In Japanese, these are all possible sounds in the language. Each is represented by 1 kana character.

a i u e o
ka ki ku ke ko
ga gi gu ge go
sa shi su se so
za ji zu ze zo
ta chi tsu te to
da ji zu de do
na ni nu ne no
ma mi mu me mo
ha hi hu he ho
ba bi bu be bo
ra ri ru re ro
ya yu yo
wa wo

Note that "hu" is often romanized as "fu", but in JSL it is written "hu". The actual sound is in between.

There is also the "p" series:

pa  pi  pu  pe  po

I would also add the following ones, since they are not exact sums of the previous sounds. They are written with two kana, where the second one is smaller:

kya kyu kyo
gya gyu gyo
sha shu sho
ja  ju  jo
cha chu cho
nya nyu nyo
mya myu myo
hya hyu hyo
bya byu byo
pya pyu pyo
rya ryu ryo

You could also add special syllables which were created for foreign words and are usually written with katakana, such as vi, fe, etc.

More correctly, the fifty sounds refer only to the fifty basic characters of the Japanese syllabary. These sounds are the ones that do NOT contain the "nigori", or "han-nigori" sounds, i.e., the ones which are made by adding strokes to basic characters to form the hard sounds, e.g., hu -> pu -> bu.

These sounds are traditionally ordered in a poem, created by a buddhist monk named Kukai. I'm posting the poem and translation on the hiragana node.

The "fifty sounds" are not really fifty in number. There are only about forty-six of them, and in modern times they are arranged as follows, in an order called "gojuuonzu" ("fifty-sound table"):

 a   i   u   e   o
ka  ki  ku  ke  ko
sa shi  su  se  so
ta chi tsu  te  to
na  ni  nu  ne  no
ha  hi  fu  he  ho
ma  mi  mu  me  mo
ya      yu      yo
ra  ri  ru  re  ro
wa              wo

The above table is to be read by rows. It starts a, i, u, e, o, ka, ki,...
This is NOT a complete list of sounds in the Japanese language!

A term for the Japanese syllables represented in the fifty sounds chart. The chart itself is a five by ten matrix of syllables. The columns are ordered from right to left, and the sounds in each column are ordered from top to bottom. While the kana ん (cf. syllabic N in Japanese) exists in the syllabary, since it isn't in the chart, it's not considered one of the fifty sounds. Also, despite being called the fifty sounds chart, it never contained fifty unique sounds in the chart.
The modern 50 sounds chart in hiragana

わ ら や ま は な た さ か あ
  り   み ひ に ち し き い
  る ゆ む ふ ぬ つ す く う
  れ   め へ ね て せ け え
を ろ よ も ほ の と そ こ お

Tongpoo's romanization *

 wa  ra  ya  ma  ha  na  ta  sa  ka  a
     ri      mi  ci  ni chi shi  ki  i
     ru  yu  mu  fu  nu tsu  su  ku  u
     re      me  he  ne  te  se  ke  e
wo/o ro  yo  mo  ho  no  to  so  ko  o

pronounciation key **

letter   example                     linguistic term              IPA
  c      h in heat                   unvoiced palatal fricative    ç
  f      wh in which                 unvoiced bilabial fricative   φ
  r      t in water, Brooklin accent alveolar tap/flap             ɾ
  u      like blowing out a candle   close back unrounded vowel    ɯ
* The above romanization scheme I just made up. For common romanization schemes of the Japanese language, see: Japanese romanization.

** Sounds not mentioned in the pronounciation key sound like what they look like. Today, 'wo' is most often pronounced as 'o' instead. A survey ( suggests about a third of the population of Japanese speakers are pronouncers of 'wo.'

The early 50 sounds chart

わ ら や ま は な た さ か あ
ゐ り い み ひ に ち し き い
う る ゆ む ふ ぬ つ す く う
ゑ れ え め へ ね て せ け え
を ろ よ も ほ の と そ こ お
ワ ラ ヤ マ ハ ナ タ サ カ ア
ヰ リ イ ミ ヒ ニ チ シ キ イ
ウ ル ユ ム フ ヌ ツ ス ク ウ
ヱ レ エ メ ヘ ネ テ セ ケ エ
ヲ ロ ヨ モ ホ ノ ト ソ コ オ
(tongpoo's romanization)
wa  ra  ya  ma  fa  na  ta  sa  ka  a
wi  ri  yi  mi  fi  ni chi shi  ki  i
 u  ru  yu  mu  fu  nu tcu  su  ku  u
we  re  ye  me  fe  ne  te she  ke  e
wo  ro  yo  mo  fo  no  to  so  ko  o
The above chart resembles the earliest 50 sounds chart known, which developed during 11th century Heian era Japan. Until the 20th century, Japan wrote everything from right to left. The chart is thought to have been developed by buddhists who knew the works of an Indian linguist Siddham. Eventually, the more organized 50 sounds chart replaced the iroha (developed around 10th-11th century) in popularity as the preferred order of syllables.

The ha column

The modern version of this column has a non-homogenous set of consonants. The evolution of this column has been:
  1. pa→fa→ha
  2. pa→fa→wa
  3. pi→fi→hi→ci (palatization by "i")
  4. pu→fu
  5. pe→fe→he
  6. po→fo→ho
Where the f + (vowel) stage lasted mostly during the nara era (710 - 780 A.D.). The transition from 'p' to 'f' was from an unvoiced bilabial plosive to an unvoiced bilabial fricative. By the edo era, the column has changed into mostly h's. Okinawan dialects still often preserve the 'p' sounds. As you see in item 2, in modern days は('ha') is sometimes pronounced as は('wa').

The a, wa, and ya column

  1. e/ye→ye
  2. ye/we→ye→e
  3. i/wi→i
  4. o/wo→wo
As you see here, while ゐ('wi') changed to ゐ('i') directly, ゑ('we') changed to ゑ('ye') before becoming an 'e' sound.

The dropping of ゐ and ゑ

Noting that pronounciation differences between ゐ and い, ゑ and え doesn't exist anymore, the government ordered in 1946 to stop the use of letters ゐ and ゑ. In 1986, the rules were relaxed to allow for historical usage of these characters.

All of my sources are in Japanese.
Copies of the orders to drop ゐ and ゑ can be found here:
I obtained most of the information for this w/u from: "日本語の起源"

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