Inrtoduction

part of the tongue

tense/lax

roundness

Example:
/a/ – [low, +back, lax, -round] /u/ – [+high, +back, tense, +round]
Tense/lax
These terms are often used interchangeably with Advanced and Retracted Tongue Root (ATR/RTR), however this distinction involves more than simply the root of the tongue. Tenseness can be more generally distributed throughout the oral cavity as well as in a greater area of the tongue.
ATR/RTR (advanced and retracted tongue root) – when the base of the tongue is forward, lowering the larynx, or retracting the base of the tongue
Other features of vowels in Standard American English include:

1. 
   nasal: vowels that precede nasal consonants, e.g., [sĩn] ‘seen’
   
2. 
   devoiced: vowels found flanked by voiceless obstruents, e.g., [t^h ḁp] ‘top’
   
3. 
   long: vowel that precede voiced phonemes are slightly lengthened in contrast to those that precede voiceless sounds, e.g., [sit] ‘seat’ vs. [si:d] ‘seed’
   

Differences between the IPA and North American vowel charts

The IPA chart uses the terms open and close instead of low and high.

The IPA chart has four height levels instead of three. The mid level in the tic-tac-chart is split into two levels in the IPA chart:

• 
  tense mid is called close-mid
  
• 
  lax mid is called open-mid
  

[] and [] are demoted to second-class citizens (non-cardinal vowels) and pushed to the inside of the chart. This makes the implicit claim that the difference between [e] and [] is mainly one of height, not of some extra "tenseness" property.

• 
  Acoustically, the IPA chart's arrangement is closer to real F1-by-F2 graphs. The acoustic difference between [e] and [] is not parallel to the acoustic difference between [i] and [].
  
• 
  Articulatorily, there is variation between languages and even between speakers within a language. For some languages/speakers, the tongue-body positions look very like what the IPA chart suggests. For other languages/speakers, the tongue-body positions are similar for [i] and [] and for [e] and [], as suggested by the tic-tac-toe chart, and the tense/lax distinction is made primarily by moving the tongue root forward or backward in the pharyngeal cavity. (See the section of the textbook on Advanced Tongue Root.)
  
• 
  Phonologically, many languages act as if the vowel space were organized as in the tic-tac-toe chart, even if the IPA chart is more physically accurate. (E.g., English, many languages of Africa)
  

[a] is a front vowel in the IPA chart and central in the tic-tac-toe chart. In the IPA chart, [æ] is a second-class citizen squeezed in between [a] and [].

• 
  Both charts are about equally close to the physical reality.
  
• 
  [a] is more common than [æ] cross-linguistically, so there is some justification for treating it as more "important".
  
• 
  Phonologically, most languages don't make a difference between front and back low vowels -- whether their single low vowel [a] should be treated as front or central or back is not a pressing question. Those languages that do make a front/back distinction in the low vowels will often use the exaggeratedly front [æ] for the front one. The rare languages which contrast three degrees of frontness for low vowels usually act the way the tic-tac-toe chart suggests.
  

The IPA chart treats [] as an unrounded back vowel. The tic-tac-toe chart treats it as a central vowel.
As suggested by the layout of the IPA chart, back unrounded vowels will tend to be somewhat more central than rounded ones, both acoustically and articulatorily. But English [] is clearly central, more so that we can explain away like this.
Unlike the earlier differences, this is a case of using the same symbol for two different sounds. From the viewpoint of strict standardization, English linguists are just plain wrong to use [] for the vowel of cup. Especially in narrow transcriptions, it would be more accurate to add a diacritic indicating that the vowel is advanced or centralized, or to use the symbol [] for a lower-mid central vowel (approved in 1996). From a more realistic viewpoint, English linguists are simply following the IPA-sanctioned practice for broad transcriptions of using the symbol for the nearest cardinal vowel whenever practical. (If we were writing a grammar of English, we'd have a moral obligation to put in a footnote explaining how our broad use of [] differs from the standard for cardinal vowel 12.)
In the tic-tac-toe chart, the mid-central region is a box like any other. In the version of the IPA chart printed in the textbook, um, well, it's anybody's guess what's going on there.
The cardinal vowels are placed around the clearly defined borders, and the cardinal vowel system is dependent on the four corners of the vowel space. The centre section of the chart may just as well have been marked with a "Here there be dragons" sign, according to the IPA, which has a tendency to view it as a vague no-land. man's In order to represent rounded and unrounded central vowels at each of its four height levels, the IPA finally bit the bullet in 1996. (Except low). When you don't want to fret over whether a vowel is higher-mid or lower-mid, tense or lax, you can use the cover sign schwa for any unrounded mid middle vowel. More central vowel symbols are present here than we will ever need. But at least it is dependable.