In their APA chapter, Braun & Clarke (2102)  say:

"How many themes are enough or too many? For our data set, we generated six themes; for brevity, only four are summarized in Exhibit 4.3. Unfortunately, there is no magic formula that states that if you have X amount of data, and you are writing a report of Y length, you should have Z number of themes. The more data you have, the more codes and thus themes, you will likely generate; if you are writing a longer report, you will have space to discuss more themes. But with more themes, your analysis can lose coherence. What is essential is that your themes are presented in sufficient depth and detail to convey the richness and complexity of your data— you are unlikely to achieve this if you report more than six or seven themes in a 10,000-word report. Your themes will likely be “thin.” If you are trying to provide a meaningful overview of your data, one to two themes are likely insufficient; however, they may be sufficient for an in-depth analysis of one aspect of the data. In an 8,000- to 10,000-word article, we typically report two to six themes." (p. 65)

For my students, I recommend 3-5 themes, simply because it is difficult for the reader to keep track of. more than 5 things.

Hello Joash.

I do thematic analysis research regularly in my research agenda. Sample size as it is related to Saturation, is what you are asking about here. Below you will find an text done by Strauss and Glaser on the topic of Saturation in Thematic Analysis. It is not about minimum number of themes that is acceptable it is about have you achieved Saturation. Depending on your studies population (whomever the media files are from) that informs your studies consideration of Saturation. In terms of analyzing qualitative or natural language data, I use Raven's Eye data analytic software. It has proven to really help me with my research.  I know you can upload data in a lot of different formats to Raven's Eye.

Glaser, B. G. & Strauss, A. L. The discovery of grounded theory: Strategies for qualitative research. Piscataway, New Jersey: Transaction.

This is another good research methods text that speaks to Thematic Analysis and Saturation.

Creswell, J. W. Qualitative inquiry and research design: Choosing among five traditions. Thousand Oaks, CA: Sage Publications.

Hope this helps.

In their APA chapter, Braun & Clarke (2102)  say:

"How many themes are enough or too many? For our data set, we generated six themes; for brevity, only four are summarized in Exhibit 4.3. Unfortunately, there is no magic formula that states that if you have X amount of data, and you are writing a report of Y length, you should have Z number of themes. The more data you have, the more codes and thus themes, you will likely generate; if you are writing a longer report, you will have space to discuss more themes. But with more themes, your analysis can lose coherence. What is essential is that your themes are presented in sufficient depth and detail to convey the richness and complexity of your data— you are unlikely to achieve this if you report more than six or seven themes in a 10,000-word report. Your themes will likely be “thin.” If you are trying to provide a meaningful overview of your data, one to two themes are likely insufficient; however, they may be sufficient for an in-depth analysis of one aspect of the data. In an 8,000- to 10,000-word article, we typically report two to six themes." (p. 65)

For my students, I recommend 3-5 themes, simply because it is difficult for the reader to keep track of. more than 5 things.

I agree in what professor Morgan says about number of themes. I would like to ad that the themes need to cover your data in order to represent them. I think that the most common mistake made by my students is to create to many themes to capture "everything" and the themes accordingly gets to "thin" since they don't have enough data to support them all if the descriptions are to be "thick enough". So my most common advice is to merge themes together to create more rich themes. By doing this you also often reach a higher abstraction level in your analysis.

The following papers should also be helpful, particularly:

• Attride-stirling (2001, p. 392) said that there are no hard and fast rules about how many themes should make a network but, from a practical stance, more than 15 may be too many to handle….…. 
• Braun and Clarke (2006, p. 82) viewed that the ‘keyness’ of a theme is not necessarily dependent on quantifiable measures but rather on whether it captures something important in relation to the overall research question. Here, in Victoria’s research on representations of lesbians and gay parents, six key themes were identified….... 
• Attride-stirling, J. (2001). Thematic networks: an analytic tool for qualitative research. Qualitative Research, 1, 3, pp. 385-405.
• Braun, V. and Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3, 2, pp. 77-101.
• Clarke, V. and Braun, V. (2013). Teaching thematic analysis: Overcoming challenges and developing strategies for effective learning. The psychologist, 26, 2, pp. 120-123.
• Ryan, G. W. and Bernard, H. R. (2003). Techniques to Identify Themes. Field Methods, 15, 1, pp. 85-109. 

The only thing that I would add is to remember to keep in mind the question(s) you've asked, your original starting point.  It could be easy to let unrelated themes get away from your analysis.  However, I wouldn't discard it altogether those unexpected themes may lead you to new areas of study.

Hello Joash, if these 'media files' concern research participant understanding of their experience, it's useful - I suggest - to remember that considered as a practice from a hermeneutic perspective, the process of understanding has a priori, core, or essential features, which could be considered a 'minimum number of themes' to which the less abstract, more contextually related themes in recorded data, can then be related. I've attached a slide set from a recent London conference making this argument. Regards, Tony.

 I'm not sure there is a minimum/maximum number of themes. I used thematic template analysis in my PhD which was a qualitative study and I had over 50 themes! If this is of interest to you please look at the work of Prof. Nigel King (see link below).

http://onlineqda.hud.ac.uk/_REQUALLO/FR/Template_Analysis/Index.php 

As a reader, I personally would fine 50 themes to be overwhelming. The usual advice when you have so many themes is to group them into categories and then to diagram those in the form of a concept model. The original Braun & Clarke (2006) has some useful examples of creating models, as does the Attride-Sterling article cited earlier.