 1’s, etc. It is important to distinguish between
analog and formal (or logical, or abstract) representation.

An analog/physical representation is a drawing, a varying voltage
level, a shape (for a letter).

A logical/formal/abstract representation is a mathematical
representation with abstract graphs, strings of symbols, or other
mathematical entities.

Though some information may originally be analog, we usually
convert it to a logical representation so as to be able to define precisely
its processing by mathematical means, or by people.

Conversely, we dealing with logical representation using physical
devices, such as computer or transmitters, we need to give an
analog form to the logical representation.

For the purpose of this analysis, the only analog form we consider
is that used for transmission, as described in the standard. But even
then, we will consider that the first step is to interpret this
analog representation as a direct implementation of an identically
structured logical representation, on which we build our analysis of
what kind of code Morse code may be. Code theory is a mathematical body
of knowledge based on the analysis of logical representations.

However we shall come back on the analog/logical transition in the
discussion at the end.

## Codes: definitions

Our logical view is that the code is used to translate sources strings
on an source alphabet $S$ to a target alphabet $T$. It is
often the case that both alphabets are identical, usually binary, when
the purpose is to add some extra property to the representation of
information, such as making it more resistant to errors (error
detection and correction), or making the representation smaller by
removing redundancy (lossless code compression) and possibly with
carefully controled loss of some information (lossy compression).

However, the purpose of Morse code is to provide only a way to
represent strings on a large alphabet, into strings based on a much
smaller alphabet (actually binary), using an intermediate alphabet
almost binary (dots and dashes) to better adapted to human perception
and manipulative abilities. This is achieved by what is called
variable-length code:

Using terms from formal language theory, the precise mathematical
definition is as follows: Let $S$ and $T$ be two finite sets, called the
source and target alphabets, respectively. A code $C: S \to T^*$ is
a total function mapping each symbol from $S$ to a sequence of
symbols over $T$, and the extension of $C$ to a homomorphism of
$S^*$ into $T^*$, which naturally maps each sequence of source
symbols to a sequence of target symbols, is referred to as its
extension.

We call codeword the image $C(s)\in T^*$ of a symbol $s\in S$.

A variable-length code $C$ is uniquely decodable if the
corresponding homomorphism of $S^*$ into $T^*$ is injective. That
means that any string in $T^*$ can be the image of at most one string
in $S^*$. We also say that the code is unambiguous, meaning that
any string can be unambiguously decoded, if at all.

A variable-length code is a prefix code if no codeword is the prefix
of another. It is also alled instantaneous code, or context-free
code
. The reason for these names is that, when reading a target string
that begins with a codeword $w$ of a prefix code, you recognize the
end of the codeword as soon as you read its last symbol, without
having to know/read the next symbol. As a consequence, prefix codes
are unambiguous and very easy to decode fast.

It is easily shown that unique decodability and the prefix property
are closed under composition of codes.

Note that the definition as a homomorphism implies that there is no
special separation between codewords.
It is their structure, such as
the prefix property, that allows identifying them unambiguously.

Indeed, if there were such separation symbols, they would have to be
part of the target alphabet, since they would be necessary to decode
string from the target alphabet. Then it would be quite simple to
revert to the theoretical model of variable-length code by appending
the separator to the preceding code word. If that were to raise
contextual difficulty (due for example to multiple separators), that
would only be a hint that the code is more complex than apparent.
This is a good reason to stick to the theoretical model described
above.

## The Morse code

The Morse code is described in the standard at three levels:

• 3 . it is intended to provide an encoding of natural language text,
using 57 characters (27 letters, 10 digits, 20 synbols and
ponctuations) and an inter-word space to cut the character string into
words. The inter-word space is used like a special character, that can
be mixed with the others, which I shall note SEP.

• 2 . all of these characters are to be encoded as successions of dash
and dot, using an inter-letter space, which I shall note sep, to
separate the dash and dot of one letter from those of the next
letter.

• 1 . The dash and dot, as well as sep are to be encoded as signal
or absence of signal (called spacing) with length precisely defined
in terms of some accepted unit. In particular, the dash and dot
encoding a letter must be separated by an inter-element space, that I
shall note σ.

This already calls for a few conclusions.

The message to be transmitted and received in analog form is a
successions of length units
(space length or time length), such that a
signal is on of off for the whole duration of each unit as specified
in the Annex 1, Part I, section 2 of the standard:

2   Spacing and length of the signals
2.1 A dash is equal to three dots.
2.2 The space between the signals forming the same letter is equal to one dot.
2.3 The space between two letters is equal to three dots.
2.4 The space between two words is equal to seven dots.


This is clearly an analog encoding in what is known as a bit
stream, which can be logically represented in binary notation by a
string of 0 ans 1, standing for the analog off and on.

In order to abstract away issues related to analog representation,
we can thus consider that Morse code messages are transmitted as bit
strings, that we shall note with 0 and 1.

Hence the the above excerpt from the standard can be expressed
logically as:

• 0 . A dot is represented by 1.
• 1 . A dash is represented by 111.
• 2 . An inter-element space σ is represented by 0.
• 3 . An inter-letter space sep is represented by 000.
• 4 . An inter-word space SEP is represented by 0000000.

So we could see Morse code as using 5 code words in binary to encode
these 5 symbols. Except for the fact that this is not quite how the
system is described, there is some more to it, and it is not the most
convenient way it can be thought of, from a naive or a mathematical
point of view.

Note also that this description is intended for laymen, not code
theory specialists. For that reason it describes more the visible
appearance than the internal structure that justifies it. It has no
reason to preclude other descriptions that are compatible with this
one, though mathematically more structured, to emphasize the
properties of the code.

But first, we should note that the complete description of the code
involves 3 levels of representation, immediately recognizable:

• 3 . The text, composed of a string of characters, including SEP.
• 2 . The encoding of a letter string as a string of dot, dash and sep.
• 1 . The encoding of a level 2 string of these three symbols as a binary string.

We may possibly discuss as to what symbols is encoded in what, but it
is an essential aspect of Morse code that it has these three levels
of representation, with characters at the top, dots and dashes in the
middle, and bits 0 and 1 at the bottom.

This implies that there are necessarily two codes, one from level 3 to
level 2, and the other from level 2 to level 1.

## Analysing the three levels of representation

In order to have a consistent analysis of this 3-tiers coding system,
we should first analyse what kind of information is relevant at each
level.

• 1 . The bit string, by definition, and by necessity of its analog
representation, is composed only of 0 and 1.

• 3 . At the text level, we need and alphabet of 58 symbols, including
the 57 characters and the inter-word space SEP. All 58 of them have
to have ultimately a binary encoding. But, though the Morse code
standard specifies these 57+1 characters, it does not specifies how
they should be used to encode information. That is the role of
English and other natural languages. The Morse code provides other
system with an alphabet of 58 symbols, on which they could build some
58-ary code, but Morse code is not itself a 58-ary code.

• 2 . At the dot and dash level, all we need is these two symbols in
order to code the 57 characters, i.e. provide a codeword for each as a
string of dot and dash, together with some separator sep to mark
when one letter finished, and another start. We also need some means
of encoding the inter-word space SEP. We might try to provide for it
directly at leavel 1, but this would mess-up the otherwise tier-structured
organization of the code.

Indeed, the description of the standard might rightly be criticized
for doing just that. But the authors may have thought that their presentation
would be simpler to grasp for the average user. Also it follows a
traditional description of Morse code, that predates this kind of
mathematical analysis.

This calls for several remarks:

• at level 3, the letter level, the inter-letter space sep is no
longer meaningful. This is quite normal, since it has no more
meaning in the universe of letters than the space separating two
written characters on paper. It is necessary at level 2 to recognize
codewords representing the letters, but that is all.

• similarly at level 2, the inter-element space σ is no longer
meaningful. It has no meaning in the world of dot and dash, but
is only necessary at level 1 to identify the binary code words
representing dot, dash. But at level 1, it is not
distinguishable from the bit 0.

So the inter-element space σ is no longer anything special. It is
just one use of 0.

However, as explained previously, if the code $\Sigma_2^*\to\Sigma_1^*$ is
to be analyzed using knowledge of variable length codes, separators
should be appended into the codewords they follow, so as to define the
code as a simple string homomorphism.

This implies the following partial specification of the code:
dot$\to$10 and dash$\to$1110

The level 2 alphabet $\Sigma_2$ needs at least one other symbol, the
inter-letter space noted sep, which should be 000 according to the
letter of the standard. However, the definition of the variable length
code as a homomorphism required appending the inter-element space 0
to each codeword for dot and dash. Hence we must have only 00 as
codeword for sep, so that toghether with the ending 0 from the
preceeding dot or dash, it makes 3 0 as required by the
standard. This always work since there is no provision in the standard
for having two inter-letter separators following each other.

This is enough to encode the alphabet $\Sigma_2=${dot, dash, sep} with a homomorphic code $C_{2\to 1} : \Sigma_2\to\Sigma_1^*$ defined as follow:

• dot$\to$10

• dash$\to$1110

• sep$\to$00

And we have the good surprise to discover that no codeword is a prefix
of another. Hence we have a prefix code, which is unambiguous and easy
to decode.

We can now proceed similarly to define the code $C_{3\to 2}: \Sigma_3\to\Sigma_2^*$.

The standard uses strings of dot and dash as codewords for the
characters in $\Sigma_3$, in the way given by the tables of the
standard for example dot dot dash dot to represent the letter
$f$.

Again, these codewords are separated by inter-letter spaces. In order
to define the code as a homomorphism, we must include the separator in
the codewords, so that the definition of the homomorphism becomes
rather: $f\to$ dot dot dash dot sep

This applies to each of the 57 characters in the alphabet $\Sigma_3$.
But again we also need the word separator SEP, which, according to
the standard, is 0000000. We first note that already 3 bits 0 are
provided by the code, 2 by the sep that ends the last letter of the
word, and 1 by the 0 bit that end the last dot or dash of the
encoding of that last letter. Hence SEP must ultimately be coded as
the remaining 0000.

But to respect the tiered approach, SEP should be encoded in some
codeword from $\Sigma_2^*$. Since sep is binary encoded as 00, it
follow that SEP can be encoded as sep sep.

Hence we can encode the alphabet
$\Sigma_3=\{A,B,\dots,Z,0,1,\dots,9,?,=,\dots,\times,@,$ SEP$\}$,
with a homomorphic code $C_{3\to 2} : \Sigma_3\to\Sigma_2^*$
defined as follows:

• $A \to$ dot dash sep

• $B \to$ dash dot dot dot sep

• $Z \to$ dash dash dot dot sep

• $7 \to$ dash dash dot dot dot sep
• SEP $\to$ sep sep (for the word separator)

And we have the further surprise to see that no codeword is a prefix
of another. Hence the code $C_{3\to 2}$ is a prefix code too.

Since the prefix property is closed under composition of codes, the
Morse code $C_{Morse}= C_{2\to 1}\circ C_{3\to 2}$ is a prefix code.

We can thus conclude that the Morse code can be understood, and easily
analyzed, as the composition of a prefix binary encoding of a 3
symbols alphabet {dot, dash, sep} into a binary alphabet, and a
prefix encoding of a 58 symbol alphabet (57 characters and one space)
into the 3 letters alphabet.

The composition itself is a prefix encoding of the 58 symbols into a
binary representation.

## Remarks on this analysis.

It is always difficult to establish that a presentation of a structure
is the best one can come up with. It seems however that the above
analysis meets the criteria set up at the beginning of this answer:
closeness to the 3-tiered definition, formally presented according
to current coding theory, simplicity, and evidencing the main
properties of the code.

Note that there is little point in looking for error correction
properties. The Morse code may not even detect a single bit error as
it may simply change two dot into one dash. However, it causes
only local errors.

Regarding compression, the ternary encoding was designed to
approximately reduce the number of dots and dashes, in an
approximative kind of Huffman coding. But the two composed codes

Regarding the size of alphabets, there is no choice for the binary and
the 58 symbols alphabet. The intermediate alphabet could contains more
symbols, but what would be the purpose?

However, some people would be inclined to recognize the space DET at
level 2, thus making the alphabet quaternary, then using it directly
at level 3, encoded as itself in level 2.

This would meet the standard definition, for DET encoded in binary
as 0000. But it would prevent the analysis of the binary encoding
$C_{2\to 1}$ as a prefix code, making it harder to show that
$C_{Morse}$ is a prefix code, hence unambiguous.

Indeed, such a choice would make the binary string 0000 ambiguous,
decodable as either SEP or as sep sep. The ambiguity would have
to be resolved with a contextual rule that sep cannot follow itself,
making the formalization more complex.

## The importance of analog to logical transition.

This analysis relies heavily on the fact that the decomposition of the
on/off signal into units of equal lengths indicates clearly an
analog representation of a binary string. Furthermore, the lengths
in units are exactly right for the above analysis, which seems
unlikely to have happened by chance (though it is possible).

However, from a (too cursory) look at the original patent 1647, it does not
seem to have been that precise, with sentences such as (on top of page
2):

The sign of a distinct numeral, or of a compound numeral when used in
a sentence of words or of numerals, consists of a distance or space of
separation between the characters of greater extent than the distance
used in separating the characters that compose any such distinct or
compound numeral.

People who were later sending by hand or receiving by ear were also
unlikely to be that precise either. Indeed, their fist, i.e. their
timing, was often recognizable. This view is also supported by the
fact that spacing lengths are not always respected, particularly when
learning Morse code.

These situations correspond to an analog view of the code as short
signal (dot), medium signal (dash), and short, medium and long
pause. Direct transposition into a logical alphabet would naturally
give a quinary alphabet, into which the 58 symbols have to be
coded. This of course is no longer a 3-tiered presentation of the
Morse code.

However, in order to make sense (and possibly avoid ambiguity), this
alphabet should be used with the constraint that two signal symbols
(dot or dash) cannot follow each other, and that pause symbols
cannot follow each other either. Analysis of the code and its
properties would be made more complex, and the natural way to simplify
it would be to do what was done: introduce proper timings to turn it
into the composition of two codes, leading to the fairly simple
analysis given above (remember that it includes showing the code is
prefix).

Furthermore, it is not strictly necessary to follow exact timings in the
analog representation. Since the decoder of the analog
translation can distinguish short, medium and long pauses, by whatever
means, it should just mimic what was done in the binary case. Hence
short and medium signal (necessarily followed by a pause) are
recognized as logical dot or dash. Short pauses are forgotten, as
only serving to mark the end of dot or dash. Medium pauses are
recognized as sep, and long pauses are recognized as two sep in
succession. Hence the analog signal is represented in a ternary
alphabet, which can be used as before to encode the 58 symbols
alphabet. Our initial analysis can be used even when timings are not
strictly respected.

Alternatively, the signal-pause alternance could be used to turn this
quinary alphabet into a ternary one, keeping only the three durations
as symbols of the alphabet, and using contextual analysis to determine
whether a given duration is signal or pause. But this is again a bit
complex to analyze.

This just shows that there are many ways to look at things, but they
are not necessarily convenient, and may not all lend themselves easily to
analysis with the mathematical tools that have been developed to
analyze codes.

More references to the patents can be found on the Internet.

# Conclusion

Given the precise timings of the standard, a good answer seems to be
to consider Morse code as the composition of a ternary prefix
encoding (of 58 characters) into a 3 symbols alphabet, composed with
a binary prefix encoding of these three symbol.

Without the precise timing of the standard, the binary level can no
longer be considered. Then the analog to logical decoding naturally
takes place at the level of the intermediate alphabet of dot and
dash. However, the analog to logical decoder can stil decode to
the previous 3 symbols alphabet, thus preserving the applicability of
our analysis.