Good question. "Warm" for you is probably not "warm" for me .... Maybe you say a tone is "warm" but I would say "round" or "soft" .... ????
Since 2003 there has been an study/project going on here in Sweden called "Modelling Perceptual Dimensions of Saxophone Sounds" by Arne Nykänen, Örjan Johansson, Jan Lundberg and Jan Berg all from Luleå University of Technology, Sweden. The study is now ready and it's published in "Acta Acustica United With Acustica vol. 95 (2009) s 539-549. The summary below is taken from the presentation.
In the past, musical instruments were developed over long periods of time by skilled craftsmen. Today, most instruments are mass-produced. Design of musical instruments as mass-produced products requires using strategies which make it easier to identify customer needs and develop exact specifications. To develop useful specifications it is necessary to convert general descriptions into something which can be commonly understood and also be interpretable in terms of acoustic metrics. In this study, methods for analysis and specification of steady state parts of alto saxophone sounds were developed. Saxophonists use of verbal descriptions of saxophone sounds was investigated. Sound stimuli were binaurally recorded. Judgements upon perceived qualities were made by saxophonists and non-saxophonists using the method of verbal attribute magnitude estimation. Perceptual dimensions were identified using principal component analysis of listening test data. Three prominent dimensions were found and described using the verbal attributes: l) warm/soft, 2) back vowel analogues and 3) sharp/rough. The perceptual dimensions were modelled as linear functions of acoustic metrics. The results were validated through listening tests with new subjects and new stimuli. Based on the findings, The method was seen as an approach which can enhance the musical instrument design process.
The study is less about the physics of the saxophone.