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The definition of a path is: A virtuous mind conjoined with renunciation.
As it says in Maitreya/Asanga's Ornament of Clear Realization:
A path is a virtuous mind conjoined with spontaneous renunciation. Boundaries are from the path of accumulation through the buddha ground. The synonyms of path are: knower, exalted wisdom, exalted knower and clear realization.
Thus, all paths are 'paths of renunciation' in this sense, although I have never come across the expression 'path of renunciation' before.
A Mahayana path is a virtuous mind conjoined with bodhicitta (which renunciation). For instance, generosity in the continuum of a bodhisattva is a Mahayana path. Generosity in the continuum of someone who is on a so-called Hinayana path is a path.
Traditionally, we say that tantra must be practiced in dependence upon the three principles of the path.
Therefore, it is aimed at bodhisattvas that are at least on the path of preparation, having realized emptiness by way of an inferential cognizer.
There are five paths:
The path of accumulation, where one becomes a bodhisattva by way of generating the spontaneous mind of enlightenment (bodhicitta)
The path of preparation. One enters this path by way of a first inferential (not direct) realization of emptiness, that is a union of samata and vipassana realizing emptiness.
The path of seeing. One enters the path of seeing by way of a first direct realization of emptiness, whereby all dualistic appearances vanish. This wisdom directly realizing emptiness acts as an antidote to the grossest levels of afflictions, therefore it is called the first true path. It is also an arya path. It is the first ground (which continues on to the path of meditation). We say that the ultimate Dharma jewel is true cessation and true path in the continuum of an arya being. This echoes your statements with respect to "path to cessation". Here, the first true cessation occurs upon entering the path of seeing. The buddha-ground or 'path of no-more-learning' is the final true cessation. There are various true cessations before.
The path of meditation. It has the ten grounds (bhumis) but there are also divisions into 11, 13, and 16. At the level of the 8th ground, a bodhisattva is free from all afflictions (just like a so-called Hinayana arhat). He still has to abandon "knowledge obscurations" which are the imprints of the afflictions (like the scent left by the garlic even after it's been removed from a container).
The path of no-more-learning or "buddha ground". One enters it when he is free from the two obscurations: afflictive and knowledge obscurations. From there, one obtains the four kayas (which can also be divided into 2, 3, and 5).
The Mahayana path is called 'path of perfection' because we perfect the perfections (paramitas) on each of the grounds.
From a tantric viewpoint, Tantra could be practiced from the generation of the three principles of the path, or as late as the last moment of ten bhumi. From a tantric point of view, one cannot achieve enlightenment (abandoning knowledge obscurations) without tantra.
The five paths are common to Mahayana traditions, including Zen, Tiantai, etc. It is classic Mahayana Abhidharma.
For instance:
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Non-renunciation means that Vajrayana might freely use defilements and desires rather than dispelling them or considering them to be renounced.
Vajrayana also uses sense objects in their meditation skilfully, like Guru Yoga or Deity yoga. Mahayana does not use such objects in such way (however, one exemption might be Pure Land Buddhism).
The analogy often given here is snake venom used to create antidotes. And obviously, the end result is renunciation itself! But by different means.
In the end it aims to transform these into skilful thing and positive (i.e. karma) instead of rejecting such attachments. The practitioner might utilise objects of pleasure and desire for the purpose of transforming poison into medicine - like compassionate energy, all, obviously, within reasonable code of conduct.
In Tantra desire is the ornament of enlightenment. Here is a comparison of sutric and tantric approach, some points are arguable, but it is the general picture.
And to summarise, Tsongkhapa:
There is no spiritual path without renunciation as the very definition of a spiritual path is an exalted awareness conjoined with non-fabricated renunciation, so of course the Vajrayana is a path of renunciation - it's a supramundane path if practised properly.
Objects are not contaminated from their own side, so if you remember emptiness, your enjoyment is not a cause of samsara.
As for ChrisW's request, here's the excerpt from the book Introduction to Tantra, The Transformation of Desire – Thubten Yeshe:
Tantra’s approach is very different. Instead of viewing pleasure and desire as something to be avoided at all costs, tantra recognizes the powerful energy aroused by our desires to be an indispensable resource for the spiritual path. Because the goal is nothing less than the realization of our highest human potential, tantra seeks to transform every experience — no matter how “unreligious” it may appear — into the path of fulfillment. It is precisely because our present life is so inseparably linked with desire that we must make use of desire’s tremendous energy if we wish to transform our life into something transcendental.
Thus the logic of tantra is really very simple: our experience of ordinary pleasure can be used as the resource for attaining the supremely pleasurable experience of totality, or enlightenment. It is natural that qualities of the mind, when cultivated, produce something similar, not opposite, to themselves. This is true for both positive and negative states of mind. In the same way that dissatisfaction itself can never become satisfaction, misery does not naturally evolve into happiness. According to tantra, we cannot hope to attain our goal of universal and complete happiness by systematically making ourselves more and more miserable. This is contrary to the way things actually work. It is only by cultivating small experiences of calm and satisfaction now that we will be able to achieve our ultimate goal of peace and tranquility in the future. And similarly, it is only through the skillful use of desirous energy and by building up the habit of experiencing what we might call true pleasure that we can hope to achieve the everlasting bliss and joy of full illumination.
From your example, playing music for my own enjoyment could make me happy and exalted enough so I can utilise this very energy in order to help other beings alleviate suffering in a joyful and courageous manner, and help in cultivating my own Pāramitās such as enthusiasm better. And so on, but that also requires great intelligence what HH Dalai Lama always highlights. Someone else could probably find a better examples though.