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I've posted a detailed answer to this interesting question on the Universalis blog. Universalis reproduces exactly what the official books are doing: the Latin Liturgia horarum and its English equivalents.
The reason isn't given anywhere in the books, as far as I know, but it is this: if you used the complementary psalms at Sext on Monday in the third week of the four-week cycle, you'd recite Psalm 70(71) at Sext. But Psalm 70(71) is already being used at Vespers on that day, so you'd end up saying it twice. That is why the books say "don't use the complementary psalms on that day at that hour". It's to avoid repetitions.
There are seven days in the four-week cycle in which this sort of thing happens.
(I don't hang out in Stack Exchange: Andrew Leach kindly pointed out this question to me. Please, if you have any other questions to do with Universalis, could you ask directly? You can comment on a blog post or just email me at [email protected]).
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The idea of praying to God regularly is to build your trust and faith in him. How can you do that if you are reciting someone else's prayer? Philippians 4:6
"Do not be anxious over anything, but in everything by prayer and supplication along with thanksgiving, let your petitions be made known to God"
"Draw close to God, and he will draw close to you. Cleanse your hands, you sinners, and purify your hearts, you indecisive ones."
Prayer to God should always be from your heart. You cannot cultivate a relationship with God by merely reciting words that really have no meaning to you. Doing so takes away from the real intention of prayer. The Jew's made the mistake of doing things out of habit and routine. That is not what God want's from us and even warns against it at Matthew 6:7-8
"When praying, do not say the same things over and over again as the people of the nations do, for they imagine they will get a hearing for their use of many words. So do not be like them, for your Father knows what you need even before you ask him."
You can look at those prayers in order to gain an understanding of what is proper to ask for. Even the prayer that Jesus gave us in Matthew 6 is just a model.
I think the purpose of using the psalms of the day is to get people to read the bible and to keep them thinking about spiritual things. Doing so is a good precursor to making regular prayer and maybe that is the goal of this schedule. Good motive, but what the bible teaches us to do is make our own regular prayers to God. 1 Thessalonians 5:17-18
Pray constantly. Give thanks for everything. This is Godβs will for you in Christ Jesus.
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I alerted Martin Kochanski, the author of Universalis, to this question, and he's written a blog post specially for it, to which readers are referred.
The simple answer is: on certain days in the four-week cycle, the Breviary has a small and slightly obscure rubric whose effect is that the complementary psalms aren't allowed at a certain daytime Hour.
No reason is ever given, but when I was programming Universalis I finally managed to work it out. On those days, some of the complementary psalms in question appear in one of the main Hours. For instance, on Monday of the third week of the four-week cycle, Psalm 70(71) is recited at Vespers. Psalm 70(71) is also one of the complementary psalms for Sext β so, to avoid repetition, the Breviary says "don't use the complementary psalms at Sext".
Similar things can happen at Lauds with the Invitatory psalm, and even in the Anglican Book of Common Prayer, where some psalms which are alternatives to canticles are prohibited where they occur in the ordinary course of reciting the psalter.