There is a Wide and Valid range of Minhaggim on this
The SA O"Ch 551:9 mentions 3
[Note: I'm reversing the order]
1. From 17th Tammuz on - "The 3 Weeks"
2. From Rosh Hodesh Av.
"The 9 Days" with Rosh Hodesh itself a subject of dispute with regard to eating
3. Shavu'a Shehol bo Tisha b'Av.
4. [Which is not in the SA] is to follow the Mishnah - Minhag Teiman - where the only prohibition is during the s'udah mafseketh.
The rationale for the Minhaggim seems intuitively obvious
1. Batlu Tamid v'Niskeihem
2. Mishenichnas Av M'ma'atin b'Simchah, and Ein Simcha Ela b'Vossor v'Yayin
3. Matches most other Talmudic and S'phardic strictures
4. Dina diGmara.
B'er HaGolah #5-7
Bei'ur HaGra #41-43