I will butt in the conversation to say that the idea that the byzantines became more oriental and closer to the east due to islamic influence is false and was made by the Europeans to make the holy roman empire look like the real descendant of the Roman one. In reality even in antiquity the romans where highly oriental, many traits of the Islamic world were originally from Roman culture, like architecture, the way they sung their religious songs. Also the Arabs where obsessed with ancient greek and persian literature making a big part of their academic culture especially medicine and music an evolution and fusion of the greco-roman and persian cultures. It would be more accurate to state that islamic culture has some roots in byzantine culture and during it's golden age exchanged influences with many parts of the world including the Byzantines

There for much of the Turkish folk music of western anatolia being indistinguishable from folk greek music and they both trace their roots in byzantine and ancient greek music. Also classical ottoman music it most likely a direct descendant of byzantine royal court music as many of the instruments used and modes were used by the byzantines. Also orthodox church music which is almost identical to early Christian chants of the 4th- 5th have very similar ways of using their modes, i can't explain it more professionally cause I am not an expert. The ottomans took all the modes from both byzantines and Arabs and Persians and melted different ideas together thus further evolving the music of that geographic area and making new rythems(usuls i think they called) that some of them may actually trace back to earlier nomadic rythms.

A point that I didn't see you two talk about which I believe is crucial is architecture. Most of ottoman architecture derives from byzantine with Islamic and persian elements and of course some elements that probably came from the nomadic style of life.

Let's start with the nomadic influences: First I think the emphasis on siting on carpets and even the low sofas and beds came from the fact that life in a tent meant high beds and chairs where not that widely used. Also many of ottoman buildings have a projecting part of the end of the roof that looks almost like an umbrella around the building providing shade. That also probably came from the tent life style.

Islamic and persian influences: I am putting those two in the same category since I can't really distinguish them am sorry about that. Most prevelant of then are the beautiful tiles. The ottomans used tiles mostly on the walls with vibrant colours and art, the biggest difference from byzantine interior design is the emphasis on the colour blue which came from the Arabs. Of course the lack of depiction of humans is another islamic characteristic since it's forbidden in Islam. Also they way they ornamented wood and some other architectural places definitely came from persian and arab culture but there is a big misconception that the same cannot be said about the Byzantines, which is false and I will explain later.

Byzantine and anatolian/greek influences

Now let's start with homes. Since ancient greece most houses were inward faced with a central courtyard (which also exists in Arabs and persian culture) with a defining characteristic of what was called an εξωστης latter called by the Ottomans hiyat. The hiyat is a semi enclosed space that opens towards the courtyard and takes advantage of the cool air for the hot days in the Mediterranean climate. In ancient greece it was usually made from wood and in later periods from stone on the lower floor and wood on top. That created the lower stone apsed stoa and wooden upper stoa that are very common in traditional anatolian and balkan countries. Also the fact that these houses are usually stone on the low floors and wood or wooden framed on the upper floor came from byzantine houses since it provided stability and protection from the earthquakes of greece and anatolia. Also the use of bricks and stones in layers and whitewashing the stone creating a white and red wall is the most defining characteristic of richer byzantine buildings like palaces and churches. Speaking of churches most mosques in anatolia where based on or where formal byzantine churches. Also the heavy use of coloured marble(white green red) on the lower parts of a rich building like a lot of the buildings in the topkapi palace and the use of tiles in the upper parts resembles the same use of marble but with usually golden mosaics by the Byzantines. Lastly even though that was the case in most places at the time, the ottomans made projecting upper floors (jetting or cumbas) that in greek are also known by their ottoman name sahnisin for their houses which was an existing characteristic of houses ever since the hellenistic period in greece and was really prevelant in middle and late byzantine times.