Ok, agree with you on slope, that was just the simplest example. I still believe that in the definition and calculation of trend (however one chooses), the time frame determines what the trend will be, and not vice versa. My opinion of course, but that is what makes markets.
The measurement of a trend is entirely dependent on sampling rate and sample size. The sampling rate (aka timeframe) is arbitrary but the trader usually has a pretty good idea which timeframe he wants to trade in. However the sample size (aka period length) can be chosen to maximize trendiness or any other quantifiable pattern characteristic within a given timeframe.
In forex if something really starts to move up on 5 min time frame and you look at 4h and daily and they are all trending higher you can start to really fire into it with confidence
Thats 4 hour uptrend, but im trading the 5 min... even if 5 min fails the 4 hour usually backs you up and brings 5 min into line