Part 1 of this essay briefly introduced the trop, followed by a study of its significance in some local contexts, concluding with some evidence of trop’s rabbinic origin. Part 2 looks at trop in its global context, structuring the two parts of most pesukim, until and after the word containing an etnahta. The process by which the trop operates demonstrates its recursive nature, providing a very early example of recursion in a musical context.
Trop contains 4 levels of separators (mafsikim) and a single set of connectors (meshartim). All trop symbols are either separators or connectors. The first level separators (often referred to as keisarim, Caesars) are the sof pasuk, which ends the sentence, and the etnahta, which divides the pasuk into two parts, analogous to a semi-colon. Both parts of the sentence, before and after the etnahta, are treated identically by the rules of trop. The second level of separators (often referred to as melakhim, kings), the zakeif katan, zakeif gadol, segol, shalshelet and tipha, define the major structure of the pasuk. Pashta, revi’i, and tevir, an additional level lower, are common third level separators, while darga, pazeir, and telisha gedolah are common fourth level separators.
Munah, merha, mahapah, and kadma are common connectors; there should not be an apparent pause between the reading of words where they appear and the following word.
Trop identifies the pasuk’s structure both at a global / macro level (the entire pasuk, or its two components divided by the etnahta) and at a local / micro level (each individual phrase).
Trop is Recursive
Recursion is primarily a mathematical notion which operates on an entity, dividing that entity into parts where at least one part is operated on by the identical process. One can think of this as an arbitrary number of Russian matryoshka (often called Babushka) dolls, each embedded in another.
In a brilliant book, Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter shows that recursion, which in mathematics was brought to its ultimate use by Kurt Gödel, was also present in painting (such as by Escher) and music (such as by Bach). In music, recursion involves a (completely or partially) identical pattern that repeats (iteratively) within a pattern. Recursion was present in the trop 1,000 years before its occurrence in Bach’s music, albeit with trop’s much less intricate musical scope.
Trop’s global operation
Except for short pesukim, the vast majority of pesukim contain one etnahta that divides the pasuk into its two principal parts. Going forward, we refer to either a short pasuk or to either of the two parts of a longer pasuk as an initial segment. Trop operates independently on each individual segment. Note that all initial segments end with a first level mafsik, either a sof pasuk or an etnahta. The lower level mafsikim (listed above) further divide the pasuk into smaller segments.
Trop’s operation on a segment is governed by the following rules:
- Read the segment (from right to left) until the first mafsik one level lower than the mafsik on which the segment ends is encountered.
- If such a mafsik is encountered, divide the segment into two, with the mafsik acting as the separator. Those two segments are then operated on again by the rule.
- If a mafsik one level lower is not found, the segment is not further divisible, and no further operation is performed.
Since all pesukim are of finite length, part 3 of the rule will eventually occur either because
- the mafsik at the end of the segment is at level 4 (and there are no mafsikim of a lower level), or
- even though the segment ends with a mafsik of levels 1, 2, or 3, no mafsik one level lower is present.
One of the fundamental rules of trop forbids the presence of a mafsik of lower level than the level being sought. For example, if a segment ends with a second level mafsik and there is no third level mafsik earlier in the segment, one can be certain that a fourth level mafsik will also not be present.
When operating with the rules of trop on any segment, the rule will divide that segment into two parts, providing it finds a mafsik one level lower; the part to the right ends on the word containing the mafsik, and the part to the left is the remainder of the original segment. This pattern repeats on any segment, regardless of length. The rule’s identical repetition on both segments demonstrates its recursiveness.
Consider the second pasuk in Ki Tavo (Deuteronomy 26:2). The first part of the pasuk
וְלָקַחְתָּ֞ מֵרֵאשִׁ֣ית ׀ כָּל־פְּרִ֣י הָאֲדָמָ֗ה אֲשֶׁ֨ר תָּבִ֧יא מֵֽאַרְצְךָ֛ אֲשֶׁ֨ר יְהוָ֧ה אֱלֹהֶ֛יךָ נֹתֵ֥ן לָ֖ךְ וְשַׂמְתָּ֣ בַטֶּ֑נֶא
encounters its first melekh, a tipha, on the word לָ֖ךְ. Note that this symbol accurately divides the first section into two parts; the first part tells us what should be taken, and the second part tells us where it should be placed.The second part of the pasuk,
וְהָֽלַכְתָּ֙ אֶל־הַמָּק֔וֹם אֲשֶׁ֤ר יִבְחַר֙ יְהוָ֣ה אֱלֹהֶ֔יךָ לְשַׁכֵּ֥ן שְׁמ֖וֹ שָֽׁם
encounters its first melekh, a zakeif katan, on the word הַמָּק֔וֹם. The pasuk tells us to travel to the place, and then provides a further description of the place.
The segment comprising the second half of the pasuk succinctly illustrates a critical detail that can cause some confusion when separating a pasuk into its constituent parts. Consider the two subdivisions of this half-pasuk, one up to and including the word הַמָּק֔וֹם and one after it. The second subdivision can be further divided by a second level mafsik, the zakeif katan on the word אֱלֹהֶ֔יךָ. However, the first subdivision is further divided by a third level mafsik, the pashta on the word וְהָֽלַכְתָּ֙. Note that it is not the level of a mafsik, but its role in the trop’s division of a segment, that determines a pasuk’s syntax.
Syntax only, not semantics
As noted in Part 1, since trop provides only syntax, it can
- provide likely support for a specific interpretation or
- be conclusively inconsistent with a specific interpretation.
The following examples, more complex than those covered in Part 1, all contain a separator / connector where the other might be expected, and therefore support dramatically different interpretations.
וַיַּכֵּ֣ר יְהוּדָ֗ה וַיֹּ֙אמֶר֙ צָֽדְקָ֣ה מִמֶּ֔נִּי כִּֽי־עַל־כֵּ֥ן לֹא־נְתַתִּ֖יהָ לְשֵׁלָ֣ה בְנִ֑י
The first part of the pasuk ends on the word מִמֶּ֔נִּי, which contains a zakaif katan. The word צָֽדְקָ֣ה has a munah, linking it to the word מִמֶּ֔נִּי. The trop is seemingly in accordance with the interpretation given by those such as Rashbam where Yehudah admits that “she is more righteous than I.” On the other hand, the trop is inconsistent with an alternative interpretation, “she is righteous; the child is mine,” which is the interpretation given by Onkelos, Rashi, and others. For that interpretation to be tenable, the word צָֽדְקָ֣ה would require a mafsik.
Often the syntax can provide (nearly) equal support for two alternative interpretations. Consider the brief pasuk in Genesis (49:18) with which Yaakov ends his berakhah to Dan:
לִֽישׁוּעָתְךָ֖ קִוִּ֥יתִי יְהוָֽה׃
An interpretation like: “I await for Your deliverance, O Lord,” as translated by JPS, is inconsistent with the trop. Such an explanation would require placing the tipha one word further, at קִוִּ֥יתִי. This interpretation is also hard to reconcile with the context, unless God’s deliverance is awaited not on behalf of Yaakov but on behalf of Dan. However, as written, the trop is consistent with various semantic alternatives. The sentence can mean “For deliverance by You, I have prayed to the Lord,” without stating explicitly for whom deliverance is prayed for. Again, the context more likely implies that Yaakov is praying for Dan’s (or his descendant’s) deliverance. Alternatively, directly addressing Dan, Yaakov tells him that he prays to the Lord for his deliverance. This explanation is given by Rashbam.
On occasion, dramatically different semantic interpretations are both possible given the trop. In both of the following pesukim the trop is consistent with either interpretation. First let’s consider Exodus 8:19:
:וְשַׂמְתִּ֣י פְדֻ֔ת בֵּ֥ין עַמִּ֖י וּבֵ֣ין עַמֶּ֑ךָ לְמָחָ֥ר יִהְיֶ֖ה הָאֹ֥ת הַזֶּֽה
Does פְדֻ֔ת mean a separation or a salvation? Both interpretations likely agree that God will create a separation between the Israelites, who will receive salvation, and the Egyptians, who will be afflicted. The argument is about the meaning of the word פְדֻ֔ת, either a separation or a salvation, making one word explicit and the other implied. Onkelos interprets פְדֻ֔ת as salvation, more consistent with its typical meaning; most commentators prefer separation, more consistent with the context of this pasuk.
Next, let’s look at Exodus 17:16:
וַיֹּ֗אמֶר כִּֽי־יָד֙ עַל־כֵּ֣ס יָ֔הּ מִלְחָמָ֥ה לַיהוָ֖ה בַּֽעֲמָלֵ֑ק מִדֹּ֖ר דֹּֽר:
Are we taking an oath, or referring to a time when there is a monarchy? The term יָד֙ עַל־כֵּ֣ס יָ֔הּ is ambiguous. It could mean that one’s hand is on God’s throne, as might happen as one is holding a religious object while taking an oath. This explanation is given by Rav Saadyah Gaon, and likely Onkelos as well. Alternatively, as posited in Sanhedrin 20b, it could be indicating that the command to obliterate Amalek refers to an era when a king is leading a religious monarchy. Which explanation is correct is disputed by the classical commentaries, some proposing both possibilities.
Dealing with lists
In numerous places, the trop deals with the individual elements in a list of items.
- וְאִם־הֵנִ֨יא אָבִ֣יהָ אֹתָהּ֮ בְּי֣וֹם שָׁמְעוֹ֒ כָּל־נְדָרֶ֗יהָ וֶֽאֱסָרֶ֛יהָ אֲשֶׁר־אָסְרָ֥ה עַל־נַפְשָׁ֖הּ לֹ֣א יָק֑וּם
- וָאֵרָ֗א אֶל־אַבְרָהָ֛ם אֶל־יִצְחָ֥ק וְאֶֽל־יַעֲקֹ֖ב בְּאֵ֣ל שַׁדָּ֑י
The first example divides the segment on the word שָׁמְעוֹ֒ – if the father objects on the day when he first hears. The second segment then lists two types of restrictions:
- vows; and
- self-imposed restrictions.
The second example divides the segment first on the word יַעֲקֹ֖ב and then on the word וָאֵרָ֗א. God declares he appeared, and then lists the three people to whom He appeared. In both examples, the action applies to all items on the list.
The next example from Numbers (30:3) contains a similar pattern but in reverse, with the list occurring first.
אִישׁ֩ כִּֽי־יִדֹּ֨ר נֶ֜דֶר לַֽיהוָ֗ה אֽוֹ־הִשָּׁ֤בַע שְׁבֻעָה֙ לֶאְסֹ֤ר אִסָּר֙ עַל־נַפְשׁ֔וֹ לֹ֥א יַחֵ֖ל דְּבָר֑וֹ
The segment divides on the word נַפְשׁ֔ו, with the first part listing vows and restrictions and the second half admonishing the listener not to profane them.
In the following three examples, only certain elements of the list link to the verb in the opening phrase.
- Numbers (6:14): וְהִקְרִ֣יב אֶת־קָרְבָּנ֣וֹ לַיהוָ֡ה כֶּבֶשׂ֩ בֶּן־שְׁנָת֨וֹ תָמִ֤ים אֶחָד֙ לְעֹלָ֔ה וְכַבְשָׂ֨ה אַחַ֧ת בַּת־שְׁנָתָ֛הּ תְּמִימָ֖ה לְחַטָּ֑את וְאַֽיִל־אֶחָ֥ד תָּמִ֖ים לִשְׁלָמִֽים
- Exodus (1:6): וַיָּ֤מָת יוֹסֵף֙ וְכָל־אֶחָ֔יו וְכֹ֖ל הַדּ֥וֹר הַהֽוּא
- Exodus (1:14): וַיְמָרְר֨וּ אֶת־חַיֵּיהֶ֜ם בַּעֲבֹדָ֣ה קָשָׁ֗ה בְּחֹ֙מֶר֙ וּבִלְבֵנִ֔ים וּבְכָל־עֲבֹדָ֖ה בַּשָּׂדֶ֑ה
In each case, one can assume the verb applies to all elements of the list, despite being syntactically linked only to the first element. In Numbers (6:14) the opening phrase וְהִקְרִ֣יב אֶת־קָרְבָּנ֣וֹ לַיהוָ֡ה presumably applies to the two other elements in the list, even the element occurring in the next segment, after the etnahta. The pasuk may be read as if the phrase is implicitly assumed to be repeated.
The reasons for this syntactic choice may often be semantic or stylistic. In the second example above, it is highly plausible that the pasuk is ranking the people mentioned: Joseph is most important, followed by his brothers, and finally other members of his generation. There are many other examples, sometimes with a less compelling assumed ranking among list members. The last example may link to the most prevalent work performed. Many other examples that occur in the Torah are less clear.
The formality introduced is necessary to guide a beginner trying to parse a sentence following the rules associated with the trop. Fortunately, almost anyone experienced with how the trop operates can look at a pasuk and directly observe the implied levels of division implied. My late father went a step further, claiming that if he assumed a particular interpretation, he could normally deduce the associated trop. I inherited my mother’s mathematical skills and not my father’s literary prowess; on occasion, I still make embarrassing errors studying and teaching trop.
 The book of Esther has particularly long pesukim, providing the most involved examples.
 These alternatives would be clearer if there was a ל before Hashem.
 Considerations based on length, potential rhythm, dramatic impact, etc. might influence the sequence of trop symbols chosen. A semantic reason can also on occasion be linked to a midrashic source, as the genre of seforim like Ve-Yavinu ba-Mikre by R. Naftali Tzvi Yehudah Gettinger on occasion attempts to demonstrate.