Solemnity and joyfulness are two emotional states that are normally exclusive to one another. However, in the month of Av we experience both as a part of the calendar.
On the subject of the aspect and enormity of the month's solemn nature, Av begins in the middle of the three weeks, a period of mourning from the 17th day of Tammuz until the 9th day of Av. This period of time is also called "Bein haMetzarim" ("between the straits"). The significance of this time is that it, as we discussed last month, was the return of the scouts or spies who sent into Canaan (Numbers 13). It is traditionally accepted that their denial of HaShem's power to bring them into the land of Israel, and consequently their 40 year punishment, began on the 9th of Av. This tragedy, along with the destruction of both Holy Temples, and other horrific events took place on this day. Therefore the day is marked with a solemn and complete full-day fast (roughly from sunset to sunset).
Conversely, there are numerous occurrences of good that happened on Tu B'Av (15th of Av). The punishment of the generation that was to die in the wilderness for failing to believe in HaShem's might ended on this day. Such was the good of this day that it was upon other good events of this day that the fourth blessing of the Grace After Meals was added. The fourth blessing goes into detail in mentioning the compassion that the Holy One has displayed to his people.
It is normal, beginning with the 15th of Av, to try to increase one's introspection of themselves. To accomplish this, one tries to spend more time in devotion to the Holy One by prayer and study. During this same timeframe, specifically beginning with the first Sabbath after the 9th of Av, begins the Seven Weeks of Consolations, which marks the beginning of the period of consolation, where HaShem beckons his people to repent and to unreservedly turn to him in spotlessness of heart and fidelity of mind. The end of the Seven Weeks will bring us to Rosh Hashannah.
For a follower of Messiah Yeshua, we can see a picture of our own walk in this month of the calendar. Due to our own sin and inability to remain in perfect faithfulness to HaShem, we, like the generation in the wilderness, were given a sentence of death. Only through the mercy of the Holy One did he call us to repentance. Once we answered the call of repentance, we were changed, no longer destined to death and separation from God, but unto life and nearness to him. Further, once we turned to him and were granted atonement through the work of Messiah, we were given the promise that the punishment of death had been lifted from us. Finally, with great joy, we walk boldly in proclaiming the truth of the Holy One and by our transformed lives we call others to his kingdom.