There is data on this matter. http://www.zainea.com/f0_m%26f.pdf present data showing that English speakers who study Mandarin typically use more tonal range in Mandarin than in English but substantially less than native Mandarin speakers do. Hardly surprising. And it is known that musical training helps.
Chen (1974) compared the pitch range between English and Chinese
speakers. The results showed that Mandarin needs a pitch range 1.5
times wider than English, and that native English speakers do not have
this range naturally. The author claimed that native speakers of a
non-tonal language need to widen their pitch range to successfully
acquire a tonal language. Wong and Perrachione (2006) studied a group
learning Mandarin tones, and found that “large individual differences
were observed. Learning success was found to be associated with the
learner’s ability to perceive pitch patterns in a nonlexical context
and their previous musical experience”. Experienced teachers have also
reported that students with musical training learn Mandarin faster.
The only advice I found was for students to try harder. Too often, though, "trying harder," without explicit guidance, only reinforces habits you already have. No one that I found suggested either
1) students with musical experience should see the known data on what pitches native speakers actually use, so the students can aim for a much closer match to native pronunciation than most ever hear about today. As monalisa says, a native speaker's pitches vary. Data shows even a single first tone can drop its pitch slightly at the end! But research shows the physical pitch in the sense of "fundamental frequency" of the sounds is truly a key part of perceived Mandarin tones.
2) students without much musical experience should spend some time using tuning forks or a keyboard to learn to hear pitch better.
I think this is neglected because too many teachers in the US are just convinced American students can never have very good Mandarin pronunciation anyway, so they don't want to worry about "fine points."
Tuning forks are helping me now, and i will pass on the little bit I have learned. It seems best to focus on one note for the high level tone. This is not surprising: Studies show people are consistently better at using the low part of their vocal range and need more training to extend their high end. A tuning fork is extremely convenient. Pitch sources free on-line and on phone apps sounded bad to me -- I did not try paying for better ones. Electronic tuners seem like overkill. I don't have a piano. Pitch pipes make less pure notes than a natural human voice (they produce more harmonics), so they are a bad model.
Left to myself, I radically misjudged both the pitches i was using and what I should do about them. My voice is not really very low, only I stay very much in the low end of it in English (this may indeed be due to Clint Eastwood). Overreacting to this, I initially aimed far too high for high tones (A440).
The standard textbook advice is: find the high end of your natural speaking range. They do not say how, and anyway this is wrong. No one outside of Chinese opera uses the actual extreme high end of their natural (i.e. non-falsetto) range in speech.
Data on page 210 of https://books.google.com/books?isbn=1139452150 suggests I should better aim for something close to G190 for my first tones, more than an octave below what I had first tried. The more I practice, the better that pitch sounds to me.
Tuning forks are surprisingly hard to find so I will point out that many adjustable forks cover the whole range that https://books.google.com/books?isbn=1139452150 found for one male voice, and do not go as high as Middle C (256--261 depending on which scale it is for). But most sets of non-adjustable tuning forks start at middle C and go up. What https://books.google.com/books?isbn=1139452150 found for one female voice uses tones above and below middle C. Of course keyboards cover all this range and much more.
Of course this is training, it is artificial. In actual speech i will not (and should not) have a fixed pitch for first tones or any other tones. But today I spoke with Chinese friends for the first time since I began work with the tuning forks. I heard their tones more accurately than I had before. And I could copy their tones more accurately, or adapt them more appropriately to my own vocal register, than I did before.