cheerfulness / clear skies
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cheerfulness
A state of cheerfulness, good mood, and calmness that often shows itself in laughter or smiles.
Der Witz des Komikers sorgte für allgemeine Heiterkeit im Publikum.
The comedian's joke caused general cheerfulness in the audience.
'Heiterkeit' (cheerfulness / clear skies) is the noun corresponding to the adjective 'heiter' (cheerful / clear). The adjective has two meanings: 'fröhlich' (cheerful) and 'wolkenlos/klar' (cloudless/clear, for weather). The noun 'Heiterkeit' (cheerfulness / clear skies) takes on exactly these two meanings.
Ein heiterer Mensch lacht viel.
A cheerful person laughs a lot.
When it is about cheerfulness, 'Heiterkeit' (cheerfulness) is often used with verbs like 'auslösen' (to trigger/cause) or 'sorgen für' (to create/cause). A typical phrase is also 'allgemeine Heiterkeit' (general amusement/cheerfulness), when an entire group is laughing.
Seine lustige Bemerkung löste große Heiterkeit aus.
His funny remark caused great cheerfulness.
The meaning 'klares Wetter' (clear weather) is much rarer and is mostly limited to the weather report or very elevated, literary language. If people, jokes, or an audience are being talked about, cheerfulness is always meant.
Der Wetterbericht meldet für das Wochenende viel Sonne und Heiterkeit.
The weather report is forecasting lots of sun and clear skies for the weekend.
fun / joke
'der Spaß' and 'die Heiterkeit' can both express joy and good spirits. 'Spaß' is very everyday and fits situations where you enjoy doing something or laugh. 'Heiterkeit' describes more of a cheerful, light mood in a person or a group. The word sounds somewhat more elevated and calmer than 'Spaß'. So you would be more likely to say 'Das macht Spaß' (That is fun), but 'Im Raum herrschte Heiterkeit' (There was cheerfulness in the room) when many people are laughing or smiling in a friendly, relaxed way.
Nach dem lustigen Kommentar der Lehrerin war im ganzen Kurs Heiterkeit zu spüren.
After the teacher's funny comment, cheerfulness could be felt throughout the whole class.
Der Ausflug mit der Klasse macht den Kindern großen Spaß.
The class trip is a lot of fun for the children.