Medium Transport; Low Wing; 2 Turbojet Eng
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Issued Advisory / Alert; Air Traffic Control Provided Assistance; Flight Crew Became Reoriented; Flight Crew Returned To Clearance
Human Factors; Procedure
We departed TEB and we're just in the departure phase of flight. We were very busy vectoring etc.; and at some point the controller said to go direct to GAYEL (he said the name not the phonic spelling). I was the non-flying pilot so I looked in our FMS route for the fix but it wasn't showing as one of our route fixes. I looked in the clearance we got before we left and we did have it as part of our clearance so we are not sure why it wasn't there. To get started on course; I typed GAYLE in the FMS which was what I had written for the clearance I received on the ground (In the printed version of our flight plan it showed GAYEL). The plane started tracking to GAYLE which was only a very slight turn to the left (so didn't stand out as being wrong fix) and then I went to plug in the Q airway that we had from GAYEL to CFB [VOR]. The FMS was not accepting the airway (I found out later because I had GAYLE and not GAYEL as the connecting fix). About that time ATC told us again to go direct to GAYEL (he used the name again and not the phonetic spelling). I told him we are going direct to GAYLE and he replied no you're not. Right as he started saying that; I looked on the printed version and saw that I had the [wrong spelling] typed into the FMS. I then put in the correct fix and we were on course. The controller never said anymore about the issue nor did he say to call. I think the biggest thing I can do to make sure this doesn't happen again is to pull up the PLAN view on the MFD to see the big picture. If I would have done this I would have seen that [the] intersection is quite a ways off our route to where we were going. Another catch would have been to notice on the FMS LEGS at the distance to that fix. It really points out to me how much I need to always question what's going on and not let complacency creep in as letting such a subtle thing as switching 2 letters around to spell the same name can drive you to the wrong fix. One of the biggest flags was when the FMS wasn't accepting the airway off of GAYLE.
After departing TEB on the TEB9 departure; we were given a heading of 270 and climb to 10;000. Shortly after we were instructed to proceed direct GAYEL. No heading was given to the fix. When the Captain tried to enter direct GAYEL; he became flustered as it wasn't in the FMC; or couldn't find it in the legs page. Captain pulled the written clearance for reference and the written clearance was transposed as GAYLE; not the correct fix spelling [GAYEL]. So when he reentered the incorrect spelling; the fix was a similar to our heading; slight left turn. As we proceeded direct for about 5 miles the controller asked us to go direct GAYEL and we realized the spelling error and corrected it. The new heading was about 50 degrees to our right; to GAYEL.  No traffic conflicts occurred.Started duty at XA:00 after a 10 hour rest; day 3. My body clock time is [off]; however I felt fairly rested. During preflight we reviewed the legs page and all points were correct. We did get a route modification while speaking with clearance delivery. The Captain wrote down the clearance. In a transposed error; the incorrect spelling was written for GAYEL. Initially; the correct spelled fix GAYEL was entered in the legs page; but for unknown reasons this fix dropped out after takeoff; or was not located in the FMC by my Captain. I was busy as Pilot Flying dealing with ice; ATC instructions and did not watch the FMC as my Captain worked the issue. We finally inputted the correct fix and continued our route. The only things I think could have alerted us of the error is checking the spelling closer; although it was so similar GAYEL vs GAYLE - that this is difficult to catch. The mileage distance to the fix should have been checked also after we selected direct to. The error was compounded by us knowing we had the correct fix before takeoff; but somehow it was dropped; so inputting the incorrect spelling didn't raise any red flags.
Medium Transport aircraft flight crew reported entering the RNAV waypoint GAYLE instead of GAYEL in the FMS during departure form TEB; resulting in a heading deviation. First Officer stated the waypoint was misspelled on the flight release.