Stockhausen's original sketch of the form plan.
At bottom are the longest MANTRA themes,
which are overlaid by the shorter versions above them.

No. 32: MANTRA
for 2 pianos with 12 antique cymbals, 1 woodblock and 2 ring modulators (and shortwave radio/tape)
1970 [65'-72']

     During Stockhausen's stay in Japan performing open-form pieces like SPIRAL at the World's Fair Expo Pavilion, he worked on a duo piece for the pianists Aloys & Alfons Kontarsky.  The genesis of this work came from a melody which surfaced spontaneously while traveling by car many years before, as well as from his interest in refining ring modulation techniques . 
     It is reported that some of his intuitive music collaborators were feeling unhappy about the open nature of the "free" music they were playing and beginning to contest ownership of the composition rights.  On the other hand, Stockhausen has been described as being unhappy with the "dilettantish interpretations" he was getting from his players.  When a performer was in a physically or mentally distracted state (sick or depressed), the music suffered as well.  This "fragility" of intuitive music would not be as present in fully-notated music.  MANTRA is completely composed down to the last tenth of second, though in a sense it is "intuitively-composed" as the product of one person's intuitively-based musical deliberations (at a much slower pace..).

     This epic piece is a big one and a little hard to stay with for its full duration for some (including myself!). In fact it's probably the longest single piano work I have in my music collection.  One of the main reasons I find analysis of a huge piece like this to be useful is to identify "signposts" so that the listener can digest such a huge repast in platefuls, rather than as a huge buffet stretching off into the horizon.  However after finishing collating all the data and superimposing the theoretical design structure onto MANTRA, I actually found the theory to be kind of...distracting.  In fact, the structural breaks that Stockhausen himself marked in the score (and used to create CD tracks) usually don't adhere to the structural design.
     What I think happened here is that Stockhausen's manipulation of the elements unregulated by the mathematics (as well as a fair share of "cheating") was done with so much panache that his stylistic and musical "personality" became a bit more important than the underlying mathematical break points.  So why study the math?  Admittedly there are a few tables in the analysis below, which at first don't seem to add much to the listening experience.  But Stockhausen gave lectures on these theories because he believed that learning of the techniques involved gave the listener an opportunity to expand one's consciousness on some level (musically and otherwise) and would enrich the listening experience.  For the most part I totally agree, but in this case the MANTRA formula is so melodically, rhythmically and structurally warped that it's apparently easier to follow it as a narrative/fantasia, rather than as a formula-variation form.  Also it's worth noting that in the same lecture, Stockhausen maintains that theoretical understanding of MANTRA is no more necessary towards its enjoyment than knowledge about astrophysics is necessary in order to be moved by the stars in the night sky. 
     So, in any case I first present an analysis of MANTRA as a formula with multiple poly-structural layers, and then afterwards as a "highlights" narrative at the end.

Form Structure
     MANTRA is basically a giant crystal created from a small snowflake.  In other words, this (ca.) 70-minute work is a melodic and thematic explosion, derived from a 1-minute tune.
The MANTRA formula.
     The original vinyl album cover shows the MANTRA "formula".  The top staff has circled numbers, 1 through 13, which label the parts of the formula and their special properties (the pink numbers are duration measurements).  Basically, Stockhausen takes this 8-bar theme (the left hand is actually a permutated inversion of the right hand part), stretches it vertically (harmonically) and horizontally (temporally) until he has (roughly) 156 versions, and then layers them into a poly-thematic 3-level ziggurat (pyramid) as sketched out at the top of the page.

     The full MANTRA work has 13 large sections (derived from the 13 parts/notes of the MANTRA formula above) with each section focusing on one of the special "musical behaviors" below (also derived from the MANTRA itself).

Musical Behavior From Bar Beginning
(root) note
1. Regular Repetition 12 A
2. Accent at the End 61 B
3. "Normal" 89 G#
4. Grace note group around a Central Tone 125 E
5. "Tremolo"  205 F
6. Chord (accented) 282 D
7. Accent at Beginning  445 G#
8. "Chromatic" Connection 488 Eb
9. Staccatto 538 Db
10. Irregular Repetition (Morse Code) 587 C
11. Trills 611 Bb
12. Sforzando accent 641 Gb
13. Arpeggio-Connection 673 A

     The piece starts out with a 4 chord intro (made from the notes of the formula), then states the MANTRA formula in it's "plain" form (bar 3-10), and then begins the first of the 13 large sections.  These sections each have 12 MANTRA formula repetitions (13 x 12 = 156), each expanded/compressed in 1 of 12 ways (see below) and superimposed on each other.  In fact, the entry sequence of the first set of 12 Expansion forms goes like this: 1 - 3 - 12 - 8 - 9 - 6 - 11 - 7 - 5 - 4 - 2 - 10, which also happens to be the relative pitch sequence of the original MANTRA (though the sequence is different for each large section).  However as indicated before, these Expansions are not single-threaded, they are actually played on top of each other, creating up to 3 simultaneous voices:

(3)-------------------------------------------------------    (9)--------------------------------------                        (11)-------
        (12)------------                            (8)---------------                                                        (6)----------------------

     Each of these 12 Expansions also attempts to incorporate a different one of the 13 musical behaviors (one is actually doubled).  After the 13th and last "Arpeggio-Connection"-themed large section, MANTRA does a hyper-fast compressed melodic "revue", followed by one last statement of the formula.

MANTRA Expansions
     In order to stretch the formula vertically (from lowest note to highest note) Stockhausen created 11 additional artificial scales.  Each artificial scale's 12 notes take up more and more room on the keyboard (become wider), until the largest scale requires the entire keyboard to play just one artificial "octave".  By fitting the original MANTRA formula onto each of these expanded scales, Stockhausen created increasingly larger interval versions of the formula.  Then, by expanding or reducing the durations of each note, he created versions of the formula which were as compressed as 3 seconds and as stretched out as 3 1/2 minutes.  One other technique to create a great variety of MANTRAs was that Stockhausen would sometimes enforce the rhythm of one phrase on another phrase, filling in any missing notes with repetition or grace notes.
Table of the 156 MANTRAs (beginning bar nos.).
Top row indicates the 13 sections, 1st column indicates the 12 subdivisions in each section.
Notice that Expansion I.1 starts at the same time as Expansion I.3, and I.12 starts in the very next measure.
This matches the diagram above in red.
(from Richard Toop's "Six Lectures from the Stockhausen Courses Kurten 2002")
     The idea of compressing/expanding the pitch ranges of melodies would later be further explored in SIRIUS (through electronic manipulation), and in Xi and YPSILON (through new microtonal wind instrument fingerings).

This sequence (bar 611) includes, woodblocks (WOOD), shortwave radio (KW) and cymbals (CYMB).
     Antique cymbals (crotales) and woodblocks are used to sometimes punctuate sections and repetitions of the MANTRA in its many layers.  These percussion elements also are meant to evoke one of the 13 musical behavior types and root notes, and occur in the same frequency as the MANTRA pitch scale itself (1st iteration, 3rd, 15th (which is 3 + 12), 23rd (15 + 8), etc...).  Cymbals dominate the beginning half (12 occurrences, followed by 12 more inverted and an ending recap note (see below table), and the woodblocks occur more often in the later sections (as an inversion).  The woodblock hits also "have the function of marking and emphasizing certain attack and decay accents" (interview with Jonathan Cott).
Table 1: Cymbal Events (1st 12 on left, remaining 13 on right, inverted).
Roman numerals in column 3 indicate which pianist plays the cymbals.
(Click to enlarge)
 (from Ádám Siska: LIVE ELECTRONICS New Tools for Composition in the 20th Century (2013 thesis))

Table of Ring Modulation tones for each performer
(Click to enlarge)
 (from Ádám Siska: LIVE ELECTRONICS)
Ring Modulation
     Stockhausen also gives each piano a ring modulator effect box ("MODUL 69B", designed by Stockhausen himself).  This effect can cause timbral "roughness", distortion, or even transpose the pitch itself (or all 3).  This idea actually came to him while working on the piece KURZWELLEN MIT BEETHOVEN (Shortwaves with Beethoven) where he applied ring modulation effects to Beethoven works, including the piano sonatas.
     The ring modulator of the 1st pianist is tuned to the pitch "roots" for each of the 13 sections, and the more dissonant the note being played is to the section "tonic", the more noisy the ring modulation effect is (when playing a "root" note the effect is minimal).  The ring modulator for the 2nd pianist is tuned to the inverted MANTRA pitch sequence (mostly).  At times the ring modulation controls are also manipulated to create vibrato and glissando effects. 

I have no idea if this video below is appropriate in a 
serious analysis of MANTRA, but it's pretty entertaining 

Maestro Ring Modulator RM-1. 
Demonstration Record. 1972

The ring modulations twist and bend the sustaining chords at bar 421.
     Additionally, whenever Stockhausen felt like it, he would deviate from the form structure and insert a kind of "exploratory cadenza" of some texture or idea he felt merited further extemporization on.  These inserts are not ruled by the MANTRA formula at all, but are more like little breakout "explosions", much like the ones found in IN FREUNDSCHAFT, CARRÉ and many other places (also called "Inserts" or sometimes "Windows").  Some of these excursions occur at bars 132 (ring modulated slow oscillations), 212 (wide interval "banging" contest), 325 ("record skipping"), 422 (swooping ring modulation), 572 (looping notes), 578 (Morse code on shortwave radio), and 639 (modulated vocal chants).

     Note: Most of the information for the above analysis of MANTRA (except for the Siska tables) was culled from Richard Toop's excellent book "Six Lectures from the Stockhausen Courses Kürten 2002", available from http://www.stockhausen-verlag.comHIGHLY RECOMMENDED.

A Narrative Analysis
     So here I ignore all the scale Expansions and tone rows and just approach MANTRA as 26 "moments" (following the "official" Kontarsky Brothers' recording), which seems ideologically at odds with the MANTRA formula concept, works better for me.  Below each track "impression" I also note the 13 "musical behavior" sections, as well as if 1 of the 12 MANTRA scale Expansions occurs immediately at the beginning of the track (also Cymbal strike "types").

Track Bar Duration   Features
1 1-2 0:30  - Intro: 4 chords from the 4 phrases of the MANTRA
 - CYMBAL: Regular Repetition
2 3-10 1:13  - The MANTRA
3 11-70 3:56  - MANTRA sequence using regular repetition (tremolo), alternating pianos
 - CYMBAL: Final note accent
 - SECTION I: Regular Repetition (from bar 12)
 - SECTION II: Final note accent (from bar 61)
4 71-108 2:41  - Moderately Fast, slowing down
 - Expansion II.8 begins
 - CYMBAL: Flurries around center note (at bar 73)
 - SECTION III: Normal (from 89)

 - CYMBAL: Slow Tremolo (at 89)
5 109-120 1:51  - Slow descending/ascending scales/notes against Morse code, rolled chords
6 121-131 0:51  - "Swinging"
 - CYMBAL: Chord
 - SECTION IV: Flurries around center note (125)
7 132-150 2:05  - Held chords with ring modulated oscillations
 - Expansion IV.10 begins
8 151-188 5:06  - High bell tones, repetition and dialogue with melodic flurries
 - SECTION IV Center Note Flurries Continues
 - Expansion IV.11 begins (152)
9 189-211 1:13  - Cartoon-like, ending in slowing tremolo modulated chords
 - SECTION IV Center Note Flurries Continues
 - Expansion IV.7 begins
 - SECTION V: Slow Tremolo (205)
10 212-237 1:41  - Wide interval "banging" contest
 - Expansion V.12 begins
11 238-261 1:21  - Unstable tempos
 - SECTION V Slow Tremolo Continues
 - CYMBAL: gliss up (241)
12 262-298 5:00  - Languid, Morse code cymbals
 - SECTION V Slow Tremolo Continues
 - Expansions V.5 & V.7 begin
 - SECTION VI: Accented chord (282)
13 299-323 1:10  - Ringing cymbal with woodblock accents
 - SECTION VI Chord Accent Continues
 - Expansion VI.7 ( begins 300)
 - CYMBAL: Trilling (308), Sforzando (312)
14 324-421 2:07  - Staccato chords, "record skipping" with woodblock, changing tempo
- SECTION VI Chord Accent Continues
 - Expansion VI.10 (begins 325)
15 422-434 4:00  - Swooping ring modulation on slow chord accents
16 435-491 3:26  - Melancholic, jazzy
 - SECTION VI Chord Accent Continues
 - SECTION VII: Accent (with 1 repetition) (445)
 - SECTION VIII: Descending chromatic scale (488) 
17 492-529  5:58  - Distorted lullaby/ballad ("ala Blade Runner"), fast arpeggios
 - SECTION VIII Descending chromatic Continues
 - Expansion VIII.12 begins
18 530-577 2:58  - Playful, fast looping notes insert (from 572)
 - SECTION VIII Descending chromatic Continues
 - Expansion VIII.4 begins
 - CYMBAL: Flurries around center note

 - SECTION IX: Staccato (538)
19 578-637 3:28  - Morse code radio, syncopation with woodblocks, cymbals
 - SECTION X: Irregular repetition (587)
 - SECTION XI: Trills (611)
20 638-640 0:56  - Modulated vocal chants "JO (yo) - HO - TA!"
21 641-655 2:02  - Quiet Morse code with cymbals and high piano key
 - SECTION XII: Sharp attack (with echo)
 - Expansion XII.1 begins
 - CYMBAL: Irregular Repetition
22 656-686 3:37  - Cymbal trills, cascading low notes, etc…
- SECTION XII Sharp attack Continues
 - Expansion XII.8 begins
 - SECTION XIII: Arpeggio connection (673)
23 687-854 4:33  - Fast Compression Toccata
24 855-880 1:58  - High Rolled chords
 - SECTION XIII Arpeggio Continues
 - Expansion XIII.6 begins
25 881-883 0:28  - A Rising scale and a cadence
 - CYMBAL: Crescendo (Repetition)
26 884-887 1:13 (0:45)  - MANTRA restatement
(64:25 - Total time of Kontarsky brothers performance)

Live Performance 
Yxus Ensemble: Kadri-Ann Sumera, Age Juurikas:

The performance video below is split up into the 13 musical behavior sections:
(Performed by Bernhard Fograscher and Jennifer Hymer at Making New Waves Festival, Budapest 2008)

As a side note, the MANTRA formula makes a full appearance in Stockhausen's DIENSTAG aus LICHT opera 1st Act, JAHRESLAUF, during the 3rd saxophone solo segment.

Stockhausen's Lecture on MANTRA (Pt 1)
MANTRA (Kontarsky Brothers) samples, track listings and CD ordering
Purchase the Score
MANTRA (Live, Alack Karis & Kian Freitas,YouTube)
Sonoloco Review (Kontarsky Brothers)
Reviews (Pestova-Meyer Piano Duo) 
Ádám Siska: LIVE ELECTRONICS New Tools for Composition in the 20th Century (2013 thesis) 
Paul Miller talk on MANTRA, Library of Congress