Mandukya Upanishad

Mandukya Upanishad is one of the shortest of the Upanishads with only 12 shlokas. It is also called as Advaita Vedanta. This upanishad is the insight of Rishi Manduka. The initial part of the Ganesha Atharva comes from this Upanishad.This Upanishad describes the four states of consciousness and their relation to the Pranava (Aum).

There are four states of consciousness
Awake state or Vishvanara

in this state, outer world is perceived through senses feeding into the mind. this state is characterised by the desire to attain pleasure. The outer world in all its glory is called Virat.

Dreaming or Taijas

the outer world perceived in the awake state through the senses is reflected on the inner world through the mental  impression, this is a state of all possibilities also called as Hirnayagarbha. connection to this state brings out intuition

These two states bind the individual soul (Jivatma) to Karma (cause and effect) and are responsible for duality including birth and death. They are connected with the first four Koshas - Body, Prana, mind and intellect.

Deep sleep or Pragna - Here the consciousness which is usually spread out in various faculties in the awake or dream state becomes integrated and condensed.  The individual soul connects to the universal soul (Brahman) or the small mind merges with the big mind. This is the state of no desire or dream and free from efforts. It is beyond Karma, and linked to Bliss body - Anandamaya kosha. Joy experienced in this state is unrelated to external objects and is because of connection to the source (Self). Connection to this state brings forth higher knowledge.

Turiya - literally means the fourth state. Rishis have said - Mind cannot reach it and speech cannot describe it. It is beyond cause and effect and is the substratum of the three states and refers to the undivided consciousness or Brahman. State of total awareness and witness consciousness. Connection to his source eliminates all impressions.

The first three states are mutually exclusive while Turiya is a witness to all, so as per the Upanishadic definition, it is the only True State

Relation to AUM...

AUM has 4 Maatraas (syllable) - the final maatra (unheard syllable) - has vibrations as half and silence as the other half. 

Understanding Om gives us a idea of the creation - from silence to sound, from subtle to gross. Chanting Om takes you from sound to silence, from gross to the subtle. Silence or emptiness symbolises the undivided consciousness or Brahman. This is equivalent to the modern concept of the Black hole - space from which everything emerges and into which everything dissolves. When a desire to express arises in this consciousness, the first manifestation is in the form of pure vibrations (Purusha). From the Purusha arises the individual soul (Jivatma) which then manifests through the integration of the five elements. 

The awake state is related to the Aaa syllable (Brahma), mastery over the syllable brings about the ability to achieve all worldly desires. the Uuu syllable is related to the dream state (Vishnu) and is the source of intuition; the Mmm state is connected to deep sleep (Shiva) state and is the source of all knowledge. The unheard syllable is connected to the Turiya (Guru) state.

When AUM is chanted, all states of existence get connected, thus the harmony of the sound.