Mechitzah

Tosafot Rosh Kiddushin 52bR' Asher ben Yechiel (c. 1259-1327)
…and even when not necessary, there is no prohibition if she enters [the courtyard]…Still, it is well-stated “Whence is a woman in the azara?” Since it is uncommon for her to enter, as it is in synagogues and in batei midrash.

Hagahot Maimoniot, Hilchot Shvitat Asor, Minhagei Yom HaKippurimR' Meir ben Yekutiel of Rottenberg (13th Century)
And people have the practice to sleep in the beit hakensset since they say prayers and songs and praises for most of the night...And this was the practice of Maharam (of Rottenberg), however he would sleep in the west, far from the place of prayer, or in the women's beit haknesset when there were no women there.

Shu"t Orach Mishpat (R' Kook), Orach Chaim Section 35R' Avraham Yitzchak Kook (1865-1935)
And we have already explained that these matters regarding the prohibition of men and women mixing in the beit haknesset...they are stringent prohibitions of the Torah...and even if these were only prohibitions from the practice of our fathers, as some err and say that they are only matters of custom, even then they would be very stringent matters. And the practices of fathers is itself Torah, and it says about this in Pesachim 50b, our fathers already accepted upon themselves, as it says, "Hear my son the word of your father and do not forget the Torah of your mother".

Shu”t Tzitz Eliezer 7:8R' Eliezer Yehuda Waldenberg (1915–2006)
And the fact that the Shulchan Aruch doesn’t mention the law of putting up a mechitza between the beit haknesset and the women's section, that is because they did not have this issue at all then, because all shuls had a thick wall between the beit haknesset and the women’s section which reached from the floor to the ceiling, and the women would hear from small windows made in the wall, and so the question of whether a mechitza was required was never asked.

Perush HaMishna l’Rambam Sukka 5:1Rambam (1138-1204)A great decree, meaning that it has great value, that they created a space for the men and a space for the women. The place of the women above the men’s space so the men would not look at the women.

Chatam Sofer, Orech Chaim 5:190R' Moshe Schreiber (1762–1839), Germany
Is it right according to our Torah law to listen to the voice of a woman in shul, in a place that men congregate, and the women’s voice goes from the women’s section to the men’s section?
The reason for this, is that we believe that all prayer and praise and thanksgiving that mixes with a thought which has any improper element, even regarding one's wife, will not ascend to the ascents before God and will not be accepted before God. And because of this we separate the women from the men in beit haknesset so one will not come to improper thoughts at the time of prayer such that one's prayer will be dismissed by God. And we learn this from the simchat beit hashoeivah that is spoken about in Tractate Sukkawhere they made sure the women were above and the men below so they would not come to kalut rosh. And it is said there on 52a, as it says: “And the land shall mourn, every family apart: The family of the house of David apart, and their wives apart; the family of the house of Nathan apart, and their wives apart.” (Zecharia 12:12). And it is a kal v’chomer- here where they are talking about a eulogy where there is no evil inclination the Torah tells us that the men were separate, and here that we are talking about happiness, where there is an evil inclination, all the more so there should be a separation. See there.

Responsa Seridei Eish 1:8R' Yechiel Yaakov Weinberg (1884–1966)
And regarding girls praying loudly - there is no issue here at all. For the great authorities of Ashkenaz, among them the genius R' Azriel Hildesheimer, have already allowed men and women to sing zemirot together for two reasons: A: Two voices cannot be distinguished. B: Holy zemirot do not bring about thoughts of sin.

Psak Beit Din Takanot Meholowitz, Hungary 1866
1. Preaching in local language: It is forbidden to preach in a gentile language...
5. Partition: It is forbidden to make the partition separating the mens’ and women’s sections in a way that men can look at women. Rather, they must be made as was practiced in the past. And if it was already done, people should not enter it.

Responsa Maharam Shick OC 77Moshe Schick (1807-1879)
Now some wicked people have set out their hand to destroy in the women’s section the partition between the women’s section and the men’s section, so it will not be totally enclosed with boards, just with pillars, and they will be able to see and to be seen….And behold, God forbid that you righteous ones should be silent on the immodest deed that those brazen ones have committed, for according to the law we are obligated to make a separation between the men’s section and the women’s section as there was in the Temple the women's courtyard separate from the men's courtyard…that the men not see the women, because it leads to frivolity [kalut rosh] and other transgressions.

Piskei Rid Sukka 51bR' Isaiah di Trani (1165–1240)
They still would come to frivolity. That means that they would gaze at each other and wink their eyes. They [the sages] enacted that the women see from above and the men from below.

Responsa Divrei Yoel OC 10R' Yoel Teitelbaum (1887-1979)
…For the concern was solely gazing, for on the whole it is impossible to supervise this and to discern whether the gazing is intentional [for pleasure]. Rather in the end they [the sages at the time of simchat beit ha-sho’eiva] felt that there was gazing among the people and thoughts of sin and this is called kalut rosh…

Responsa Seridei Eish I:8R' Yechiel Yaakov Weinberg (1884–1966)
However, the rabbis of Hungary are very stringent and demand that the mechitza be higher than the heads of the women, and they go further and demand that, if the synagogue does not have a mechitza like this, it is prohibited to pray there, and it is prohibited for the women to come and pray and it is better for them to stay at home. Certainly their intention is for the good, to preserve modesty as it was practiced in prior generations, but in our times the situation has changed and nature has changed, and if women stay at home and don’t come to synagogue, they will lose their Judaism entirely, and it is certainly prohibited to push them away and distance them because of an excessive stringency that has no solid source in the Talmud or halachic authorities…In our time women are very disturbed if we distance them from houses of prayer, and going to prayer in synagogue in our time is the fulfillment of Judaism for women and for mothers.

Igrot Moshe, Orach Chaim 1:39R' Moshe Feinstein(1895–1986)I have been asked to talk about the issue of mechitza that must be between men and women in shul, and how tall it must be according to halacha, because it seems that there are places in this country that are lax in the separation between men and women in shul, but it should stay a mikdash me'at......And we can also learn from TB Sukkah that even if there is a mechitza but it is still possible to have kalut rosh (lightness of mind) there still is a biblical prohibition, because even originally women were inside and men were outside. According to Rashi this means that women were in the Ezrat Nashim (women’s courtyard) and the men were in the main courtyard on the Temple Mount; that there was a large separation between them because of the laws of separation. But since they needed to stand facing the gates in order to see the celebration, they noticed that they would have kalut rosh and they didn’t raise the divider. And if so, it was still biblically prohibited, and so to they allowed the balcony to be built to prevent this...And so we need to say that the prohibition is about kalut rosh, and they thought in the beginning that they would have kalut rosh because the men would want to look at the women...But when they saw that this brought about a lot of kalut rosh meaning that there were many conversations between the sexes, and touching of hands, etc. they made a decree that women would be behind or above so that the men would not be able to look at them (Tosfot TB Sanhedrin 20). But still they fell into kalut rosh because there was not a divided at the gates, and it was on the side of the women who were watching the men enter, and so to when we need a mechitza it must be a full divider...And it seems that we would need a balcony according to the Torah, that the women would be upstairs and the men on the main floor, because this is the only way they would not interact...And so in shuls that have both men and women, it is better that there is a balcony for the women. And if for some reason it is difficult to build a balcony there needs to be a real divider between the men and women, so they do not fall into having kalut rosh...And therefore it is not sufficient to have a mechitzah that is ten tefachim from the ground for that is nothing with regards to kalut rosh for they can talk and have much conversation with women with no difficulty and they can touch hands and there is no greater kalut rosh than this and it is considered literal mingling and biblically prohibited......rather one needs to make a mechitzah a measure higher than this so they do not come to kalut rosh. But it seems in my humble opinion it suffices to have a mechitzah that is as high as one's shoulders, for we see that a mechitzah is not a function of a prohibition of looking, as I wrote above, for this is already known from the outset and nevertheless it is permitted and there is no concern for it. And even after the great tikkun they only say there was a balcony there where planks rested on protrusions and a balcony as standard does not have a mechitzah, see Eiruvin Ch. 7 and Tosfot to 86 and Rambam's Peirush Hamishna. And we find that the women were seen since the balcony was not surrounded with a wall and there was no concern for this, and although they were not visible to the men standing under the balcony, nevertheless they would have been visible to the men standing in the middle...And therefore for a mechitzah that reaches shoulder height where it seems one will not come to kalut rosh one can permit it. And that is a height of three amot which is eighteen tefachim...And although heads are visible even when it is eighteen tefachim and they are standing, they will not come to kalut rosh. And those who are stringent to raise the mechitzah until the heads are also not visible, a blessing upon them. And particularly because many women in this country are not careful about covering their heads. But by law there is room to allow it also where the height is only eighteen tefachim and if one finds there some with uncovered heads, the Aruch Hashulchan has ruled that in our time, in our many sins, there are many like this [and] it is mutar to pray facing them, and therefore we do not protest. But if it is less than eighteen tefachim, it is prohibited and one has to protest with all one's might, for even if it is only prohibited rabbinically, one has to protest, and all the more so since in my humble opinion it is biblical, as I demonstrated above.

Igrot Moshe Orach Chaim 3:23R' Moshe Feinstein(1895–1986)
At the least a mechitza to shoulder height, and according to the measure of Rashi and Tosafot this is 18 handbreadths, and for Rashbam it is the amount of 17 handbreadths (see there), but as we have seen nowadays that shoulder height for most women is less, so one who is lenient with a mechitza of 60 inches [152 cm], which is five feet, one should not protest…There are synagogues that raised the floor of the women’s section a foot or more and don’t allow for raising the mechitza and this is not good in my humble opinion…but in any event I don’t say it is absolutely prohibited…

Igrot Moshe Orach Chaim 3:23R' Moshe Feinstein (1895-1986)
And aside from the matter of a mechitzah as an obligatory matter even where the women are exceptionally tznaua, there is the matter of a mechitzah with regards to the prohibition of looking at places that have to be covered, where there is the law of nakedness. In our time in our many sins, many women go around immodestly, and although one can be lenient with regards to those who come to beit haknesset with uncovered heads...but many women, in our many sins, go with bear arms and the like, which is definitely subject to the law of nakedness where it is forbidden to pray facing them, and for this too a mechitzah of shoulder height is effective. But relatedly, a glass mechitzah is effective for the law of mechitzah but not effectively regarding the prohibition of [saying] Keriat Shema and Tefillah facing nakedness, where [the rule is] that it is prohibited to read Keriat Shema facing nakedness [visible] through glass...

Igrot Moshe Orach Chaim 1:43R' Moshe Feinstein(1895–1986)
And I have heard that there is glass that can be seen in one direction, and it is good to use this glass so the women can see but the men cannot see.

Igrot Moshe Orach Chaim 4:32R' Moshe Feinstein(1895–1986)
…Two pretty wood panels made decoratively with many holes…in my humble opinion they are fit to make a mechitza, when through these holes it is impossible to come to kalut rosh, and the height of the mechitzah from the women's side should not be less than 60 inches.

Igrot Moshe, Orach Chaim 4:31R' Moshe Feinstein(1895–1986)
And regarding what the honoured Rabbi wrote that he saw a mechitzah in a beit haknesset that was only 70 inches from the men's side and just 30 inches from the women's side, the honoured rabbi should know that regarding the raising of the women's side, that was certainly not done on my advice, for many things are said in my name which I did not say, for certainly the measurement of the mechitzah of 60 inches has to be from the women's side. And I have heard that in some shuls they have raised the women's place, but it is not correct to do so. But I do not judge the rabbis unfavourably, maybe they could not do more and were lenient with themselves on the grounds of the great challenge of parnasah, and maybe a little for the sake of heaven, for they thought that if they leave there it will turn to conservatism, God forbid.

Shu"t Seridei Eish, Part 1 Section 8R' Yechiel Yaakov Weinberg (1884–1966)
And with regards to the height of a mechitza, a dispute has emerged in the United States. The Great Rav Moshe Feinstein, great in knowledge and adhered to in his final rulings, ruled that it is enough for it to be 18 tefachim (60 in.) a little below the shoulders, in order to reduce the possibility of frivolous behavior, even though the men and women can still see each other. Against him, the Rebbe from Satmar, along with all of the Hungarian Rabbanim forbade such a mechitza, as they held the mechitza must be high enough to cover the heads of the people. And they based themselves on Maimonides in his explanation of the Mishna, that the prohibition is that men cannot look at women.
To me, the words of Rav Moshe Feinstein and his proofs are clear and powerful, and in the words written against him, I don’t see anything which would push aside his words. And it is clear, that regarding what Rambam writes in his explanation of the Mishna that men should not look at women, his intent is that they should not come to impropriety, looking at them for the purposes of pleasure. And this is clear from the language of the Gemara in Succah 51b: The Rabbis taught, originally the women were inside and the men outside and they came to kalut rosh, they established that the women sit on the outside and the men on the inside and they still came to kalut rosh, they established that the women sit above and the men below...
And with a mechitzah of 18 tefachim it is impossible to come to kalut rosh, and there is no concern that at the time of prayer [men] will come to look for the purposes of pleasure. And it is clear, that regarding what the Rambam writes that they should not look, that is for the purposes of pleasure, but there is no prohibition in merely glancing, as is written in the laws of Issurei Biah 21:2, that it is prohibited to look with the intent to gain pleasure.

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Shu"t Bnei Vanim, Part 1, Sections 1-2R' Yehuda Herzl Henkin (1945-2020)
I will also tell him what I saw and heard from the geniuses of the [United States] who testified truthfully that the gaonR' Yosef Dov Soloveitchik shlit"a on multiple occasions allowed prayer with a mechitzah of 10 tefachim at a time of great need.

Igrat Moshe, Orach Chaim 1:39Rav Moses Feinstein(1895–1986)
...the core of the law is that even if the men are on one side and the women on another, it is forbidden for them to be without a mechitzah [and] in my humble opinion this is a Torah-level law. And a proof comes from Sukkah 51 where the Gemara objects to the balcony...that it is forbidden to add anything to the construction of the Temple and courtyard. And Rav answers that they found a verse [which implies] that one needs to separate the men from the women, as Rashi explains, see there. And it is obvious that the intent of the answer is that it is therefore as if it is explicit that one needs to make the boards and balcony and there was no need to have Gad the Seer and Nathan the Prophet transmit this mater when they were informed by the word of God of [how to] build the [Temple]. And this was automatically within the bounds of 'All this I give you in writing'. And if it were just a rabbinic prohibition it would not make sense to say that it is automatically within the bounds of 'All this I give you in writing'; since it would not be needed biblically and would be dismissed, it would certainly not make sense to dismiss the prohibition of 'All this I give you in writing' even if it were a biblical prohibition, and all the more so a rabbinic prohibition would not override the biblical prohibition of 'All this I give you in writing'...So it would also seem from a passage in the Jerusalem Talmud, Sukkah 5, 5: Commenting on this addition which amended the Temple structure, it states, "Whence did they learn [that they should build it]? From something in the Torah." Thus, such a separation is referred to as a Biblical matter. And even though its source in the Bible—Zechariah 12:12—is a verse from the prophets (Divrei kabalah), in which case the rule is that it cannot establish any Biblical prohibition, here a Biblical law can well be derived from it: for the verse does not seek to originate any prohibition, but merely requires that mourning be observed in accordance with [apparently preexisting] Scriptural law—men separate and women separate.' We learn similarly of many Biblical laws from the actions of the Prophets Judges and Kings, out of verses quoted in passages of Oral Tradition….