Python TO Serial TO Arduino. Send a string?

Hey team, I'm struggling to send a string (multiple characters) to Arduino. Here is what I currently have in Arduino:

int my_led = 13;
String IncomingData = "";
String Temp = "";
char var;

void setup()
{
  Serial.begin(9600);
  Serial.println("Ready");
  pinMode(my_led, OUTPUT);
}
void loop()
{
  while(Serial.available())
  {
    var = Serial.read();
    Temp = String(var);
    IncomingData+= Temp;
  }
 if (IncomingData == "HHH"){
    digitalWrite(my_led, HIGH);
 }
 if (IncomingData == "LLL"){
    digitalWrite(my_led, LOW);
 }
 IncomingData = "";
}

And in Python:

import serial
from tkinter import *

def led_on():
     arduinoData.write('HHH'.encode())
def led_off():
     arduinoData.write('LLL'.encode());


led_control_window = Tk()
led_control_window.minsize(width=266, height=266)
led_control_window.maxsize(width=266, height=266)

btn = Button(led_control_window,text="led on",command=led_on)
btn2 = Button(led_control_window,text="led off",command=led_off)

btn.pack()
btn2.pack()

arduinoData = serial.Serial('com5', 9600)

mainloop()

Really hope this make sense to somebody. Basically, It works when sending a single letter ('h' instead of 'hhh'). But for my next project (piezo + note) I need to send a string... ('1C440') - Pin-Note-Pitch.
Thank for any advice you might have.

  while(Serial.available())
  {
    var = Serial.read();
    Temp = String(var);
    IncomingData+= Temp;
  }

Why do you think you need to wrap the character in a String, in order to append it to a String?

Why do you think that all the serial data will arrive at once? IT WILL NOT.

There is a reason that decent serial communications protocols use start and end markers. See Robin2's page on serial data handling: http://forum.arduino.cc/index.php?topic=396450.0

This Python - Arduino demo may also be useful. Just be aware that it was written before I wrote the tutorial in the link that @PaulS has posted.

...R