Although this analysis is far from complete, I am not sure if I have time to finish it more precisely. Although the start of the analysis was always my own, I draw heavily on Russell (1991) and Mitchell(2002) (see page with references).
The poem has four strophes, all consisting of two lines. In each strophe one line is about light, the other about a disaster that happened (because a child has died). The first strophe starts with the rising sun and contrasts this with the night when the disaster happened. The other strophes start with the disaster and contrast this in the second line with the sun, eternal light and the joyful light of the world. The development of the disaster is as follows. First, the disaster is mentioned. Then the fact is stated that it came to him alone. The third strophe makes aware that you must not let the disaster influence you completely. This provides to rephrase the disaster in a milder way as a little lamp that went out in his heart. So, the poem tells a story about dealing with the disaster.
Mahler uses the strophic structure of the poem in his music, but uses at the same time more or less a sonata form. The first strophe is used as the presentation of two themes (consistent with the two lines in the strophe), the second repeats those themes with small variations. The third strophe is the development part, while the last strophe is used as a repeat, again varied. This is consistent with the content of the strophes, where the first two strophes state the problem, the third strophe starts a movement toward acceptance by telling that you must not let the disaster dwell in your heart, and as a consequence of this you are able to rephrase the problem as the value of life (a lamp) that vanished.
The song is very simple with respect to basic harmonies: it is written in d minor that changes sometimes into D major. The harmonically interesting things are mainly in creating very subtle colors.
Let us now have a closer look at the presentation of the two themes. The orchestra starts with a very nice and subtle duet. Let's first have a look at the upper line (oboe). It starts with a motive of major second upwards, and then continues with motives of two minor upward seconds. The first and second motive are connected with a downward diminished fifth jump. The third and fourth motive are a full tone lower repetition of the first two motives, expect for starting with a minor instead of major second. So, the small two-note motives are integrated into large four-note motives. The last half of the oboe prelude has two other motives: A-Bes-A en Bes-A-G-F-G-A.
The second melody, with horn, starts with a motive that is a half-speed version of the oboe motive in bar 3. This motive is repeated twice, followed by a downward minor second parallel with the oboe, and a very standard cadence bass line at the end, that gives a IV-V-I in combination with the oboe. The whole harmonic line that is more or less implied in the duet is very standard: I-V-IV-VII-IV-V-I-IV-V-I. The position of the chords is a little bit less standard: during the first two bars the root is never in the bass line, and if it is in the chord at all, it is in the top line. This gives the feeling of not being grounded, the melody starting to develop from nothing.
When we summarize this introduction we see the important role of the half-tone motive and the diminished fifth, while the harmony gets its definition in the bar where the voice enters. When the voice enters, the harmony is clearly d-minor. Nevertheless, the voice uses the chord E-G-Bes as rest points for the long notes on the first beat of the bars. This chord has no direction, and may already symbolize the feeling of being lost. The voice melody has a clear line, starting with the main motive, the half-tone rise, and ending with an extended version of this motive, a half-tone rise, followed by a half-tone decrease (hell aufgehn), already used at the end of bar 2 by the oboe. The downward line on bars 5 and 6 may be seen as an extension of the line of the horn in bar 1 and 2. The slow pace of the voice line may suggest the slow movement of the rising sun. It may also already bring in the sad message of the second line of the text. The melody line emphasizes the words "Sonn" and "Hell", with "Hell" as its main point.
The piano just repeats the basic motive focused on D with an appoggiatura. This motive has the effect of repeated sighs, but may also symbolize the rising sun. The Cis-D motive is repeated, where in bar 7 A-Bes is used and in bar 8 Fis-G. There the chord E-G-Bes-D (E in the voice) is formed, where the E and D are a light dissonance and the chord works more or less as a dominant. The first theme ends with a condensed repetition of the phrase of the voice by the piano, in first instance without any accompaniment and ending with a sharp dissonant D-E. Have a look at the contrast between the rising sun, perhaps symbolized in the rising seconds in the piano, and the downward line of the voice.
The second theme starts with a chromatic rise of the G into a Gis that transforms the E chord back to a d minor chord. The voice takes over the E from the piano (although an octave lower), and then has a chromatic line upwards (in contrast with the sad message?) towards Bes on "Unglück, and downwards back till E after some cadence on "Nacht", emphasizing the contrasts with the word Sun in the first voice line, and finding rest in a D major chord. Be aware that the same interval Bes-E, the diminished fifth, was central in the first vocal line. The right hand piano follows roughly the voice line, accompanied by a simple figure, where the top note follows the chromatic line in the voice melody a third lower. The harmonic line that is implied amidst all this chromatics is a very simple diachronic harmony: I-V7-I-IV-V-I-IV-V-I.
If you think that the cadence on " Nacht" is just a cadence, you are missing the point. They create a very specific tension, and are the introduction of an important new motive that is immediately repeated in condensed form by the horn ending with a diminished seventh in bar 16. Later it is a very important motive in the development phase of the song. The melody in bar 16-17 is a variation on the introductory oboe line, again moving between E and Bes. The Bes-A motive at the end of bar 17 resembles the motive at the end of bar 19 that is a very central motive in the accompaniment of the first line of the last strophe (below "Lämplein"). It is also the motive C-Bes-A of the horn at the end of bar 2, and transforms the line of the oboe towards new dimensions, although the general line that follows is consistent with what happens in bar 3 and 4, at a lower pitch, after its transformation. The harp diatonic accompaniment provides consolation, as it does in the accompaniment of the second line of the song, where the A7 chord increases tension (in contrast with the original situation in bar 3-4), that is released in a clear d minor chord on bar 18. With the line at the end of bar 18 a IV (g minor) reassures this tonic, to go back to the d minor in m. 19.
A chromatic rise towards E suggests a transformation towards an other chord (as in bar 17-18), but it transforms to a plain d minor chord, where it started, a very open d chord. The first presentation of the themes ends with a very sparse orchestration where the clocks, i.e. the top D's have a very important function. Also the consolation of the diatonic harp is present here. It creates the very simple world of a child, and has at the same moment some suggestion of death.
For the second strophe, starting in bar 22, the first bar of the introduction in the first strophe is skipped and the melody immediately starts with the second bar. In bar 23 the lower line is changed into a downward line over a diminished fifth, the last part chromatic. The voice line is a variation of the first theme, where the rhythm is changed and the accentuated note is on the first beat only on "mir", giving an emphasis on this word. The new rhythm gives more drama to the melodic line, consistent with the "Unglück". In the accompaniment the flute is added following the voice an octave higher, and the Cis-D motive continuing in the fagot bass line. The harmony is more dissonant than in the first couplet, consistent with the change in meaning of the text. The oboe interlude becomes much more extended, and starts already when the voice has to finish its melody. The oboe forms a canon with the clarinet, with in its second half a variation that resembles the horn motive in double speed in bars 2 and 3. All this increases the expressiveness.
The transitory Gis towards the second theme is off-beat and therefore has much less emphasis. This is consistent with the meaning of the text; in the first strophe it was the introduction to the line of the night, while in the second strophe is the introduction of the sun. The rest of the second theme is exactly the same as in the first strophe for the piano version, where in the orchestra second violins are added. Recognize that in the poem the words "die Sonne" are repeated, and therefore get an emphasis.
The postlude of the second strophe has an E instead of an Fis in the horn, and is pp repeated, first accompanied by a raised major chord, then a d minor chord. This is followed by a variation of the cadence on "allgemein" and "Nacht", also repeated, and then attacked by the introduction of the oboe in m. 40 with its starting motive, although the order of the two motives differs; first the one with C-Fis, then the one with D-Gis. This rising line continues with a sequence that ends with the Cis-D motive that first was a deep accompaniment of the first voice line. The melody of the first voice line is now in the horns, where the voice comes in at the moment in the interlude started in the first two strophes. And the voice takes this oboe interlude in countermovement and repeats this while the oboe is doing the motive of the postlude of the second strophe. Peter Russell (p. 71) suggests that this is a combination of the light and darkness. Then the violins take over the melody of the oboe using a motive that resembles the cadence motive, as also used by the horn in m. 15-16, and used also in m. 39. The voice has again a rise over a diminished fifth (A-Es), but starts relatively high (A) and has a long melisma on "ewige". The harp accompanies this with chords build around d, an with a lot of coloring in bars 53-55 (changing a moment from d into D on the first "ewige Licht") .
The postlude gives according to Mitchell the real climax of the development phase. The chords in m. 59 are yet unstable between minor and major, and then continues with a development of the "cadence" motive of m. 14 ("Nacht", "allgemein") that is omitted from the voice line, but becomes central in the transformation process from grief towards eternal light. In bar 63 the flute starts with a variation of the introductory oboe line, immediately completed with a variation of this oboe line by the oboe at a higher pitch than the original one, that ends with the original accompaniment of the second half of the second introduction of bar 23. The flute gives a feeling of consolation, that is the essence of the fourth strophe of the poem.
In the fourth strophe, the recapitulation of the theme, the voice melody is again varied, but now with the eight notes, that are also used in the sigh-motive in the orchestra. The interlude is again a variation on the original interlude, with a rhythm consistent with the sigh-motive. The second line is again a repetition of the first line, but has a clear A-D bass line in m. 76-77. The postlude is also a repetition of the postlude of the first strophe. However, in bar 79 the voice joins in the postlude by repeating the last sentence, and the song ends with the childish glockenspiel. Recognize that in bar 81 there are no accents, in contrasts with m. 19, while the sforzando is on "Welt", m. 82, giving a different meaning to the last three bars of the song compared with bars 20-21. The ending of the harp on an A gives the feeling of not really having found a resolution, as did also the emotional tension in the interlude before the fourth strophe. The final resolution is at the end of the last song of the cycle. But before that, we have to see what happens in the songs in between.