(Sati has two meanings - please see note at the end)

or here

All the suttas below have parallels in Chinese, Sanskrit or Tibetan.

"With mindfulness as his gate-keeper, the disciple of the ones abandons what is unskillful, develops what is skillful, abandons what is blameworthy, develops what is blameless, and looks after himself with purity."
AN 7.6


“The abandoning of both
sensual perceptions and dejection;
the dispelling of the lack of mental strength,
the forestalling of under-doing (note: that usually brings remorse);

  • obtention (of citta) and purified equanimity, preceded by reflection on the Dhamma: this, I say, is emancipation by final knowledge, the breaking up of ignorance.”

Pahānaṃ kāmacchandānaṃ,
Domanassāna cūbhayaṃ;
Thinassa ca panūdanaṃ,
Kukkuccānaṃ nivāraṇaṃ.

Aññāvimokkhaṃ pabrūmi,
avijjāya pabhedanaṃ.

Snp 5.13 or 5.14 (Udaya) - AN 3.33
How sati fits within the different progressions?:
Faculties + Powers
Confidence > energy > sati (obtention [of citta]) > establishment [in citta] > discernment.
Noble eightfold path
Right view > right thought > right speech > right action > right livelihood > right effort > right obtention (sammā sati) > right establishment.
Awakening factors
Sati (obtention) > investigation-of-dhammas (dhamma-vicaya) > energy > joy > tranquillity > establishment [in (almost liberated) citta] > equanimity.
Contemplation of the dhammas (dhammānupassanā) emphasizes contemplation of the Hindrances [Sensory desire - Ill-will - Sloth-torpor - over-doing/under-doing - Doubt,] & the contemplation of the Awakening Factors.

Mindfulness/Sati is the means to a reflection on the Teaching - gate-keeping any bad (or even new) knowledge and phenomena from entering; and allowing in samatha and vipassana (SN 35.245 [a.k.a. 35.204]).
But it is also the obtention of the establishment in citta, through the four ways to the obtention (Cattaro Satipaṭṭhānā). 

"The four ways to the obtention/acquisition [of citta] (cattaro satipaṭṭhāna,) too, I say, have a nutriment; they are not without nutriment. And what is the nutriment for the four ways to the obtention [of citta]? It should be said: the three kinds of good conduct."
AN 10.61
"Well then, Bahiya, purify the very starting point of wholesome states. And what is the starting point of wholesome states? Virtue that is well purified and view that is straight. Then, Bahiya, when your virtue is well purified and your view is straight, based upon virtue, established upon virtue, you should develop the four ways to the obtention (of citta) ."
SN 47.15
In other words, the more you have previously given concern to: confidence > energy > & > right view > right thought > right speech > right action > right livelihood > right effort; the better the obtention/sati will be; the better the dhammānupassanā and dhammāvicaya will be - (and "purified equanimity and obtention/acquisition (of citta)" will ensue - AN 3.33 above).
"When arousing the enlightenment factor of the obtainment (of citta), friend, a bhikkhu understands: ‘My citta is well liberated; I have uprooted sloth and torpor and thoroughly removed overdoing and underdoing (the latter being source of remorse). My energy has been aroused. I attend as a matter of vital concern, not sluggishly.’"
SN 46.8
When one noble disciple intellectualises thoroughly with the emotional intellect [mano], from the origin (yoniso manasikaroto), bhikkhus, unarisen sensual desire does not arise and arisen sensual desire is abandoned. When one attends carefully, unarisen ill will … sloth and torpor … overdoing and underdoing ... doubt does not arise and arisen doubt is abandoned. Also, the unarisen enlightenment factor of the obtention (of citta) arises and the arisen enlightenment factor of the obtention (of citta) comes to fulfilment by development.
SN 46.24
The enlightenment factor of the obtention (of citta) , bhikkhus, is a nonobstruction, nonhindrances, noncorruptions of the mind (citta).
SN 46.34
The enlightenment factor of the obtention (of citta) is a maker of vision, maker of knowledge, promoting the growth of wisdom (discernment ), free from vexation, leading towards Nibbāna.
SN 46.40
There are, bhikkhus, things (dhammas) that are the basis for the enlightenment factor of the obtention (of citta):
frequently setting the mano upon them from the origin (yoniso manasikara) is the nutriment for the arising of the unarisen enlightenment factor of the
obtention (of citta) and for the fulfilment by development of the arisen enlightenment factor of the obtention (of citta).
SN 46.51
Whatever mindfulness there is of things internal is the enlightenment factor of the obtention (of citta); whatever mindfulness there is of things external is also the enlightenment factor of the obtention (of citta). (see also the other enlightenment factors).
SN 46.52
With a mind that has developed the enlightenment factor of the the obtention (of citta) he knows and sees things as they as they have come to be - yathābhūtaṃ). This is a cause for knowledge and vision; it is in this way that knowledge and vision are with cause and condition.
SN 46.56

Knowing the nutriments for each of the four ways for the  obtention [active]:
... enlightenment factor of the obtention (of citta), which is based upon seclusion, dispassion, and cessation, maturing in release.
AN 4.14


Sati (as smṛti calls the recollective function. [active]:
Here, a bhikkhu is mindful, possessing supreme mindfulness and alertness, one who remembers and recollects what was done and said long ago (cirakatampi cirabhāsitampi saritā anussaritā).
AN 8.30
See also SN 7.4, SN 7.67, SN 48.50.


Let's take Ill-will for instance. It must be called in from memory as a whole. That bad feeling in which consciousness has found its home (SN 22.3;) that particular ill-will, must unveil its entire nature. Consciousness that has settled (SN 12.39,) must bring out all the cumulated "qualities" of ill-will.

Like the cowherd in AN 10.20, sati must render (actively) the big picture.
Contemplation and investigation:
Then comes the contemplation (passive) and the investigation (active) (anupassati & vicaya).

In our particular example, the meditator contemplates the hindrances; which is a sub-category of the fourth reflection, that is the reflection on dhammas (that comes after the contemplations on body, feeling and mind (citta)). [passive]
"Come you, monk, fare along contemplating the mental states (e.g. ill-will) among mental states, but do not apply yourself to a train of thought connected with mental states (vitakkaṃ vitakkesi)."
MN 125
Then he investigates them. (And realizes them).[active]

Finally, the meditator contemplates (samanupassati) the disappearance of the five hindrances within himself.[passive]
It is because the dhammas are cognized and realized properly, that there is an abandonment of the defiling mental qualities.
As he remains thus focused on mental qualities in & of themselves, his mind becomes concentrated, his defilements are abandoned. He takes note of that fact. As a result, he is rewarded with a pleasant abiding here & now, together with obtention (of citta) & alertness. Why is that? Because the wise, experienced, skillful monk picks up on the theme of his own mind (citta)."

SN 47.
His obtention (of citta) is purified.



"Sati" has two meanings:


1- from the Vedic smṛti स्मृति (√ smṛ), meaning "thinking upon" (calling to mind, ~"mindful") [TĀr. ChUp. MBh.]. It can be being mindful of (thinking upon) the breath. Or being mindful of not letting the external akusala get in, etc.


2- from सति sati = साति sāti = सन्ति santi = gaining , obtaining, acquisition (RV.)

Sati as "obtention", "obtainment" (of citta). 


Another way to see smṛti

Smṛ or Spṛ ?
स्मृ  smṛ = vario lectio (v.l.) for √ spṛ 

स्पृ  √ spṛ  (v.l. [ stṛ ] and [ smṛ ] )

- to release , extricate or deliver from (abl.), save, gain, win  RV. TS. Br. TUp.

- Caus. [ spāráyati ] , to attract to one's self, win TBr. ; to preserve, save, rescue ib. ; 

- to gladden, delight, gratify, bestow  Dhātup.


The second meaning, viz. "obtention", is seen in the extract:
Paṇidhāya parimukhaṃ satiṃ upaṭṭhapetvā.

Intending (paṇidhāya) towards the beginning (parimukhaṃ), and having looked after (upaṭṭhapetvā) the [b] obtention[/b] (sati) [of Citta].

OR again in this extract:

Yet he is covetous - inflamed by lust for sensual pleasance - with a mind (citta) full of enmity - with intentions (mana-saṅkappo) corrupted by hate - negligent about the [b] obtention[/b] [of (a free) citta] - lacking clear comprehension - not well set (located, settled [in citta]) - (with a) scatter-mind (citta) - loose in his/her sense faculties.

So ca hoti abhijjhālu kāmesu tibbasārāgo byāpannacitto paduṭṭhamanasaṅkappo muṭṭhassati asampajāno asamāhito vibbhantacitto pākatindriyo.
SN 22.80


Usually, the first meaning of sati is used in the context of seclusion from the external. That is to say, remembering the Teaching to prevent the "attacks" from the external (see SN 35.238 & SN 35.245).

The closer one is to samādhi (https://justpaste.it/5x298), the closer one is to the second meaning of sati.
Also, it is remarkable to notice how citta has to be gladdened (as per the Dhātup definition above,) to reach cetaso ekodibhāva and cetovimutti, in ānāpānasati and jhāna (https://justpaste.it/69jp3).




To resume, sati is all about restraining the indriya (indriyesu guttadvārā), to reach a liberated citta.
This restraint is worked upon, both by guarding oneself from the external field of sensory experience, with the help of sati (as smṛti स्मृति [√ smṛ]) - but also, by a second process, through the use of the internal field of sensory experience that is mano (the emotional sensory intellect/brain-"heart",) to obtain (sati as सति) the establihment in citta.
A citta that should be devoid of the sensory experiences of the saḷāyatana fields.

As one can notice in SN 36.7 (SA 1028) (https://bit.ly/2B4HpMt) , there is quite a difference between sati and sampajāno, that is much clearer now with that second definition of sati (viz. obtention/acquisition).
Indeed, the late ("western") concept of mindfulness is just sampajāno; not sati.

Sati means:

1. recollecting and thinking upon the Teaching as a warding measure (as smṛti स्मृति).
It is to be paired with sampajāno, as an exercise of constant awareness about the internal, (viz. about the things purely done by satta, with the internal āyatana(ni)). [No one can breath for you, or walk for you, or chew for you, etc. - There is no external influences in these deeds]. 

2. the obtention/acquisition [of samadhi (https://justpaste.it/5x298)] .


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