An offspring is 23 chromosomes of mother and 23 of father, if one of the mate learns say music after the birth of their first child— will their second offspring have better music skills than former? about my knowledge of biology, so far I read about DNA, Genome, chromosomes and genetic engineering, I am an engineer by profession.
Short Answer: Probably, at least a little, much research still needs to be done. For a great paper on this see here.
Your question can be answered in two parts:
Can Epigenetics be inherited - The answer to the simple question in your title, do chromosomes change with time, is yes. There are copying errors, and carcinogenic mutations that can permanently change cells' genomes. Less permanent is epigenetics, which are temporary chemical modifications to the DNA which changes how it is expressed. The epigenetics of the cell may change for various reasons from the cell undergoing mitosis to a wildly changing diet. However, when sperm and egg cells are formed these epigentic modifications are typically wiped from the genome in order to remove any possible damaging marks. However, there is a growing body of literature which states that some marks escape the purge and change the characteristics of the progeny. Studies include scare-treating mice, 1836 famine, fatty rats, human psychological changes. While the exact genes causing the change is unknown, there is clearly a strong correlation
Can Music Skill be Captured in Epigenetics - Music skill is to difficult to measure biologically, but likely has some links in speech skills, which would be captured in the brain, and muscle skills. While there is some evidence for there being an muscle epi-memory, brain epigenetic inheritance is more tenuous (although see this exciting paper on RNA causing memory changes through epigenetics). I would say the general scientific consensus is that there is some relation between epigentics, and memory, and music skill - there is a great deal of work still to be done.
Until recently, the answer to your question would have been a definitive "no", because is smacks of the rejected Larmarckian ideas of the inheritance of acquired characteristics. And the example you provided, about musical skills, that would be "no" as well from a genetic perspective, although the hearing of music around the house may have cultural impacts on the second child. However the new field of epigenetics has revealed that DNA methylation, over the course of ones life can influence offspring of future generations. There was a community in Sweden that took very detailed health records over generations. The effects of a famine in 1836 showed up in the grandchildren of the survivors. It is further explained here: https://io9.gizmodo.com/how-an-1836-famine-altered-the-genes-of-children-born-d-1200001177